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Talk:Diffeomorphism

Resolved question concerning linking 2003
HELP : can't get the links to Manifold#Differentiable_manifolds to work. When I click (try atlases for instance) I get to the top of the Manifold page. What's wrong ? Pascalromon 14:19, 27 Nov 2003 (UTC)

Unresolved question concerning unexplained mathematical symbols
I think I followed the definition at the beginning through and including this statement:

Two manifolds M and N are diffeomorphic (symbol being usually $$\simeq$$) if there is a diffeomorphism $$f$$ from M to N''. For example''

.... So far, so good, but wait ... The very next line has this expression:

 * $$\mathbb{R}/\mathbb{Z} \simeq S^1.$$

Let me take some guesses at what this means in layperson's terms:

 * "The set of real numbers without the integers is diffeomorphic to $$S^1$$ (whatever $$S^1$$ means)" ... or is it something like:

 * "The quotient set formed by the real numbers with the integers (see also word salad (mental health)) is diffeomorphic to $$S^1$$ (whatever $$S^1$$ means)"

If in about a week or so I see no disambiguation either here or in the article, then I plan to be bold and update the article with one of the above two definitions.

Please be gentle on me, my WikiStress is higher than usual. Vonkje 16:12, 1 Jun 2005 (UTC)

It's the quotient group. You can think of it this way: take a line segment and bend it round into a circle, identifying the end points. Of course that gives you one special point, and there should be no special points at all. Still, it is the correct topological picture. Charles Matthews 16:40, 1 Jun 2005 (UTC)

 * set minus is usually written with a backslash (or at least looks like one), not a forward slash. So the reals minus the integers would be $$\mathbb{R}\setminus \mathbb{Z}$$.  As Charles points out, what we have here is$$\mathbb{R}/\mathbb{Z}$$, the quotient group of the additive group of real numbers modulo the normal subgroup of integers.  This quotient group is set of all cosets of $$\mathbb{Z}$$.  Cosets will look like {..., -2.5, -1.5, -0.5, 0.5, 1.5, 2.5,...}, which is the coset $$0.5+\mathbb{Z}$$.  Notice that the cosets $$0+\mathbb{Z}$$ and $$1+\mathbb{Z}$$ are actually the same coset, so that this quotient space eventually gets back to where it started, just like a circle.  Speaking of circles, $$S^n$$ or sometimes $$\mathbb{S}^n$$ is the n-dimensional sphere.  A 1-dimensional sphere is also called a circle.  So the observation that the quotient group above comes back to where it starts is the starting point of the observation that that quotient group is like a circle.  If you see your circle as the set of complex numbers of modulus 1, then your diffeomorphism can be given by the formula $$x\mapsto e^{2\pi ix}$$.  Then you just have to check that this is a homeomorphism, is smooth, and has smooth inverse.  It will also turn out to be an isomorphism of groups, so that it is actually an isomorphism of Lie groups.  Hope that helps. -Lethe | Talk 18:36, Jun 1, 2005 (UTC)

 * Thank you so much ... your added explanation in the article, and your patient explanation on this page helped. ... So where the cosets $$0+\mathbb{Z}$$ and $$1+\mathbb{Z}$$ are the same would refer to that "special point" that Charles was mentioning where the two ends of the line segment forming the circle meet?  Vonkje 15:50, 6 Jun 2005 (UTC)
 * Charles's description has a special point because he started with a line segment. A line segment has two special points: its endpoints.  When you join the two endpoints with glue, they become a single point, and a real circle doesn't have any special points, from a topological standpoint.  But the true construction R/Z doesn't suffer this defect.  Like I mentioned, 0+Z and 1+Z are the same cosets, and so the same points of R/Z, however, they are not special.  It's also true that 0.5+Z and 1.5+Z are the same points, and 0.9+Z and 1.9+Z are the same points, etc.  Since I started with an infinite line R, instead of a line segment, there are no special points anywhere, at least not from a topological point of view.  When you recall that R and Z are groups, both have a special point, 0, the identity.  R/Z then has a special point as well, but this is OK for groups.  You could get rid of this special point by viewing R as an affine space with a Z action, and R/Z the set of Z-orbits. -lethe talk 21:23, 29 November 2005 (UTC)

Newcomer to the subject here, but isn't the diffeomorphism in the example above rather given by $$x \mapsto e^{2\pi i x}$$? Thanks, 155.198.196.187 11:03, 31 May 2006 (UTC)
 * Yes. -lethe talk [ +] 12:25, 31 May 2006 (UTC)

Any chance of a lay example?
I'm linking into here from guage theory, trying to wrap my head around some of these topics. However, IMHO, all of them are written by math nerds for other math nerds. Can't anyone provide a single example of these concepts for us non-math-nerds? The second sentance of ANY article should not start "the mathematical definition is as follows"!

Maury 22:26, 28 November 2005 (UTC)
 * The definition is very simple. Just read carefully, it says a function which is invertible, and both the function and the inverse are smooth. I assume you know what the words "invertible function" and "smooth" mean. Otherwise, this article will not be helpful to you. Oleg Alexandrov (talk) 22:33, 28 November 2005 (UTC)
 * And there is an example there, and a good discussion at ==Local description==. Oleg Alexandrov (talk) 22:34, 28 November 2005 (UTC)

it says a function which is invertible, and both the function and the inverse are smooth

But it's buried. If this is the definition, it should be the second sentance of the article. The definition should be blocked off into its own para. Like this... Maury 13:06, 29 November 2005 (UTC)

 * I don't think it was buried. It was the third sentence in the article (if you count the wording "The mathematical definition is as follows." as a sentence). But I do agree that it looks better now. Oleg Alexandrov (talk) 20:18, 29 November 2005 (UTC)

R/Z : Divide any real number by any integer; throw away the integer part of the answer; what remains is a number, x, between -1 and +1: -1 < x < +1; this set of numbers is R/Z. Now use the formula, z(x) = exp( i*pi*x ); "exp" = "e to the"; "*" = multiply ); for any value of x in our set, R/Z, z(x) is a point on the unit circle in the Complex Plane. This is the circle centered on 0 (the origin), and having radius, 1.  For example, the x-value 0 maps onto the point z=1; the x-value 0.5 maps onto the point z=i, the x-value 1 (we'll allow it to sneak in) maps onto the point z=-1, and the x-value -0.5 maps onto the point z=-i. So using this sequence of x-values we have traversed the unit circle counterclockwise, E, N, W, S. If we allowed the x-value, x=-1, it would also map to z=-1, since exp( i*pi*1) = exp( -i*pi*1) = -1 (Euler's identity, q.v.).  This unit circle is called S1, because a circle is a 1-dimensional "sphere" (embedded in 2-dimensional space), a sphere (S2) is a 2-dimensional object (embedded in 3-dimensional space), etc.  The integer part of the cosets (the "Z" part), when inserted into the formula, sends you to z=+1, z=-1, (E and W), again and again, contributing nothing new. So we have mapped a line segment (-1 to +1) on the Real axis, onto the unit circle, of which it is a diameter. If this segment is traversed from -1 to +1, then the unit circle will be traced from -1, counterclockwise, all the way around to -1 again.

 * I've added three nice examples. The example about the reals modulo the integers was too algebraic, and not very informative. Since all manifolds are locally diffeomorphic to real space we can do all practical calculations in coordinate charts. Thus I have given three examples of maps from two-space to two-space. For any calculation I have ever had to make (and I work in applications of singularity theory to differential geometry) I have always used the Jacobian method in my three examples. It's all well and good saying that a map and it's inverse must be differentiable, but how does one compute an explicit inverse for a map? With the Jacobian method it's automatic: it tells you that the map is differentiable, and that there exists a differentiable inverse. Dharma6662000 (talk) 19:24, 16 August 2008 (UTC)

Manifolds or just normed vector spaces?
The current definition of a diffeomorphism is given in terms of a map between manifolds, but a diffeomorphism can be defined over much more general spaces (namely, normed vector spaces - see e.g. Marsden, Ratiu, & Abraham, "Manifolds, Tensor Analysis, and Applications"). Why is the more specific case used here? Trevorgoodchild (talk) 05:23, 29 January 2008 (UTC)
 * I think the most general case is even more general than that, you can have diffeomorphisms over infinite-dimensional manifolds (which look locally like a normed vector space). But I guess going that general could make the article hard to understand. The most used cases are in manifolds over R^n anyway. If you want, you can add somewhere at the bottom a section on generalizations, but I'd prefer that at least the first several sections in the article be about the "usual" R^n manifolds case. Oleg Alexandrov (talk) 15:44, 29 January 2008 (UTC)

Relationship with homoeomorphisms
The "model example" section says that the diffeomorphism $$f$$ "happens to be" a homoeomorphism. From the definition, it seems to me that all diffeomorphisms are homoeomorphisms, since all differentiable functions are continuous. If that is the case, "happens to be" seems misleading. LachlanA (talk) 22:45, 6 May 2008 (UTC)

 * Yeah you're totally right: every diffeomorphism is a homeomorphism, but the converse is false. I'm sure you knew, but just to make sure: a function and its inverse must be differentiable (resp. continuous) for it to be a diffeomorphism (resp. homeomorphism). I've changed the article, and made your point clear. I replaced the sentence in question with: "f being a diffeomorphism is a stronger condition than f being a homeomorphism. For a diffeomorphism we need f and its inverse to be differentiable. For a homeomorphism we only require that f and its inverse be continuous. Dharma6662000 (talk) 19:28, 16 August 2008 (UTC)

Diffeomorphic manifolds=a smooth bijection with smooth inverse between them
I am quite sure that two manifolds are diffeomorphic iff there is a smooth bijective function between them that also has a smooth inverse (not as in the article where only a bijective differentiable function with differentiable inverse is required). For example, think about it this way: two mathematical objects are equivalent (and considered the same) iff there is an isomorphism between them. According to the definition given in the article, two differentiable manifolds may be equivalent without really having the same differentiable structure.

