A Lefèvre

R Schott

M Chave

Field experiments assessing companion plants for crop health in horticulture provide actionable knowledge

Keywords: agronomic experiment, multi-species system, vegetable, orchards, ornamental plants, agroecological practices, pest regulation

Fostering plant biodiversity using companion plants (CP) is a promising alternative for the sustainable management of crop health in horticultural systems. To evaluate the contribution to pest and disease control, CPs are tested in field experiments carried in conditions close to those used on commercial farms. Such experiments provide scientific knowledge on the achievement of the targeted function, but also raise many operational issues. We therefore consider that experimenters who design and test systems with CPs support the production of experience-based knowledge. Our objective is to highlight and characterize this knowledge. We interviewed experiment leaders, covering 21 locations in metropolitan France and the West Indies. We defined a "modality of use of companion plants" (MU) as the combination of the CP, its technical management, including spatial and temporal arrangement in the field, and the main expected functional processes assigned to CPs. 46 MUs were investigated. MU scale was relevant for experimenters to report on interactions between practices and CPs, and to properly assess CP success or failure. We showed that experimenters reflect on both anticipated barriers and on barriers they actually faced when implementing CPs in the field. The diverse obstacles encountered were mainly related to CP growth, labour, crop management, or sanitary control difficulties. Experimenters also identified improvements. This knowledge from practical experiments completes scientific knowledge on ecological processes and should be useful to stakeholders intending to adopt CPs. Building on such actionable knowledge would be valuable for the sustainable management of crop health in horticulture.

INTRODUCTION

Designing and implementing horticultural systems with higher plant diversity is a promising way to sustain production and limit crop damage due to pests and diseases [START_REF] Ratnadass | Synergies and trade-offs in natural regulation of crop pests and diseases under plant species diversification[END_REF]. At field scale, enhancing plant diversity is a strategic practice to combine with other agroecological practices in a systemic approach [START_REF] Schut | Systems approaches to innovation in crop protection. A systematic literature review[END_REF]. Thus, plant pest and disease regulation services can be delivered in horticultural cropping systems where plant diversity is practised over time and space [START_REF] Deguine | Integrated pest management: good intentions, hard realities[END_REF][START_REF] Gaba | Multiple cropping systems as drivers for providing multiple ecosystem services: from concepts to design[END_REF]. Companion plants (CPs) introduced in the agrosystem are known to stimulate ecological a E-mail: amelie.lefevre@inrae.fr processes in order to directly or indirectly regulate pests. A growing body of research focuses on the potential of CPs in crop health management [START_REF] Ratnadass | Synergies and trade-offs in natural regulation of crop pests and diseases under plant species diversification[END_REF][START_REF] Alaphilippe | The use of agronomic practices in integrated pest management programmes in horticulture[END_REF]. Considering laboratory results, CPs represent a promising lever for agroecology. Yet their efficient use in the field remains uncertain. More widespread adoption on farms requires a better understanding of how the functional processes expected from CP are actually achieved. However, performing suitable agricultural experiments to assess the processes involved with CPs is a challenge [START_REF] Alaphilippe | The use of agronomic practices in integrated pest management programmes in horticulture[END_REF]. In addition, specific operational knowledge on how to implement these CPs is essential to guarantee effective regulatory functions as well as the compatibility with other practices and practical applicability on farms. The challenge of designing more complex horticultural systems that include CPs makes it more necessary to jointly produce and apply both scientific-based and experience-based knowledge [START_REF] Girard | The interplays between singularity and genericity of agroecological knowledge in a network of livestock farmers[END_REF].