In fact, C0 functions are just continuous, so according to the article, two manifolds are diffeomorphic if they are homeomorphic. I am therefore changing this. If you think otherwise, please give your reasons.

Topology Expert (talk) 14:03, 6 October 2008 (UTC)

For connected manifolds, the diffeomorphism group acts transitively on the manifold???
We need some experts on diffeomorphism groups. Currently, the content has no meaning and is incorrect. For instance, the article claims the title of this section. I wanted to believe it but it seems somewhat dodgy. Here is a counterexample:

The line with two origins or the bug-eyed line; see. This space is trivially connected. However, there is no homeomorphism carrying (0,b) -> [(5,b), (5,a)] (for instance). Suppose such a homeomorphism exists; let y denote the image of (0,a) under this homeomorphism and x denote the image of (0,b) under this homeomorphism. Clearly, x is not y and there are disjoint neighbourhoods about x and y respectively (the space is Hausdorff apart from (0,a) and (0,b)). Continuity of the homemorphism means that there are disjoint neighbourhoods about (0,a) and (0,b); a contradiction. Since every diffeomorphism is a homeomorphism, this is a counterexample to the claim in the article.

I am not an expert on the subject and I don't know of many of the important results to do with diffeomorphism groups. If someone knows, could he/she please include them in the article (provided there is a reference)?

Another issue is the factual accuracy of the article. The article claimed (or said in effect that):

 * A curve with a cusp, in particular the curve, x^2 - y^3 = 0, is not difeomorphic to a straight line. Wrong, since the projection map yields a diffeomorphism (if you consider the natural differential structures on the two manifolds)

 * A fibre bundle is a space (which is wrong); fibre bundles are ordered quadruples (if you don't consider structure groups acting on them)

I fixed both these things and I will remove the statement given as the title of this section. If there are any objections, please post your reasons here.

Topology Expert (talk) 03:16, 9 October 2008 (UTC)

Hi there. Manifolds in this article (as standard on Wikipedia unless otherwise noted) are considered to be 2nd countable Hausdorff. Although the diffeomorphism groups of non-Hausdorff manifolds are occasionally considered in research literature, more often than not they're considered beside-the-point. Generally speaking the people who work on diffeomorphism groups are living in the Hausdorff world, and if they do consider non-Hausdorff manifolds they make a point of it (rather than the other way around that you're considering).

 * Hi Rybu,

There was quite a big discussion on what manifolds should be according to Wikipedia here. Also see manifold (the lede). In Wikipedia (in general at least), unless it is stated that manifolds must be Hausdorff in a given article, one can assume otherwise (I actually would prefer to assume that a manifold is second countable and Hausdorff but Wikipedia has separate articles for seperate assumptions of what a manifold is). If you assume the Hausdorff condition, just add it to the lede and I can add back that example.

Topology Expert (talk) 02:23, 10 October 2008 (UTC)

FYI, the curve $$x^2 - y^3 = 0$$ is not diffeomorphic to the real line, because this curve is not a submanifold. The proof follows from the implicit function theorem -- if it was a submanifold, you'd be able to write it as the graph of a smooth function $$y=f(x)$$ or $$x=f(y)$$ near the origin, but in the first case $$f$$ is not smooth, and in the 2nd case $$f$$ would have to be multi-valued.. contradiction. Did you erase that example? If so it'd be best to change it back as it's an important example. You've been making so many modifications it's hard for me to keep track. I'll see if I can find it in the revision history. Rybu (talk) 19:46, 9 October 2008 (UTC)

 * I agree, the curve is trivially not an immersed submanifold (at (0,0), the differential is not defined). Nor can it be an embedded submanifold. But the article wrote that this curve is not diffeomorphic to the straightline and according to your reason, this does not make any sense (since you can't speak of whether two curves are diffeomorphic unless they are both smooth manifolds).

I might be confused, but I am quite sure that it is still possible to give the curve a differentiable structure since the projection map yields a local homeomorphism with the curve and the reals (and the transition maps are smooth since there is only one (the identity map of R)). So with this (natural) differentiable structure, the curve is diffeomorphic to the straight line.

 * Any continuum has a smooth structure making it diffeomorphic to $$\mathbb R^n$$ but it's not a natural smooth structure. In this sense the smooth structure you're putting on the above curve would be considered unnatural.  The natural smooth structure on a subset of a manifold is the one coming from the ambient manifold.  This is a common notion for sets as well as submanifolds.  ie: A function $$f: X \to Y$$ is smooth (for $$X,Y$$ subsets of manifolds) if for every point in $$X$$, $$f$$ admits a local smooth extension to an open neighbourhood of the point in the manifold.  I elaborated on this in diffeomorphisms of subsets of manifolds under the main definition in the article.  If you'd like a reference, see for example the Guillemin and Pollack textbook Differential topology. Rybu (talk) 04:58, 10 October 2008 (UTC)

Topology Expert (talk) 02:23, 10 October 2008 (UTC)

By the way, I noticed that you added that example back. But with manifolds, the definition of a map d being smooth is when f o d o f^(-1) is smooth (where f and f^(-1) are charts over a particular open set). The curve in question is a smooth 1-manifold so shouldn't the projection map's inverse be smooth? You seem to assume that unless the curve is a submanifold of R2 it can't be a smooth manifold (which is not true from what I know (I could be wrong though)). Also, the article contradicts itself since it says that for dimensions 1, 2 and 3, homeomorphic smooth manifolds must be diffeomorphic (, this is a contradiction if you assume that the curve is a smooth manifold which it is with the differentiable structure that I mentioned). But first could you please tell me why that curve is not a smooth manifold with that differentiable structure? Topology Expert (talk) 02:35, 10 October 2008 (UTC)

 * There's a bit of a problem with conventions going on here. As I mentioned above, smoothness of a map is something defined for arbitrary subsets of manifolds.  I mention the curve is not a submanifold, I do not say that it's impossible to make it into a manifold via a creative choice of a homeomorphism with $$\mathbb R$$.  Being a submanifold and being possible to turn a set into a manifold, are two completely different things. The 1st is relevant, the 2nd is not. The submanifold needs some cleaning up, but according to Wikipedia I'm using the word submanifold in the intrinsic embedded submanifold sense.  Rybu (talk) 04:58, 10 October 2008 (UTC)

Taking a standard definition of manifolds, where 1-manifolds are either intervals or the circle (cf Milnor's "Topology from the Differentiable Viewpoint", Guillemin & Pollack, etc), the statement is that in dimension > 1 the group of diffeomorphisms of compact support acts n-transitively on the manifold. This is discussed in Augustin Banyaga's book for example. For the circle, ordering of points must be taken into account. A quick search of mathscinet gave the generalisations of Boothy, Hatakeyama and Michor & Vizman to symplectic and contact manifolds. Here it seems that, rather than taking an exotic definition of smooth manifold, compact or not, the assumptions of the original authors in the literature need to be unearthed. What is true is that the old article does not give page numbers in the sources, or any direct citations. This would help avoid situations like this.

Banyaga's book is available here. The transitivity lemma is stated and proved on Page 29. It gives the n-transitivity result for smooth manifolds of dimension >1. The assumptions on the manifolds are given in the first chapter of the book. An analogous result for symplectomorphisms appears on page 109. Mathsci (talk) 07:30, 10 October 2008 (UTC)
 * I have added a Notes section in the article (for some strange reason the Hirsh and Smale references had been left orphaned), together with a direct link to Banyaga for the point on transitivity. I added the comments on Choquet and Rado, having sourced them in the literature. These days I would actually add the exact references for both statements, time permitting. Mathsci (talk) 07:39, 10 October 2008 (UTC)

Dear Rybu,

I am not disagreeing with you but I usually assume differentiable manifolds to take any appropriate differentiable structure. I think that this is the point of view taken by most mathematicians. Of course, in the context of submanifolds, this may be different. However, prior to me deleting the example (the one with the curve having a cusp), the article had not mentioned anywhere that the reason that this curve is not diffeomorphic to a straight line is because one is not a smooth manifold. Furthermore, they did not state why the curve with the cusp is not a smooth manifold by referring to submanifolds. Now what you have written has the appropriate hypothesis so I am not going to remove it again.

I have got to go know (I have more to say but I might not be able to respond very soon; if I do I will let you know on your talk page).

Topology Expert (talk) 11:02, 10 October 2008 (UTC)

Accoding to Rybu, the curve with a cusp is not a submanifold of $$R^2$$? It is definitely an imbedded submanifold if not an immersed one. Furthermore, according to Rybu, the projection map from the curve with a the cusp to the x-axis (along wih the obvious atlas) is not natural? Why not (it would be 'natural' if we embedded the 'curve with a cusp' in itself (with the previous mentioned differential structure))? To determine whether two manifolds M and N are diffeomorphic, we should not have to know where they are imbedded in. I don't think that the reason given in the article is appropriate; the curve with the cusp is a submanifold of $$R^2$$ (although not an immersed one). So remove the example altogether (the current reason is bogus):

Saying that two spaces are not diffeomorphic because one of the spaces is not even a manifold is useless and unecessary. If one of them is not a manifold, there should not be any question as two whether they are diffeomorphic. Therefore, I will remove that example.