Agronomic experiments can meet various complementary objectives, ranging from isolating biological mechanisms in the laboratory, to identifying the conditions for activating mechanisms in the field, and to designing practicable cropping systems compatible with the expected functional processes. Agronomic experiments at field scale ("field experiments", in short) can be performed in a tightly controlled and monitored environment or in agronomic and socio-economic conditions similar to those found on local commercial farms. We focused on the latter type of field experiments carried out in 'almost' commercial conditions, where the common objective was to assess the effect of CPs on pest and disease control for agroecological crop protection in horticultural systems. We assume that these forms of agronomic experiments provide relevant support for generating agroecological actionable knowledge [START_REF] Cardona | Les stations expérimentales comme lieux de production des savoirs agronomiques semi-confinés. Enquête dans deux stations INRAE engagées dans l'agro-écologie[END_REF][START_REF] Navarrete | Intertwining deterministic and open-ended perspectives in the experimentation of agroecological production systems: a challenge for agronomy researchers[END_REF]. Actionable knowledge refers to context-specific knowledge that assists stakeholders (farmers, agricultural advisors, etc.) in their decision making and their action, for instance to implement innovative cropping systems [START_REF] Geertsema | Actionable knowledge for ecological intensification of agriculture[END_REF]. It is required to support stakeholders' exploration and adaptation of an agroecological practice tested elsewhere. In this respect, making explicit the agronomic logic, that is, the relationships between the objectives reached, the practices implemented, and the observations made in the field, is a useful resource [START_REF] Verret | Tracking on-farm innovative practices to support crop mixture design: The case of annual mixtures including a legume crop[END_REF][START_REF] Quinio | Cognitive resources to promote exploration in agroecological systems design[END_REF].

The aim of this article was to highlight how field experiments implemented in agronomic and socio-economic conditions similar to commercial conditions, are also providing actionable knowledge to support the adoption of agroecological practices. We illustrated and characterized knowledge related to decisions and actions that experimenters are able to report back. From these experiences, we identified insights and significant points related to the introduction of companion plants for crop health management in horticultural systems.

MATERIALS AND METHODS

Context of the study, sample and level of data collection

We performed our study on field experiments (FEs) dedicated to assess the effect of CP introduction in horticultural systems to directly or indirectly control pests, diseases or weeds. These FEs were headed by agronomists whose main function was experiment leader (EL). They worked for public or private agricultural institutions involved in research and professional support. We surveyed FEs which had already been implemented in the field. FEs were carried out in experimental stations or farms in conditions close to those of commercial farms, according to the EL interviewed (regarding the cash crop management, environmental conditions, inputs and technical means). We thus surveyed 21 on-going or recently completed FEs. The study was performed on a diversity of perennial or annual horticultural productions: 38% in open-field market-gardening, 19% in fruit orchards, 19% in protected marketgardening, 14% in off-soil vegetable greenhouses, and 9% in off-soil ornamental plants. We covered diverse French pedoclimatic areas that accounted for 38%, 29% and 9% of the FEs in the Mediterranean, oceanic and continental metropolitan regions, respectively, and 24% in the tropical climate in the West Indies.

The CPs targeted various objectives, such as agroecological management of aerial pests, soil-borne pathogens, and weeds, in respectively 76%, 19% and 5% of the FEs. Each FE could include one or several tested options that were specified according to the functional objectives assigned to the CPs. Options could also vary according to the bioregulation target (Figure 1).

We developed an approach to identify the type of questions related to the 'agronomic logic' inspired by that of [START_REF] Verret | Tracking on-farm innovative practices to support crop mixture design: The case of annual mixtures including a legume crop[END_REF], and to specify the level of precision to be described by the interviewed EL. Thus, finding the relevant informative level for data collection was essential to ensure that the EL would be able to precisely reflect on what had been done and learned from this practical experience. We assumed that the CPs' characteristics and their expected functions regarding pests or diseases were closely interrelated with the way a CP was practically integrated into the field plot. As a result, we performed data collection per FE on one or several modalities of use (MU) of agroecological CPs. A MU is defined according to: (i) the selected CP (or a mix of CPs); (ii) the annual or perennial status of CPs that constrains design and practices; (iii) technical management including spatial and temporal arrangements around or within the cash crop (field margin, strip, mixed or rotation), adapting typologies of multispecies systems from [START_REF] Brooker | Improving intercropping: a synthesis of research in agronomy, plant physiology and ecology[END_REF] and [START_REF] Gaba | Multiple cropping systems as drivers for providing multiple ecosystem services: from concepts to design[END_REF]; and (iv) the main expected functional process(es) assigned to CPs to deliver the targeted regulation services. We finally selected and investigated 46 MUs from the 21 FEs, covering a diversity of situations at the MU level (Figure 1). . Distribution of the 46 surveyed modalities of use of agroecological companion plants (CP) according to the spatial and temporal arrangement (CP arranged at the edge of the field margin, in strips with the cash crop, in rotation between two cash crops, or mixed with the cash crop) and according to the characteristics of CPs: functions expected in the field (pest repellent, trap, soil-cover to control weeds, soil sanitizing, banker, providing food or habitat resources for natural enemies or mycorrhizal CP) and organisms directly targeted (pests or diseases, including weeds, natural enemies populations or soil microorganisms).