Topology Expert (talk) 08:58, 15 November 2008 (UTC)

 * This is an issue of standards. From the point of view of Differential Topology, your curve with a cusp has no natural manifold structure.  You can make it a manifold in many ways, the way you've chosen (writing it as the graph of a non-smooth function and then mandating that it has the differentiable structure to make the function a diffeomorphism onto its image) allows one to make sense of a smooth structure, but it's in no way natural, because if you make a different choice of bijection with the reals, you can get a different smooth structure. To be precise -- in your terminology, there is no obvious atlas.  What is the obvious atlas on a Cantor set, or any continuum?  If you have no standards for what the words you use mean, it's not clear they belong in a mathematics article.  I suppose undefined terms have a place, but here we're talking about something technical, not heuristic. Ah well.  Somebody else can clean this up. Rybu (talk) 10:02, 15 November 2008 (UTC)

Maybe I 'clouded up' my main point with other discussion. My main point was:

As a subspace of itelf (with the projection map as a chart over the whole set), the 'curve with the cusp' is an immersed submanifold. Surely this is natural (and it is imbedded in itelf as it is in the natural manifold $$R^2$$)?

Could you please comment?

Topology Expert (talk) 12:45, 15 November 2008 (UTC)

Possible error in text
A differentiable bijection is not necessarily a diffeomorphism, e.g. $$f(x)=x^3$$ is not a diffeomorphism from to itself because its derivative vanishes at 0 (and hence its inverse is not differentiable at 0). This is an example of a homeomorphism that is not a diffeomorphism.

Possibly this should be:

A differentiable bijection is not necessarily a diffeomorphism, e.g. $$f(x)=x^3$$ is not a diffeomorphism from its image to itself because its derivative equals 0 at x = 0 (and hence its inverse is not differentiable at x = 0). This is an example of a homeomorphism that is not a diffeomorphism. See the inverse function theorem.

--Alephtwo (talk) 17:52, 7 May 2009 (UTC)

 * The text is fine, you're misquoting it though. Apparently the blackboard bold R is not showing up on your end -- some kind of rendering problem in your browser, maybe?   $$\mathbb R$$ (\mathbb R) is the TeX code. Rybu (talk) 20:03, 7 May 2009 (UTC)

 * Rybu, Thanks for catching that problem on my side, my browser did not render the blackboard bold R. The technical part of the statement that I question, is the statement: "derivative vanishes at 0".  I believe that the derivative of  $$f(x)=x^3$$ is defined at 0 ( f '(0) = 0 ).  It is the derivative of the inverse function that cannot be defined at zero, because the zero value of the derivative of the original function f ', is in the denominator of the of derivative of the the inverse function.  From inverse function theorem :

 * Statement of the theorem
 * For functions of a single variable, the theorem states that, if $$f$$ is a continuously differentiable function and $$f$$ has nonzero derivative at a, then $$f$$ is invertible in a neighborhood of a, the inverse is continuously differentiable, ...

 * (italics above added), which is why I suggested the link.

 * (aside, how do I make the derivative apostrophe display more clearly: $$f'$$ ?
 * thanks
 * --Alephtwo (talk) 15:05, 8 May 2009 (UTC)
 * The derivative a diffeomorphism has to be invertible at every point. Vanishing in this elementary case precludes that possibility. As for the TeX question: $$f^\prime$$.I don't know if this helps. Mathsci (talk) 15:21, 8 May 2009 (UTC)

 * Alephtwo, there's no problem in the text. Perhaps it's a terminology issue.  When someone says the derivative vanishes they mean the derivative exists but is zero.  This implies (via the inverse function theorem) that f is not a diffeomorphism.  So I'm happy with the text as-is.  Rybu (talk) 16:39, 8 May 2009 (UTC)

Dubious
I flagged the statement
 * In both cases, the diffeomorphism group is locally homeomorphic to the space of Cr vector fields on the manifold (where r is the order of differentiability considered).

as dubious. To the best of my knowledge, there are diffeomorphisms in any neighborhood of the identity that are not in the image of the exponential map. Sławomir Biały (talk) 14:13, 4 December 2009 (UTC)

 * In any case, it seems unlikely that this can be true in both cases, as asserted in the article: the strong topology is not locally homeomorphic to the weak topology on a non-compact manifold. It seems like a way forward would be to write a short section that describes each topology a little bit more carefully, making sure also to provide references in order to verify the content and also to avoid possible misunderstandings.   Sławomir Biały  (talk) 16:44, 4 December 2009 (UTC)

 * I think I have resolved the problem. The article wasn't claiming that the local homeomorphism came from the exponential mapping of the group.  Sławomir Biały  (talk) 02:53, 5 December 2009 (UTC)

Recent addition
The following content was recently added to the text. I feel that it is out of place there because the first section does not clearly say what it is trying to do (namely that the Lie algebra of the diffeomorphism group is the space of vector fields, which can be mentioned in a better way elsewhere). The addition of a fermion is poorly explained, and does not show how one gets a "group" out of the resulting Lie superalgebra. The second section seems to be about diffeomorphism groups in a particular context (relativity theory maybe), and so again seems poorly placed and poorly explained. I'm leaving this here for discussion, since it's possible some of this content should be worked into the article in some other way. Sławomir Biały (talk) 11:52, 22 April 2012 (UTC)

 * Update: I have incorporated some of the content about the Lie algebra into an earlier paragraph of the Diffeomorphism group section. There should be a separate article on superdiffeomorphism group that includes noncommutativity.  I think this has little place in an article about the diffeomorphism group of an (ordinary) manifold.   Sławomir Biały  (talk) 12:00, 22 April 2012 (UTC)

By making a small change to the coordinate x at each point in space:
 * === Representations ===
 * $$x^{\mu} \rightarrow x^{\mu} + \varepsilon h^{\mu}(x) $$

a simple representation of the diffeomorphism group is given by the (infinitesimal) generators is found:

 * $$ L_{h} = h^{\mu}(x)\frac{\partial}{\partial x_\mu} $$

which form a group since:

 * $$ \left[ L_{h}, L_{f} \right] = L_{h.\nabla f - f.\nabla h } $$

There is also a `vector' matrix representation:

 * $$ (L_{h})^{\mu}_{\nu} = \delta^{\mu}_{\nu} h^{\tau}(x)\frac{\partial}{\partial x_\tau} - \frac{\partial h^{\mu}(x)}{\partial x_\nu} $$

and a `spinor' matrix representation:

 * $$ (L_{h})^{\alpha \beta} = \delta^{\alpha \beta} h^{\tau}(x)\frac{\partial}{\partial x_\tau} - \frac{\partial h^{\mu}(x)}{\partial x_\nu} \sigma_{\mu \nu}^{\alpha \beta}$$

and there are representations for tensors and mixed indices. The diffeomorphism group can be extended by supersymmetry by adding a new fermion parameter in which case the rule is:

 * $$ \left[ L_{h,\psi}, L_{f,\eta} \right] = L_{h.\nabla f - f.\nabla h + \psi \Gamma \eta , h.\nabla \eta - f.\nabla \psi} $$

Subgroups of the diffeomorphism group include the conformal group and (in 4 dimensions) the Poincare group. For example the algbera for the generators of the Poincare group M and P can be found by substituting in:
 * === Subgroups ===

 * $$ h^{\mu}(x) = M^{\mu}_{\nu} x^{\nu} + P^{\mu} $$

Question on notation
I know very little about this topic, but hope to learn more. As a user of Wikipedia, I would like to know what the following notation means: $$g_{|U \cap X} = f_{|U \cap X}$$

I mention this, not so much to get my question answered (I know, this is not a discussion group) but to point out that Wikipedia apparently offers no help with notation. I cannot click on a symbol and get its definition. I have been wondering if there might be some way to close up that gap. Dratman (talk) 20:05, 19 September 2012 (UTC)

Smooth manifolds vs differentiable manifolds
This article ambiguously switches between the categories of differentiable manifolds and smooth manifolds at various points. I looked at past revisions and there seems to have been a bit of a mini edit war concerning the adjectives "differentiable", "continuously differentiable", and "smooth" in the definition.

I believe the best course of action is to clarify that there are two notions of diffeomorphism, a diffeomorphism of smooth manifolds and a diffeomorphism of differentiable manifolds, and that the respective notion of diffeomorphism is the isomorphism in each category. We could also mention $C^r$ manifolds as a generalization.

This should ideally be done soon, given the mishmash state of the article. As an example, as of 2/15/24 the first two sentences contradict each other, with the first saying "smooth manifolds" and the second stating the definition for differentiable manifolds. 2601:644:400:8430:C5F:BA8C:DFC9:46DF (talk) 09:31, 15 February 2024 (UTC)

 * I think you are just fighting the terminology used in the field of manifold theory. There are a lot of words that are near synonyms.  Often they are synonyms, depending on the context.  We aren't writing a textbook here, just giving the basic ideas and pointing people to references.  If one makes torturous revisions to adhere to terminology that is more specific than researchers use in practice, I don't think we'd be helping anyone. Rybu (talk) 01:36, 18 February 2024 (UTC)
 * Rybu, I don't mean to confuse anyone by being overly pedantic, but I believe that a diffeomorphism of smooth manifolds is not a synonym for diffeomorphism of C^1 manifolds. I'm simply worried that people unfamiliar with the material may get tripped up on the unstated fact there are multiple different notions of diffeomorphism.
 * What do you suggest would be an appropriate edit to clarify that there are multiple notions of diffeomorphism depending on the notion of manifolds one considers? 2601:644:400:8430:1937:78A9:EDC5:C955 (talk) 04:03, 23 February 2024 (UTC)
 * The article has many issues, even after my recent edits. The main issue was that, in the first line of the lead it defined diffeomorphism of non-differentiable manifolds. In my edit, I get rid of the ambiguous word "smooth", which can be understood either as "differentiable" or "indefinitely differentiable. I hope that this solves the problem. D.Lazard (talk) 12:00, 23 February 2024 (UTC)

title: Talk:Diffeomorphism


text:
User talk:Quist

Hey there! Welcome to Wikipedia! I hope you like this place--I sure do--and want to stay. If you need help on how to title new articles check out Naming conventions, and for help on formatting the pages visit the manual of style. If you need help look at Help and The FAQ, plus if you can't find your answer there, check The Village pump or The Reference Desk! Happy wiki-ing! Alexandros

October 24th
Hi Quist,

Just to let you know; you inadvertently removed the bulk of the entry for October 24, when you made adding a birth entry. I've restored the page and your edit, but could you make sure you check again before hitting save page in future?

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title: User talk:Quist


text:
University of Toronto Faculty of Dentistry

The University of Toronto Faculty of Dentistry is a dental school located in Toronto, Ontario, Canada. It is one of the ten dental schools in Canada. It is the largest dental school in Canada with a range of undergraduate and graduate level programs with a total enrolment in the range of 560. The faculty is located at the heart of Downtown Toronto's Discovery District, a neighbourhood with a high concentration of hospitals and research institutes, just south of the University of Toronto's St. George campus. In 2014, the Faculty of Dentistry joined the Toronto Addis Ababa Academic Collaboration (TAAAC), providing support in building capacity for oral health in Ethiopia by creating collaborative teaching opportunities.

History
In 1868, the Province of Ontario passed the "Act Respecting Dentistry". This Act granted the Royal College of Dental Surgeons of Ontario (RCDSO) dual responsibilities of licensing and dental education in Ontario. In 1875, the Royal College of Dental Surgeons was founded, with its first semester schedule to start on November 3, 1875. Under the Act, the RCDS could grant a Licentiate of Dental Surgery (LDS), which was not a formal degree, but training that granted those interested in a license to practice dentistry in Ontario. James Branston Willmott was the first Dean of the school. Dr. Willmott, joined by Dr. Luke Teskey were the only two professors guiding the first class of eleven students in 1875–1876. The RCDS established an affiliation with the University of Toronto in 1888, but it was not until 1889 that it granted the Doctor of Dental Surgery (DDS) degrees. In 1925, the RCDS became the Faculty of Dentistry at the University of Toronto.

One of the alumni of the Faculty of Dentistry, Dr. Harold Keith Box, DDS, PhD (1890–1956), was as a stakeholder in establishing the Canadian Dental Association Research Foundation (CDAF), organized by a committee of the Canadian Dental Association. The CDAF was the first of its kind in Canada, an organization that focused on providing funding for dental research projects in Canadian dental schools. Due to his contribution to the establishment of the CDAF and the various publications that emerged from his work, Dr. Harold Keith Box is known as the father of dental research in Canada. The Faculty of Dentistry continued to contribute to dental research initiatives and became a distinguished contributor at the international level. In 1921, International Association of Dental Research had four chapters, one of which was in Toronto (in addition to Chicago, Boston and New York). In 1927, the IADR Toronto Chapter was successful in receiving funds for the first hospital internship that ensured students had four free hours per day so that they could devote time to research.

In 1945, after years of negotiations, three University of Toronto Faculty of Dentistry professors, Dr. Frank Lott, Dr. Harvey Reid and Dr. Roy G. Ellis, in their role as members of the CDA Research Committee, were able to secure funding for dental research from the National Research Council of Canada. This ensured that generations of dental researchers had access to funding. With the continuous flow of funding, the Faculty of Dentistry was able to establish the first Dental Public Health specialty in 1946, and with its inception began a long history of effort in research and initiatives that would influence oral health policies in Canada.

Ice hockey team
The faculty of dentistry formerly operated a senior ice hockey team known as the Toronto Dentals, which competed in the Ontario Hockey Association and won the J. Ross Robertson Cup in 1917. The Toronto Dentals also became Canadian national champions with their victory at the 1917 Allan Cup. Alumni of the Toronto Dentals hockey team include Stan Brown, Bert McCaffrey, Glenn Smith, Rod Smylie, Charles Stewart and Harry Watson.

Curriculum
The Faculty offers a number of programs. The undergraduate programs include a four-year internationally recognized dental degree Doctor of Dental Surgery (DDS), and an International Dentist Advanced Placement Program (IDAPP). Graduate studies programs include MSc or PhD with a research option or in one of ten dental specialty training options, including Dental Anesthesia, Dental Public Health, Endodontics, Oral and Maxillofacial Radiology, Oral and Maxillofacial Surgery, Oral and Maxillofacial Pathology and/or Oral Medicine, Orthodontics and Dentofacial Orthopedics, Paediatric Dentistry, Periodontics, and Prosthodontics. Approximately 13 yearly residencies for the dental specialty training are available at several of Toronto's teaching hospitals, namely The Hospital for Sick Children, Mount Sinai Hospital, and Sunnybrook Health Sciences Centre. The Faculty also hosts ongoing onsite and online courses and workshops for dental professionals through their Continuing Dental Education unit.

Research
The Faculty of Dentistry has various research foci, including Biomaterials and Biomedical Engineering, Connective Tissues and Regenerative Medicine, Dental Public Health, Education Research, Microbiology, Oral Pathology and Cancer, and Pain and Neuroscience. In addition, the Faculty of Dentistry, in collaboration with the Faculty of Medicine and the Faculty of Applied Science and Engineering, has established the world-renowned Institute of Biomaterials and Biomedical Engineering (IBBME).

Imaging research facility

In 2019, the Faculty of Dentistry opened the Collaborative Advanced Microscopy Laboratories of Dentistry (Camilod), an imaging facility available to researchers in the Toronto area. Camilod uses atomic force microscopy, light and hard electron microscopy, confocal, sliding electron microscopy, epifluorescence microscopes, and wide field microscopy.


text:
Cockburn Sound

Cockburn Sound is an inlet of the Indian Ocean on the coast of Western Australia. It extends from the south of the mouth of the Swan River at Fremantle for about 25 km to Point Peron near Rockingham. The total area of the sound is about 100 km2.

It is bounded on the east by the mainland council areas of Fremantle, Cockburn, Kwinana and Rockingham, on the west by Garden Island and Carnac Island, and includes several rocky outcrops and reefs. Gage Roads lies to the north.

The sound was named in 1827 by Captain James Stirling, probably after Admiral Sir George Cockburn.

The Perth Seawater Desalination Plant can be found here.

Water pollution
Several media reports have been made on pollution of the water in Cockburn Sound, where in late 2015 an estimated 2,100 fish died as a result of algal blooms through poor local water quality.

Reports from 2010 suggest the monitoring of pollutants was not adequate, using degradation of sea-grass since 2005 as a key indicator.

Media coverage in December 2016 indicates the leaking of firefighting chemicals from the local defence base at Garden Island could be a factor, however the report was only made available in September 2016. Prior official reports indicate the water quality was improving. However, there is doubt towards the scope of the report, as it does not include factors such as thermal stress from increased water temperature and oxygen stress as a result of climate change and localised desalination.

title: Cockburn Sound


text:
Richard Donner

Richard Donner (born Richard Donald Schwartzberg; April 24, 1930 – July 5, 2021) was an American filmmaker. Described as "one of Hollywood's most reliable makers of action blockbusters," Donner directed some of the most financially-successful films of the 1970s and 1980s. His 50-year career crossed genres and influenced trends among filmmakers across the world.

Donner began his career in 1957 as a television director. In the 1960s, he directed episodes of the series The Rifleman, The Man from U.N.C.L.E., The Fugitive, The Twilight Zone, The Banana Splits, and many others. Donner made his film debut with the low-budget aviation drama X-15 in 1961, but had his critical and commercial breakthrough with the horror film The Omen in 1976. He directed the landmark superhero film Superman in 1978, which provided an inspiration for the fantasy film genre to eventually gain artistic respectability and commercial dominance. Donner later went on to direct films in the 1980s such as The Goonies and Scrooged, while reinvigorating the buddy cop film genre with the Lethal Weapon series.

Donner and his wife, Lauren, owned a production company, The Donners' Company, which is most successful for producing the Free Willy and X-Men film franchises. Donner also produced Tales from the Crypt and co-wrote several comic books for Superman publisher DC Comics. In 2000, Donner received the President's Award from the Academy of Science Fiction, Fantasy and Horror Films.

Early years
Richard Donald Schwartzberg was born in the Bronx, New York City, to Russian Jewish parents, Hattie (Horowitz) and Fred Schwartzberg. His father owned a small furniture-manufacturing business and his mother was a stay at home mother. He had a sister, Joan. His grandfather owned a movie theatre in Brooklyn, which had an early influence on his perception of film.

After graduating from high school, Donner served in the U.S. Navy, becoming an aerial photographer. He briefly attended New York University but left to pursue a career in acting, relocating to Los Angeles and adopting the stage name "Richard Donner".

Television
Initially, Donner wanted to develop a career as an actor. He gained a bit part in a television program directed by Martin Ritt, who encouraged Donner to become a director instead. Ritt hired Donner as his assistant. Later, Donner was on the staff of Desilu, where he directed commercials. He made the transition into television dramas in the late 1950s, directing some episodes of the Steve McQueen western serial Wanted Dead or Alive and the Chuck Connors western The Rifleman.

During his early career as a director he worked on over twenty-five television series, including Have Gun – Will Travel, The Fugitive, Combat!, Get Smart, The Man from U.N.C.L.E., The Wild Wild West, Gilligan's Island, Kojak, and The Twilight Zone (such as the episodes "Nightmare at 20,000 Feet" starring William Shatner, The Jeopardy Room starring Martin Landau and "From Agnes—With Love" starring Wally Cox), as well as the serial Danger Island from the children's program The Banana Splits.

In the 1990s, Donner served as executive producer of the HBO anthology horror series Tales from the Crypt, based on the EC Comics series of the same name. He directed three episodes during its run ("Dig That Cat... He's Real Gone", "The Ventriloquist's Dummy", "Showdown"), and also produced three feature film spinoffs; Demon Knight, Bordello of Blood, and Ritual.