Information collected related to action and data analysis

We collected data from April to July 2021. First, the EL provided all published or internal documents presenting the FE and related project(s). The required data were then identified from documents. We thus prepared the experimenter's interview and were able to preselect from the FE the MUs to be surveyed. Third, a face-to-face meeting at the experimental site with the EL began validating the selection of MUs. A 60-to 90-minute semi-structured interview with the EL was held to complete factual data and explore the remaining points. The answers collected were fully transcribed and then encoded into a set of qualitative and quantitative variables.

The data collected were intended first to identify knowledge that emerged from the stage of agrosystem design and experiment preparation: (i) which CPs were integrated into the cropping system and how, which practices were possibly modified compared to the current system with no CP, and which contrasting options were explored at the design stage; (ii) the functional objective assigned to CPs; and (iii) the types of difficulties that experimenters anticipated before the system was tested. A second group of questions specified knowledge produced during and through evaluation of the experiment: (iv) crop management practices implemented, modifications made during the test stage, and explanations for such adjustments; (v) experimenters' analysis of success and failure of the systems on which experiments were run, regarding targeted functional objectives as well as agrosystem management and the indicators they had used for it; (vi) barriers actually faced; and (vii) solutions identified or tested to improve functional or management dimensions of the agrosystem that integrated CP. We quantified the variables to analyse the 46 MUs regarding anticipated difficulties, barriers actually faced to implement CP, and proposals by ELs for improving solutions.

RESULTS and DISCUSSION

The data collected on FE and related MUs revealed agronomic logics, meaning links between expected functional objectives and the practices implemented, under precisely described and analysed conditions. We illustrate these results by highlighting first the experience-based knowledge from 3 contrasting FEs and secondly, by a transversal synthesis of the 46 MUs.

Actionable knowledge emerged from the modalities of use of CPs: examples from 3 FEs 1. Context and objectives of the design of the system with CP(s)

Three FEs are presented to highlight the type of actionable knowledge revealed by the surveys. The authors of this article are familiar with selected FE2, FE3 and FE14 implemented at INRAE experimental stations, in Pyrénées-Orientales, Drôme and Guadeloupe, respectively.

FE2 studied the potential of Calendula officinalis L. as a banker plant for a mirid predatory bug Macrolophus pygmaeus R. (Homoptera: Miridae) on protected vegetable system [START_REF] Perrin | Implementing and assessing the efficacy of Calendula officinalis L. (marigold) as banker plant for Macrolophus pygmaeus (Rambur) in protected vegetable production[END_REF]. The objective assigned to this CP was to provide alternative food, refuge and shelter to generalist natural enemies able to control the most prevalent pests on tomato. C. officinalis was expected to provide trophic resources and habitat to keep M. pygmaeus populations high in the winter period so that they could colonize cash crops early the next year from this local rearing. Within FE2, four MUs were studied. MUs differed in their spatial arrangement within the tomato crop -whether they were arranged in strips or mixed intercropping -, and in the way C. officinalis was introduced (sown, planted or in pots). MUs were designed to consider farms' diversity of production constraints, such as whether they practised soil solarization or not, and were selected depending on whether the CP remained in the field at the end of the tomato cycle or was removed. Prior to the trials, the EL feared the MUs would come with excessive costs, difficulties in work organization, and additional sanitary risks, and would display uncertainty as regards pest management efficiency.

FE3 studied the potential of rosemary Rosmarinus officinalis L. as a perennial CP in an apple orchard. The objectives assigned to this CP were to repel and disrupt the apple rosy aphid Dysaphis plantaginea Passerini (Homoptera: Aphididae) [START_REF] Dardouri | Behavioural response of green peach aphid Myzus persicae (Sulzer) to volatiles from different rosemary (Rosmarinus officinalis L.) clones[END_REF] as well as providing food resources for natural enemies of pests. R. officinalis was expected to release volatile organic compounds (VOCs) likely to directly affect the development of and the tree colonization by D. plantaginea in spring and autumn, respectively. Within FE3, two MUs were studied. MUs differed in their spatial arrangement within the orchard plots (strip or mixed arrangement). Interviewees pointed out that R. officinalis was a relevant choice of CP in the orchard as a perennial plant and emphasized its hardiness and low water requirements due to its Mediterranean origin. MUs were designed to explore the potential to maximize VOC emissions while being a practicable system for machinery. The original orchard was modified for the MU with mixed cropping since a few apple trees were removed to densify the CPs in the orchard. Prior to the trials, EL feared MUs would induce difficulties in pest management efficiency, labour organization and intensity, and that they would be an obstacle to regular orchard practices. They also feared a lack of knowledge about CP management and regulatory bottlenecks such as the prohibition of organic pesticide applications when flowers are present in the orchard.

FE14 studied the potential of Crotalaria juncea as a sanitizing and mycorrhizal plant in diversified open-field market-gardening systems under tropical conditions [START_REF] Deberdt | Crotalaria spectabilis and Raphanus sativus as Previous Crops Show Promise for the Control of Bacterial Wilt of Tomato Without Reducing Bacterial Populations[END_REF][START_REF] Chave | Rhizophagus irregularis MUCL 41833 transitorily reduces tomato bacterial wilt incidence caused by Ralstonia solanacearum under in vitro conditions[END_REF]. The objectives assigned to CPs were to reduce the incidence of two soilborne pests, bacterial wilt and root-knot nematodes, and to increase tomato mycorrhization. C. juncea used as a cover crop was also expected to contribute to weed control and to nitrogen provision. Interviewees highlighted the multiple services provided by crotalaria, based on scientific knowledge and interest in this rustic, indigenous and short life-cycle plant whose lowcost seeds are easily available. Within FE14, two MUs were studied. MUs differed in spatial and temporal arrangements: in rotation as a previous crop whose aerial parts are then left as a mulch and as a field margin. Experimenters paid attention to the design step to select other agroecological practices compatible with the mycorrhization process. They feared that MUs would involve excessive costs, difficulties in work organization and a lack of suitable equipment.

Experience-based knowledge produced from the experiments and its evaluation

The experimenters cited a set of indicators to assess the functional processes involved and additional indicators to assess the system being tested in the field. In Table 1, we illustrate how the experimenters analysed their experience to identify not only points of satisfaction about each MU but also barriers faced in many dimensions. In addition, they pointed out improvements.

The 3 FEs detailed illustrate that the experimenter who designs and tests agroecological systems under conditions close to those of production produces experience-based knowledge. These 3 FEs emphasized that the functional objectives and the modalities of use of CPs must be considered together in order to properly assess CPs. To succeed in CP germination and growth: perform a specific soil preparation before sowing CP, increase water supply and CP sowing density. FE2/3 Strip CP in pots FO: ++ M: + CP growth, CP management, Labour intensity, Complexify the system management.

To limit its dryness and provide sufficient substrate for the CP along the cash crop cycle: use bigger pots. Pay attention to the climatic conditions that can be lethal for mirids at the time of transfer from tomato plants to CP

Transversal analysis from the 46 modalities of use for companion plants

Experimenters anticipated in average more potential difficulties than were actually faced. An average of 4.8 potential barriers per MU was anticipated, whereas 3.7 types of barriers faced per MU were reported. The main barriers feared at the system design stage related to sanitary problems: "sanitary risks" from the CP and "uncertain control of pests and diseases" using CP were associated respectively with 70% and 52% of the MUs (Figure 2). Moreover, "Work organization" complexity, "CP growth" difficulties and lack of "Technical knowledge or references" were feared for 50%, 48% and 46% of the MUs (Figure 2). The main obstacles actually faced during the implementation in field conditions related to "CP growth" (in 54% of the MUs). Work was also affected since "work organization" and "system management or monitoring" were considered as highly complex in respectively 50% and 33% of the MUs (Figure 2). Pest and disease control was not considered efficient in 35% of the MUs (Figure 2). Moreover, during the face-to-face discussion, ELs were able to cite and specify solutions to test or already tested (as illustrated in Table 1). These solutions were expected to improve the achievement of functional objectives and the management of the agrosystem in 54% and 61% of the MUs, respectively. Based on a wide range of situations we show that field experiments are supporting the production of actionable knowledge in a strategic agroecological practice. Experimenters were able to anticipate a wide range of problems, relating not only to technical aspects. The main barriers encountered during the experiment focused less on technical and sanitary problems than anticipated. This indicates that experimenters are already facing part of the real-life conditions in such FEs, including barriers at the scale of innovative agroecological systems as a whole [START_REF] Cardona | Les stations expérimentales comme lieux de production des savoirs agronomiques semi-confinés. Enquête dans deux stations INRAE engagées dans l'agro-écologie[END_REF]. Like innovative farmers, the leaders of this type of FE are designers who are continuously adjusting to find a compromise between realization of the function and operational feasibility, taking into account broader issues than those of the trial or of the field scale [START_REF] Navarrete | Intertwining deterministic and open-ended perspectives in the experimentation of agroecological production systems: a challenge for agronomy researchers[END_REF].