Films
Donner's first feature film was X-15 (1961), which starred David McLean, Charles Bronson, and Mary Tyler Moore. Seven years passed before he directed his next, Salt and Pepper (1968), with Sammy Davis Jr., and Peter Lawford. His breakthrough film was The Omen (1976). A supernatural horror made in the wake of the success of The Exorcist, the film stars Gregory Peck, David Warner, and Lee Remick. It was the fifth-highest-grossing film of 1976.

Superman (1978)
In 1978, Donner directed Superman: The Movie, starring Christopher Reeve in the title role. Donner approached the film with the concept of creating verisimilitude, an intuitive feeling evoked in the audience that makes the fantasy story feel real within the story's context. It was a hit worldwide, projecting Reeve to international fame. The film co-starred Margot Kidder as Lois Lane, Marlon Brando as Jor-El, and Gene Hackman as archvillain Lex Luthor. At the box office, it grossed $134 million domestically.

Principal photography of Superman included filming of its first sequel, Superman II. Donner filmed most of Superman II with the expectation he would complete the sequel after the release of Superman. Despite the first film's success, Donner was fired from Superman II. Throughout the filming of both Superman films, Donner had a difficult relationship with executive producers Alexander Salkind and Ilya Salkind and producer Pierre Spengler. The Salkinds refused Donner's demand that Spengler be fired; instead, the Salkinds replaced Donner as director of Superman II with Richard Lester, who had worked with the Salkinds on The Three Musketeers and The Four Musketeers and as an uncredited producer on Superman. Following Donner's dismissal, Marlon Brando's scenes were removed from Superman II and much of the film was re-shot under Lester's direction. Gene Hackman refused to return for re-shoots; consequently, all the Lex Luthor scenes that appear in Superman II were filmed by Donner, although Donner refused to be credited.

Rotten Tomatoes' summary states that "Superman II meets, if not exceeds, the standard set by its predecessor." Donner continued to promote the view that his contribution was superior to the rest of the series. Superman II: The Richard Donner Cut was released on November 28, 2006, the same date as the DVD release of the summer film Superman Returns. This version of the film features the re-insertion of Marlon Brando's scenes as Jor-El and relies on a minimum of footage shot by Richard Lester.

1980s and 1990s
Donner directed mixed commercial flops (Inside Moves, Radio Flyer) and successes (The Goonies, the Lethal Weapon series, Scrooged, and Conspiracy Theory).

Lethal Weapon (1987–1998)
Donner's next blockbuster film was the action comedy Lethal Weapon, written by Shane Black. It starred Mel Gibson as a widowed narcotics detective with a suicidal bent "who breaks every rule for the sheer joy of it". It co-starred Danny Glover as a calm homicide detective with a loving family and consideration for retirement. The film's action sequences were considered "truly spectacular" and made the film one of the year's biggest hits.

Donner directed six films starring Mel Gibson in total, creating a Lethal Weapon franchise with three sequels; the last of these was Lethal Weapon 4, released in 1998. In an interview in 2000, Gibson described his impressions of Donner:

"Uncle Dick. He's a great guy, just terrific. Extremely professional. He's an old veteran and has an understanding of film that is the culmination of years of experience. He's got his technical stuff down, his vision down. No matter what you say about Dick, it underrates him. He really loves what he's doing, loves working with actors, and he allows you freedom to explore all kinds of areas. 'All right, kid,' he'll say, and slap you on the back and let you try something, because even he doesn't know sometimes. He's just an extremely charming, talented, great fuckin' guy. I love him."

X-Men
Donner became the executive producer for the 2000 Marvel Comics film X-Men, then also an executive producer for the 2009 X-Men prequel, X-Men Origins: Wolverine. In addition, Donner's wife has produced all of the films in the X-Men film series under their Donners' Company brand.

On October 16, 2008, Donner and Lauren Shuler Donner received stars in a double ceremony on the Hollywood Walk of Fame for their achievements in motion pictures, located at 6712 Hollywood Boulevard.

Unrealized projects
Donner was the first choice to direct the "unofficial" James Bond film Never Say Never Again, but turned it down after he decided he disliked Lorenzo Semple Jr.'s script. He would have served as director of Jurassic Park had Columbia/TriStar won the bidding war for the screen rights against other studios. Donner was also considered as director for The Lost Boys, The Last Boy Scout, Memoirs of an Invisible Man, Dave, Forever Young, and The Sandlot. In 2001, Donner claimed he planned to direct a sequel to The Goonies after Timeline, and he planned to direct the fifth and final Lethal Weapon film before his death in 2021.

Books

 * Action Comics (co-writer with Geoff Johns)
 * Last Son and Escape from Bizarro World (2009)
 * You're the Director... You Figure It Out: The Life and Films of Richard Donner (2010)

Comic books
One of Donner's assistants in the late 1990s was comic book writer Geoff Johns. In October 2006, Donner, Johns, and artist Adam Kubert became the new creative team on Action Comics, the publisher's most time honored publication and one of DC Comics' two main Superman titles. Together, Johns and Donner collaborated on the stories Last Son and Escape from Bizarro World, both of which have been released in collected book form. Donner and Johns also co-wrote a story for Action Comics #1000, released in April 2018.

Biography
In 2010, Donner's authorized biography titled You're the Director... You Figure It Out: The Life and Films of Richard Donner by James Christie was published by BearManor Media. The book features a foreword by Mel Gibson.

Script Magazine described the book as an "engaging portrait of a warm-hearted (if occasionally gruff) man who can justly be considered the modern equivalent of Victor Fleming and Michael Curtiz—a highly talented, professional director of motion pictures who has thrived in the studio system and made some pretty good pictures to boot."

The Directors Guild of America called the book "a compelling study of an ebullient, ballsy risk-taker who was a director even before he was aware of it" that "ably captures Donner's joy in doing a job he loves."

Personal life and death
Donner married film producer Lauren Shuler on November 27, 1985. The two merged their respective production companies to create Shuler-Donner Productions (now The Donners' Company) the same year. Donner and Shuler remained together until the former's death.

Donner was a cousin of actor Steve Kahan, who played a policeman tracking Otis in Superman: The Movie, and played Captain Ed Murphy in the Lethal Weapon film franchise. Donner also cast Kahan in some of his other films.

Donner died on July 5, 2021, at his West Hollywood, California home, at the age of 91. The cause of death was heart failure with atherosclerosis as an underlying cause.

Film
Executive producer
 * Omen III: The Final Conflict (1981)
 * The Lost Boys (1987)
 * Delirious (1991)
 * Free Willy (1993)
 * Free Willy 2: The Adventure Home (1995)
 * Tales from the Crypt: Demon Knight (1995)
 * Tales from the Crypt: Bordello of Blood (1996)
 * Free Willy 3: The Rescue (1997)
 * Any Given Sunday (1999)
 * X-Men (2000)
 * X-Men Origins: Wolverine (2009)

title: Richard Donner


text:
Lingqu

The Lingqu is a canal in Xing'an County, near Guilin, in the northwestern corner of Guangxi, China.

It connects the Xiang River (which flows north into the Yangtze) with the Li River (which flows south into the Gui River and Xijiang), and thus is part of a historical waterway between the Yangtze and the Pearl River Delta. It was the first canal in the world to connect two river valleys and enabled boats to travel 2000 km inland from Beijing to Pearl River Delta. It is also one of the most well-preserved such projects in the world.

The canal is 36.4 km long.

History
In 214 BC, Qin Shi Huang, the First Emperor of the Qin dynasty (221–206 BC), ordered the construction of a canal connecting the Xiang and the Li rivers, in order to attack the Baiyue tribes in the south. The architect who designed the canal was Shi Lu. It is the oldest contour canal in the world, receiving its water from the Xiang. It was fitted with thirty-seven flash locks by 825 AD. There is a clear description of pound locks in the twelfth century, which were probably installed one or two centuries before. The canal also aided water conservation by diverting up to a third of the flow of the Xiang to the Li.

The canal has been placed on the UNESCO World Heritage Sites tentative list.

title: Lingqu


text: User talk:193.60.133.201

title: User talk:193.60.133.201


text:
Rainbow boa

The rainbow boa (Epicrates cenchria) is a boa species endemic to Central and South America. A semi-arboreal species (not only do they climb in the wild but also proven in captivity), it is known for its attractive iridescent/holographic sheen caused by structural coloration. Five subspecies are currently recognized, including the nominate subspecies described here.

Distribution and habitat
The rainbow boa is found in lower Central America (Costa Rica and Panama), and farther south into South America. It occurs east of the Andes, roughly reaching northern Argentina (in the provinces Chaco, Córdoba, Corrientes, Formosa, Salta, Santiago del Estero and Tucumán).

The rainbow boa's habitat generally consists of humid woodlands and rainforests, but it can also be found in open savannas.

Description
The rainbow boa is typically orange, brown, or reddish brown, with a paler belly and black markings: three parallel stripes on the head, rings down the back, and lateral blotches with a crescent over them, although there is a great deal of natural variation that may be heightened by artificial breeding. It is 4 to 6 feet in length and sexually dimorphic, as females are slightly longer and have wider abdomens. Males are stronger when compared with females of the same length and more resistant to handling.

Behavior
Rainbow boas are nocturnal and most active in the middle of the night.

This species is semi-arboreal, spending time both on the ground and in trees. They are also known to spend time in bodies of water, and are considered capable swimmers.

Mating habits
The rainbow boa reaches sexual maturity at 1.2 meters for males and 1.4 meters for females, or when they're approximately 2.5 to 4 years old. Females need to be the correct size otherwise they could have complications during and after birth. Males can mate with multiple females which can be beneficial for reptile breeding.