The first obstacle identified is the growth of the companion plant, a prerequisite for the function to be fulfilled. Particular attention must be paid to this stage by seeking a trade-off between CP management and its final added value. Labour is also one of the main obstacles identified. Additionally, agrosystems that include CPs become more complex and need to be managed differently [START_REF] Alaphilippe | The use of agronomic practices in integrated pest management programmes in horticulture[END_REF]. Multi-species systems must be redesigned, at least to some extent, to ensure that the functional as well as the productive and labour processes are compatible [START_REF] Schut | Systems approaches to innovation in crop protection. A systematic literature review[END_REF]. A larger sample of horticultural field experiments should be surveyed to detect whether specific productions or CPs faced particular barriers. Taking these barriers into account in research projects on CPs is essential if CPs are to be adopted on the farm. Moreover, building relevant cognitive resources to give visibility to this knowledge is as strategic as presenting scientific results [START_REF] Girard | The interplays between singularity and genericity of agroecological knowledge in a network of livestock farmers[END_REF][START_REF] Quinio | Cognitive resources to promote exploration in agroecological systems design[END_REF]. Such resources could present the modalities of use as options to explore, where experimenters have already gained useful experience.

CONCLUSION

In this study, we documented how field experiments carried out in conditions similar to those found on local commercial farms can provide useful actionable knowledge for managing CPs for crop health in horticultural systems. This illustrated the value of these experiments for agroecological transition, as they both support scientific knowledge on CPs, and reveal barriers and solutions in many respects. Research projects on CPs should take into consideration the barriers thus revealed in the practice, as this actionable knowledge is necessary for the adoption of CPs by stakeholders. Thus, dissemination of field experiment results should include and put forward this knowledge. Scientific-and action-based knowledge is produced jointly in agroecological field experiments and should both be applied.

  Figure1. Distribution of the 46 surveyed modalities of use of agroecological companion plants (CP) according to the spatial and temporal arrangement (CP arranged at the edge of the field margin, in strips with the cash crop, in rotation between two cash crops, or mixed with the cash crop) and according to the characteristics of CPs: functions expected in the field (pest repellent, trap, soil-cover to control weeds, soil sanitizing, banker, providing food or habitat resources for natural enemies or mycorrhizal CP) and organisms directly targeted (pests or diseases, including weeds, natural enemies populations or soil microorganisms).Information collected related to action and data analysisWe collected data from April to July 2021. First, the EL provided all published or internal documents presenting the FE and related project(s). The required data were then identified from documents. We thus prepared the experimenter's interview and were able to preselect from the FE the MUs to be surveyed. Third, a face-to-face meeting at the experimental site with the EL began validating the selection of MUs. A 60-to 90-minute semi-structured interview with the EL was held to complete factual data and explore the remaining points. The answers collected were fully transcribed and then encoded into a set of qualitative and quantitative variables.The data collected were intended first to identify knowledge that emerged from the stage of agrosystem design and experiment preparation: (i) which CPs were integrated into the cropping system and how, which practices were possibly modified compared to the current system with no CP, and which contrasting options were explored at the design stage; (ii) the functional objective assigned to CPs; and (iii) the types of difficulties that experimenters anticipated before the system was tested. A second group of questions specified knowledge produced during and through evaluation of the experiment: (iv) crop management practices implemented, modifications made during the test stage, and explanations for such adjustments; (v) experimenters' analysis of success and failure of the systems on which experiments were run, regarding targeted functional objectives as well as agrosystem management and the indicators they had used for it; (vi) barriers actually faced; and (vii) solutions identified or tested to improve functional or management dimensions of the agrosystem that integrated CP. We quantified the variables to analyse the 46 MUs regarding anticipated difficulties, barriers actually faced to implement CP, and proposals by ELs for improving solutions.