Eating habits during breeding season
Males generally go without feeding during the mating season and females tend to eat smaller portions during the breeding season. In order to decrease the probability of birthing issues, females should be fed smaller rats/mice in order to save space for proper ova development.

It is not unusual for both sexes to go without eating during the mating season.

Captivity
The most common type of rainbow boa found in the pet trade is the Brazilian rainbow boa, E. c. cenchria. During the 1980s and early 1990s, substantial numbers were exported from Suriname. Today, however, far fewer are exported, and most offered for sale are captive bred. With good care, a captive Brazilian rainbow boa can be expected to live for up to 30 years, but most live up to 20.

Etymology
The subspecific names barbouri and gaigeae are in honor of American herpetologists Thomas Barbour and Helen Beulah Thompson Gaige, respectively.

title: Rainbow boa


text:
Blubberhouses

Blubberhouses is a small village and civil parish in the Washburn Valley in North Yorkshire, England. The population as at the 2011 Census was less than 100, so details were included in the civil parish of Fewston. In 2015, North Yorkshire County Council estimated the population of the village and parish to be 40. Blubberhouses is situated to the south of the Yorkshire Dales National Park, and to the north of a Roman road and Fewston Reservoir on the A59 road linking Harrogate to Skipton.

History
The name of the village derives from the Anglo-Saxon bluberhūs = "the house(s) which is/are at the bubbling stream", with a later regularised plural; the -um form came from the Anglo-Saxon dative plural case æt bluberhūsum = "at the houses which ...". Later forms of the name on record include "Bluburgh", "Bluborrow", and "Bluburhouse". A forge was recorded at Blubberhouses in 1227, and in the 16th century, the village had metal smelters for lead and iron ore. The lead was mined locally around Fewston. The other major industry in the area was cloth-working with Westhouse mill being located just across the A59 road from the church. Westhouse opened in the 1790s to process flax, but closed in 1877 and the stone from the mill was used in the construction of the dam wall of Fewston Reservoir.

St Andrew's Church, Blubberhouses was designed by Edward Buckton Lamb and was built in 1851. It is now a grade II listed building. It is part of the ecclesiastical parish of Fewston, and is maintained by the Friends of Blubberhouses Church, after it was decided to keep the church open in 2015 in spite of the need to repair the church roof at quite a cost. The village has a cricket team who play in the Theakston Nidderdale League, and their ground is alongside the River Washburn in the village.

The road through Blubberhouses, which is now the A59, running in an east/west direction, was first turnpiked in the 1770s. The section to the west of Blubberhouses towards Skipton goes over Blubberhouses Moor, and between 1823 and 1827, a new cut was made to the south with a gentler gradient, but following the contours of the hill. The 19th century section has suffered many landslips, and so a new bypass, following the route of the original turnpike, is due to open in 2025.

In July 2014, Stage 2 of the 2014 Tour de France from York to Sheffield, passed through the village. It was also the location of the first climb of the stage, the Category 4 Côte de Blubberhouses, at the 47 km point. It was 1.8 km long at an average gradient of 6.1%. The single point for the King of the Mountains competition was claimed by Frenchman, Cyril Lemoine of Cofidis.

Blubberhouses is used as a key setting for local composer Andy Tillison's composition Le Sacre Du Travail, a five movement electric sinfonia recorded by the progressive rock ensemble The Tangent. The work is, according to the CD sleeve notes, loosely based around Igor Stravinsky's Le Sacre Du Printemps (The Rite Of Spring). One section of the second movement is named "Dawn At Blubberhouses" and the village is mentioned by name in the lyrics/libretto. A short story in the CD sleeve describes the journey of a man travelling to work through the village in the small hours of the morning. The CD album was released in 2013 by the German record label Inside Out Music.

Governance
There is no record of Blubberhouses in the Domesday Book, but it is recorded in 1172 as Bluberh(o)usum, being in the lower division of the wapentake of Claro in the West Riding of Yorkshire. It is now in the county of North Yorkshire. At the 1901 census, the size of the parish was recorded as 3,736 acre, and in 2021, it was listed as reaching to 1,524 ha. The village and parish is represented at Parliament as part of the Skipton and Ripon Constituency. In 2011, the population was estimated to be 30, which by 2015, had risen to 40.

title: Blubberhouses


text:
Clifton Chenier

Clifton Chenier (June 25, 1925 – December 12, 1987), was an American musician known as a pioneer of zydeco, a style of music that arose from Creole music, with R&B, blues, and Cajun influences. He sang and played the accordion. Chenier won a Grammy Award in 1983.

Chenier was known as the King of Zydeco, and also billed as the King of the South.

Biography
Chenier was a native of Leonville, Louisiana, near Opelousas. He spoke Louisiana French as a first language.

Chenier was exposed to music growing up, as he accompanied his father, Joseph Chenier, a farmer and player of the single-row diatonic accordion, to dances. His uncle, Morris Chenier, played fiddle. Musical influences that he cited from radio were Muddy Waters, Peetie Wheatstraw, and Lightning Hopkins, while local influences included Creole musicians Claude Faulk, Jesse and ZoZo Reynolds, and Sidney Babineaux. Clifton began playing accordion around 1947, and by 1950 was playing in a club in Basile with his brother Cleveland Chenier on rubboard. Before launching a professional music career, Chenier worked in fields and at a Gulf Oil refinery in Port Arthur, Texas, outside of whose gates he also played music with Cleveland.

Chenier began his recording career in 1954, when he signed with Elko Records and released Cliston's Blues [sic], a regional success. In 1955 he signed with Specialty Records and garnered his first national hit with his label debut "Eh, 'tite Fille" ("Hey, Little Girl", a cover of Professor Longhair's song). The release's national success led to numerous tours with popular rhythm and blues performers such as Ray Charles, Etta James, Chuck Berry, Little Richard, T-Bone Walker, and Lowell Fulson. He also toured in the early days with Clarence Garlow, billed as the Two Crazy Frenchmen. Chenier was signed with Chess Records in Chicago, followed by the Arhoolie label in the early 1960s. Arhoolie gave Chenier exposure to new audiences of blues and rock listeners across the US.

In April 1966, Chenier appeared at the Berkeley Blues Festival on the University of California campus and was subsequently described by Ralph J. Gleason, jazz critic of the San Francisco Chronicle, as "one of the most surprising musicians I have heard in some time, with a marvelously moving style of playing the accordion ... blues accordion, that's right, blues accordion." Over time, the band expanded to include saxophone and organ, and electric effects pedals, with all melody instruments taking turns at solos.

Chenier was the first act to play at Antone's, a blues club on Sixth Street in Austin, Texas. In 1976, he reached a national audience by appearing on the first season of the PBS music program Austin City Limits. In 1979 he returned to the show with his Red Hot Louisiana Band.

Chenier's popularity peaked in the 1980s, and he was recognized with a Grammy Award in 1983 for his album I'm Here! It was the first Grammy for his new label Alligator Records. Chenier followed Queen Ida as the second Louisiana Creole to win a Grammy.

Chenier is credited with redesigning the wood and crimped tin washboard into the vest frottoir, an instrument that easily hung from the shoulders. He sketched his idea for a metalworker in Port Arthur named Willie Landry, who made the first frottoir. Cleveland Chenier, Clifton's older brother, also played in the Red Hot Louisiana Band. He found popularity for his ability to manipulate the distinctive sound of the frottoir by rubbing several bottle openers (held in each hand) along its ridges.

During their prime, Chenier and his band traveled throughout the world.

Later years and death
Chenier suffered from diabetes, which eventually forced him to have a foot amputated and required dialysis because of associated kidney problems.

He died of diabetes-related kidney disease in December 1987 in Lafayette, Louisiana. His funeral took place at Immaculate Heart of Mary Church in Lafayette.

Legacy and tributes
Since 1987, his son C. J. Chenier (born Clayton Joseph Thompson) has carried on the zydeco tradition by touring with Chenier's band and recording albums. Clifton Chenier's bandmate and protégé Buckwheat Zydeco achieved national success playing the piano accordion.

Rory Gallagher wrote a song in tribute to Chenier, "The King of Zydeco". Paul Simon mentioned Chenier in his song "That Was Your Mother" on his 1986 album Graceland, calling him the "King of the Bayou." Sonny Landreth recalls growing up on the rhythm of Clifton and Cleveland and the Red Hot Louisiana Band in South of I-10, song title and name of the album released in 1995. John Mellencamp refers to "Clifton" in his song "Lafayette", about the Louisiana city where Chenier often performed, on Mellencamp's 2003 album Trouble No More. Zachary Richard mentions Chenier in his song "Clif's Zydeco" (on Richard's 2012 album Le Fou). The Squeezebox Stompers' "Zydeco Train" says, "Clifton Chenier, he's the engineer."

The jam band Phish often covers Chenier's song "My Soul" in live performances.

Chenier is the subject of Les Blank's 1973 documentary film Hot Pepper.

Awards and honors
Chenier received a 1984 National Heritage Fellowship from the National Endowment for the Arts, the U.S. government's highest honor in folk and traditional arts. He was inducted posthumously into the Blues Hall of Fame in 1989, and the Louisiana Music Hall of Fame in 2011. In 2014, he received a Grammy Lifetime Achievement Award.

In 2016, the Library of Congress deemed Chenier's album Bogalusa Boogie to be "culturally, historically, or aesthetically significant" and selected it for preservation in the National Recording Registry.

Partial discography

 * Cliston's Blues (Elko Records), 1954
 * Ay-Tete Fi (Specialty Records), 1955
 * Louisiana Blues & Zydeco (Arhoolie Records), 1965
 * Bon Ton Roulet ! (Arhoolie), 1967
 * Bogalusa Boogie (Arhoolie), 1975
 * Frenchin' the Boogie (Blue Star), 1976
 * Boogie in Black and White (with swamp pop musician Rod Bernard, Jin Records), 1976
 * Red Hot Louisiana Band (Arhoolie), 1977
 * New Orleans (GNP Crescendo Records GNP 2119) 1978
 * Boogie & Zydeco (Sonet Records SNTF 801), 1979
 * I'm Here! (Alligator Records), 1982

title: Clifton Chenier


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Semi-Tough

Semi-Tough is a 1977 American sports comedy film directed by Michael Ritchie and starring Burt Reynolds, Kris Kristofferson, Jill Clayburgh, Robert Preston, Lotte Lenya, and Bert Convy. It is set in the world of American professional football.

The plot involves a love triangle between the characters portrayed by Reynolds, Kristofferson, and Clayburgh. Semi-Tough also includes a parody of Werner Erhard's Erhard Seminars Training (est), depicted in the film as an organization called "B.E.A.T.".

The film is based on the 1972 novel of the same name by Dan Jenkins. It was adapted for the screen by writer Walter Bernstein and director Ritchie, who added a new storyline that included a satire of the self-help movement and new religions.

Semi-Tough follows the story of pro football friends Billy Clyde Puckett and Marvin "Shake" Tiller, who have a third roommate, Barbara Jane Bookman. A romance develops between Shake and Barbara Jane when he becomes self-confident after completing a self awareness course called "B.E.A.T.", at which point Billy Clyde slyly makes a play to win her for himself.

The film received mixed receptions. Some reviewers praised its parodies of the est training, Erhard and other new age movements such as Rolfing. Others criticized the script and direction, noting that some of director Ritchie's previous films had more of a personal tone. Still other reviews lamented the film's departure from the novel Semi-Tough, which dealt with football rather than the new age movement.

Plot
Wide receiver Marvin "Shake" Tiller and running back Billy Clyde Puckett are football buddies who play for a Miami pro team owned by "Big Ed" Bookman. Bookman's daughter Barbara Jane is roommates with both men, and the film depicts a subtle love triangle relationship developing between Barbara Jane and her two friends. Initially the three of them are just good friends, but she begins to have romantic feelings for Shake, who has become more self-confident after taking self-improvement training from seminar leader Friedrich Bismark. The program is called Bismark Energy Action Training, or B.E.A.T. After Shake completes his course, Barbara Jane and he have sex and start a relationship. They plan to marry after the football season ends.

Barbara Jane is not a follower of B.E.A.T., and Shake is warned by his leader Bismark that "mixed marriages don't work". Barbara Jane is determined to make it work, so she attends B.E.A.T. in an effort to "get it". At the end of the long training session, she is worn out from Bismark's "sadistic abuse, pious drivel, and sheer double talk". She also feels guilty that she did not "get it". Shake is insistent that the training has had proven results for him, noting that he has not dropped a football pass since completing B.E.A.T.

Billy Clyde also has feelings for Barbara Jane and, without telling Shake or Barbara Jean beforehand, enrolls in the same B.E.A.T. training session as Barbara Jane in order to understand what she is going through. In the training, Billy Clyde is shown coping successfully with the seminar rules that prohibit going to the bathroom or talking to other trainees. When the trainees are encouraged to express their most repressed feelings, he exhibits the strongest, most primal display of them all. Shake is surprised to see Billy Clyde emerge from Barbara Jane's B.E.A.T. training session. Billy Clyde "confesses" to Shake that he took the training so that he'd be able to mock it more effectively afterwards, but says that he actually "got it". For a time, he pretends he underwent a conversion to Bismark's way of thinking, often using B.E.A.T. terminology and clichés in everyday conversation.

Shake worries that Barbara Jane hasn't "gotten it", and Billy Clyde tries to reassure Shake that it doesn't matter, but Shake remains dubious. During the wedding, as Big Ed escorts Barbara Jane to the altar, the minister turns to Bismark and gives him some advice on how he can avoid capital gains tax in his business. When the minister asks Shake if he takes Barbara Jane to be his wife, he hesitates and finally says "no". An outraged Big Ed vows to trade Shake to the worst team in the league, then tells Bismark to "experience this" as he punches him in the face, starting a general brawl. Billy Clyde makes a disparaging remark about Bismark to Barbara Jane, causing her to realize that he had been sandbagging all along about having "gotten it" from B.E.A.T. The two of them leave in the commotion, and reveal their feelings about each other as they walk along the beach in their wedding attire.

Cast

 * Burt Reynolds as Billy Clyde Puckett
 * Kris Kristofferson as Marvin "Shake" Tiller
 * Jill Clayburgh as Barbara Jane Bookman
 * Robert Preston as Big Ed Bookman
 * Bert Convy as Friedrich Bismark
 * Roger E. Mosley as Puddin Patterson Sr.
 * Lotte Lenya as Clara Pelf
 * Richard Masur as Phillip Hooper
 * Carl Weathers as Dreamer Tatum
 * Brian Dennehy as T.J. Lambert
 * Mary Jo Catlett as Earlene Emery
 * Joe Kapp as Hose Manning
 * Ron Silver as Vlada Kostov
 * Jim McKrell as Bud McNair
 * Peter Bromilow as Kostov's Interpreter
 * Norman Alden as Coach Alvin Parks

Adaptation
Semi-Tough is based on the best-selling novel of the same name by Dan Jenkins, and the screenplay was written by Walter Bernstein. Bernstein and director Michael Ritchie used parts of Jenkins's novel and added parodies of self-help groups, new religions, and the Human Potential Movement. Ritchie was partially influenced by Adam Smith's book, Powers of Mind. Though the screenplay altered his original work, Dan Jenkins wrote that Semi-Tough "wasn't a horrible movie in my opinion". Jenkins later commented that the film adaptation of his book Baja Oklahoma  was: "a lot more faithful to the novel than Semi-Tough ever was".

Preparation
Burt Reynolds began training with Kris Kristofferson to get in shape before film production. Before work began on the film, actor Bert Convy attended one of Werner Erhard's est training sessions to prepare for his role as B.E.A.T. seminar leader Friedrich Bismark. Pro football stars were hired to give realism to the film, including John Matuszak, Paul Hornung, Joe Kapp, and Ed "Too Tall" Jones. The football scenes were shot at the Cotton Bowl.

Parodies of self-improvement, new religions
Bernstein and Ritchie's modified screenplay based on Jenkins' book includes a storyline with "satiric jabs" at new religions, self-improvement, and the Human Potential Movement. A form of Rolfing is also parodied in the film by Lotte Lenya, whose character Clara Pelf is seen as a spoof of "a Rolf-like masseuse". Big Ed Bookman is seen crawling around on all fours practicing something called "creep therapy" or "movagenics". Movagenics is seen in the film as a way for individuals to find their "lost center of consciousness". Big Ed Bookman is also shown proselytizing for "Movagenics", a fictional group in the film which TIME referred to in its review as both a cult and a new faith.

The film includes a parody with B.E.A.T. as a stand-in for est. In American Film Now, Friedrich Bismark is simply described as "the Werner Erhard character". The Grove Book of Hollywood describes Bert Convy as a "Werner Erhard-lookalike", in his portrayal of Friedrich Bismark. Barbara Jane Bookman's guilty feelings for not "getting it" after completing her B.E.A.T. seminar are seen as a reference to: "how creeds like est put nonbelievers on the defensive". The book also notes: "The film captures the peculiar mixture of spirituality and pragmatism that surrounded est", and also describes the minister's capital gains tax advice to Friedrich Bismark at Shake's wedding as "sardonic". A 1977 review in TIME refers to Friedrich Bismark's B.E.A.T. as "an est-like movement", and notes: "the Ritchie-Bernstein version of an est seminar is done with marvelous malice". Psychiatry and the Cinema characterizes the film as a "devastating parody" of the est training.

After Semi-Tough's release in 1977, Bert Convy was contacted by a number of est followers, as well as by Werner Erhard. After Convy appeared on The Tonight Show and discussed his experiences attending Erhard Seminars Training in preparation for his role as Friedrich Bismark, he received a letter from Erhard stating: "it would be great for us to get together". Of the est seminar itself, Convy recalled that when another attendee complained of a headache during the course, the group leader told him to "experience it", and when another attendee wet his pants, he was told to "experience the warmth". In a scene from the film, a woman exclaims to Friedrich Bismark in the middle of a seminar: "I peed in my pants and it felt good." During actual filming on Semi-Tough, Convy received a late-night phone call from actress Valerie Harper, known in Hollywood as a devoted student of Werner Erhard. She related to Convy that Erhard was "pleased" with the role, and she wished him success in the film. Convy suspected that her real reason for calling was to subtly pressure him to go easy on his parody of Erhard in the film. Harper is mentioned by name in the completed film.

Box office and releases
The film grossed $37,187,139 at the box office. Semi-Tough was later developed as a short-lived television series with a similar theme, and was a candidate to be picked up in the fall of 1980. The 1980 series starred Bruce McGill playing Burt Reynolds' original role, with co-star David Hasselhoff. The film has since been released in both VHS and DVD formats.

Awards
In 1978, Walter Bernstein received a nomination for a WGA Award from the Writers Guild of America in the "Best Comedy Adapted from Another Medium" category, for his adaptation of Dan Jenkins' novel.

Critical reception
The press disagreed on the satire of Erhard in the film, some praising it, some not. The Wall Street Journal criticized the film, save for its satirical nature: "The movie isn't much - an erratic ramble - but it has some pleasant moments, and a delicious send-up of the self-improvement guru Werner Erhard." The Charlotte Observer praised Bert Convy's portrayal of the self-help guru Frederick Bismark, and called Convy: "… a hilariously smug consciousness-raiser with a more than passing resemblance to EST's [sic] Werner Erhard". Magill's Survey of Cinema described the film as a chiding of American "religious fads and philosophies", and The Grove Book of Hollywood called it a "cheeky film" that poked fun at the "est" craze and other human potential fads. TIME called the film one of 1977's best comedies, and also noted that it was: "without a doubt the year's most socially useful film".

The film did not receive a positive review in Variety, where the reviewer commented: "Semi-Tough begins as a bawdy and lively romantic comedy about slap-happy pro football players, then slows down to a too-inside putdown of contemporary self-help programs." Variety noted that stars Burt Reynolds, Kris Kristofferson, and Jill Clayburgh were "excellent" within the "zigzag" script and poor direction they were given. Michael Costello of Allmovie also criticized the script, and wrote: "While much of this is quite predictable, a number of the jokes score, Burt Reynolds works his charm overtime, and Jill Clayburgh and Kris Kristofferson are effective as comic foils." In American Film Now, author James Monaco commented on director Michael Ritchie's directing style in Semi-Tough, stating that in the film Ritchie was "speaking in a professional voice". Monaco noted that Ritchie's prior films evoked a more personal tone, and had more of a sense of wonder than Semi-Tough.

Chapman's In Search of Stupidity characterized the film as: "Possibly one of the best movies ever made by Burt Reynolds", and described the film's parody of est as accurate and amusing. Leonard Maltin criticized parts of the script, stating that Reynolds' charm filled in for the film's other deficiencies. Hunsberger criticized Ritchie's screenplay adaptation in his work The Quintessential Dictionary, complaining that the game of football should have supported the film as a plot device, but was instead left to the side in favor of other stories.

title: Semi-Tough


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Professor Chronotis

Professor Urban Chronotis is a fictional character created by Douglas Adams. He was originally created for the 1979 Doctor Who serial Shada, starring Tom Baker and Lalla Ward. However, the filming of the serial was never completed due to a strike. Adams then re-used the character and many of the themes from Shada in his novel Dirk Gently's Holistic Detective Agency, first published in 1987. In both versions, Chronotis is a clandestine time-traveller, whose time machine (or TARDIS, as they are termed in Doctor Who) is disguised as his college rooms.

Shada
In 1992, Shada was completed through the use of linking narration by Tom Baker and released on home video. In this version, the character was played by Denis Carey.

In 2003, Shada was remade as an animated webcast by Big Finish Productions for the BBC. Professor Chronotis was played by James Fox in this version, which was also released as an audio play on CD.

Chronotis holds the post of Regius Professor of Chronology at Cambridge University. He is also a Fellow of the fictional St. Cedd's College, Cambridge, where he has resided for three centuries. In Shada, Chronotis is a retired Time Lord and old friend of the Doctor, living out his remaining centuries in academic seclusion. His unnaturally long life goes un-discussed by his colleagues, owing to the discretion he claims is a hallmark of the old Cambridge colleges. Owing to repeated time travel and advanced age, he is extremely forgetful and absent-minded. He often does not remember which time period he has travelled nor the reason why. Among other things, he has a liking for tea and silly jokes. It is also revealed during the course of Shada that Chronotis was a Time Lord criminal named Salyavin, who was falsely imprisoned on and escaped from the prison planetoid Shada.

Despite his being innocent, the Time Lords attempted to imprison Salyavin in Shada for fear of him using his unique mind powers to take over Gallifrey. Salyavin escaped by using his mind powers to control the guards escorting him. Salyavin escapes with his TARDIS, disguised as a bookmark. Shortly after escaping Shada, he used his powers to remove all memory of Shada from every Time Lord. In doing so, he is forced to regenerate, and take the name of Chronotis.

Dirk Gently
The Dirk Gently version of the character is almost identical to the Shada version, though the novel contains no references to Time Lords. Here Chronotis is so old and forgetful that he has no idea who or what he originally was, though he has vague memories of Cleopatra (who he claims wore outrageous earrings and reeked of cat food and death), and believes that he achieved his position after his retirement from "something pretty good". He is known as "Reg", short for Regius Professor Chronotis.

His Chair of Chronology was created by mad king George III who was terrified that if time were to start flowing backwards, all the bad experiences of his life might recur. In fact, central to the book's theme are the three questions the King asked Reg upon his appointment: if one could travel through time, if there was a reason one thing happened after another, and if there was any way of stopping it (the answers are, in order, yes, no and maybe), Reg apparently promptly realising the answer to the three questions and concluding that he could then take time off in reasonable comfort. However, at the conclusion of the novel his time machine was burned out when the phone repair man fixed Reg's telephone so that it would never go wrong again; for some reason the phone always malfunctioned whenever Reg used the time machine due to there being something fundamentally inexplicable about the British telephone system.

In 2007, Dirk Gently's Holistic Detective Agency was adapted to radio and Professor Chronotis was portrayed by Andrew Sachs (who had previously played Skagra in the Big Finish audio/BBC webcast version of Shada).

title: Professor Chronotis


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User:Interdigital

This site is extremely interesting. Now I have to make sure I don't spend too much time here. :-)

My blog

title: User:Interdigital


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Group algebra of a locally compact group

In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.

The algebra Cc(G) of continuous functions with compact support
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure. Using the Haar measure, one can define a convolution operation on the space Cc(G) of complex-valued continuous functions on G with compact support; Cc(G) can then be given any of various norms and the completion will be a group algebra.

To define the convolution operation, let f and g be two functions in Cc(G). For t in G, define

 * $$ [f * g](t) = \int_G f(s) g \left (s^{-1} t \right )\, d \mu(s).$$

The fact that $$f * g$$ is continuous is immediate from the dominated convergence theorem. Also

 * $$ \operatorname{Support}(f * g) \subseteq \operatorname{Support}(f) \cdot \operatorname{Support}(g) $$

where the dot stands for the product in G. Cc(G) also has a natural involution defined by:

 * $$ f^*(s) = \overline{f(s^{-1})} \, \Delta(s^{-1}) $$

where Δ is the modular function on G. With this involution, it is a *-algebra.

Theorem. With the norm:

 * $$ \|f\|_1 := \int_G |f(s)| \, d\mu(s), $$

Cc(G) becomes an involutive normed algebra with an approximate identity.

The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed, if V is a compact neighborhood of the identity, let fV be a non-negative continuous function supported in V such that

 * $$ \int_V f_{V}(g)\, d \mu(g) =1.$$

Then {fV}V is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the discrete topology.

Note that for discrete groups, Cc(G) is the same thing as the complex group ring C[G].

The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following

Theorem. Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, then

 * $$ \pi_U (f) = \int_G f(g) U(g)\, d \mu(g)$$

is a non-degenerate bounded *-representation of the normed algebra Cc(G). The map

 * $$ U \mapsto \pi_U$$

is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded *-representations of Cc(G). This bijection respects unitary equivalence and strong containment. In particular, $\pi$U is irreducible if and only if U is irreducible.

Non-degeneracy of a representation π of Cc(G) on a Hilbert space Hπ means that

 * $$ \left \{\pi(f) \xi : f \in \operatorname{C}_c(G), \xi \in H_\pi \right \} $$

is dense in Hπ.

The convolution algebra L1(G)
It is a standard theorem of measure theory that the completion of Cc(G) in the L1(G) norm is isomorphic to the space L1(G) of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero.

"Theorem. L1(G) is a Banach *-algebra with the convolution product and involution defined above and with the L1 norm. L1(G) also has a bounded approximate identity."

The group C*-algebra C*(G)
Let C[G] be the group ring of a discrete group G.

For a locally compact group G, the group C*-algebra C*(G) of G is defined to be the C*-enveloping algebra of L1(G), i.e. the completion of Cc(G) with respect to the largest C*-norm:

 * $$ \|f\|_{C^*} := \sup_\pi \|\pi(f)\|,$$

where π ranges over all non-degenerate *-representations of Cc(G) on Hilbert spaces. When G is discrete, it follows from the triangle inequality that, for any such π, one has:

 * $$ \|\pi (f)\| \leq \| f \|_1,$$

hence the norm is well-defined.

It follows from the definition that, when G is a discrete group, C*(G) has the following universal property: any *-homomorphism from C[G] to some B(H) (the C*-algebra of bounded operators on some Hilbert space H) factors through the inclusion map:

 * $$\mathbf{C}[G] \hookrightarrow C^*_{\max}(G).$$

The reduced group C*-algebra Cr*(G)
The reduced group C*-algebra Cr*(G) is the completion of Cc(G) with respect to the norm

 * $$ \|f\|_{C^*_r} := \sup \left \{ \|f*g\|_2: \|g\|_2 = 1 \right \},$$

where

 * $$ \|f\|_2 = \sqrt{\int_G |f|^2 \, d\mu}$$

is the L2 norm. Since the completion of Cc(G) with regard to the L2 norm is a Hilbert space, the Cr* norm is the norm of the bounded operator acting on L2(G) by convolution with f and thus a C*-norm.

Equivalently, Cr*(G) is the C*-algebra generated by the image of the left regular representation on ℓ2(G).

In general, Cr*(G) is a quotient of C*(G). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if G is amenable.

von Neumann algebras associated to groups
The group von Neumann algebra W*(G) of G is the enveloping von Neumann algebra of C*(G).

For a discrete group G, we can consider the Hilbert space ℓ2(G) for which G is an orthonormal basis. Since G operates on ℓ2(G) by permuting the basis vectors, we can identify the complex group ring C[G] with a subalgebra of the algebra of bounded operators on ℓ2(G). The weak closure of this subalgebra, NG, is a von Neumann algebra.

The center of NG can be described in terms of those elements of G whose conjugacy class is finite. In particular, if the identity element of G is the only group element with that property (that is, G has the infinite conjugacy class property), the center of NG consists only of complex multiples of the identity.

NG is isomorphic to the hyperfinite type II1 factor if and only if G is countable, amenable, and has the infinite conjugacy class property.