FE2

  between the trees within the row and a perennial CP in alleyways OR use a CP variety with a better soil-cover capacity OR re-design future orchards to optimise the arrangement of trees, CPs and drips. between tree rows: test tools already mobilized by the producers of aromatic plants. Get inspiration from agroforestry designs with wider CP strips between tree rows. s assessment for FO: Functional objectives; M: practicability and crop management; '0' unsatisfied, '-' hardly satisfied, '+' satisfied, '++' highly satisfied. FE: Field experiment; MU: Modality of use; CP: Companion plant.

Figure 2 .

 2 Figure 2. Proportion of barriers anticipated before and actually faced during the experiment with each of the 46 modalities of use of companion plants (CP). MU: Modality of use of agroecological companion plants.

  Anticipated barriers before implementing the MU in the field Faced barriers while implementing the MU in the field

Table 1 .

 1 Success or failures from experimenters' analysis, barriers faced, solutions identified from 3 field experiments.

	FE/MU	Assessment a Barriers	Examples of solutions or problems to solve
	FE2/1 Strip	FO: +	CP growth, CP management,	To prevent from weeds development where the plastic
	CP planted	M: ++	Work organization, Lack of	mulch was removed to plant CPs with no excessive
			technical knowledge.	costs or labour: adapt irrigation in the planted area,
				increase CP density. CP could be planted earlier.
	FE2/2 Strip	FO: 0	CP growth (seeds germination
	CP sown	M: 0	failure)

ACKNOWLEDGEMENTS

The authors want to thank INRAE SumCROP programme (MultiServ project) and 'GIS Fruits and PIClég' for funding this study, all the experimenters interviewed for their interest in this approach and Liz Carey Libbrecht for language editing the English version of this paper.

Literature cited


Mark Arnold
email: markgarnold@yahoo.com

Caroline Collange

David Defour
email: david.defour@univ-perp.fr

IMPLEMENTING LNS USING FILTERING UNITS OF GPUS

Keywords: Graphics processing units, logarithmic number system, texture filtering, polynomial approximation

Current GPUs offer specialized graphics hardware in addition to generic floating-point processing units. We propose a method which reuses specialized texture filtering units to perform piecewise polynomial evaluations, which helps accelerate LNS computations and can be used in combination with hardware-based transcendental functions.

INTRODUCTION

While the primary design goal of GPUs is efficient execution of graphics rendering, the massive parallelism available in these chips has opened up lately the possibility using them for more general signal-processing applications.

Current GPUs are mostly vector computers which focus on single-precision arithmetic, but they also offer graphicsrelated specialized units in hardware, such as texture filtering units and transcendental evaluation units. Development environments such as NVIDIA CUDA make it possible to program GPUs for various non-graphics applications. However, few such applications make use of texture filtering. We propose a novel approach to take advantage of these otherwiseidle texture filtering units to evaluate functions, which can be used to implement the Logarithmic Number System (LNS) on GPUs. In this paper we will consider the CUDA framework and the GT200 processor from NVIDIA, although our approach is generalizable to other GPUs.

We first give an overview of the GT200 GPU and LNS in the rest of this section, then present the proposed scheme in section 2, and discuss performance and accuracy in section 3.

Computational units in modern GPU

Figure 1 describes the hardware organization of computational units in the GT200 processor. The first level of the computational hierarchy corresponds to the Thread Processor Cluster (TPC). Up to 10 TPC are embedded in this GPU.

Each TPC is made of one load/store unit (not represented on the figure), 1 texture unit and 3 multiprocessors.

Multiprocessor units

Each multiprocessor embeds three kinds of vector units. Eight SP units perform single-precision multiply-and-adds. Two special function units (SFUs) compute reciprocals, reciprocal square roots, base-2 logarithms and exponentials, sines and cosines. Finally, one double-precision unit computes fused multiply-and-adds. All units are optimized for throughput and are accessed through 32-way SIMD instructions.

Texture Units

The texture unit or filtering unit is dedicated to accelerate memory accesses specific to graphics operations with uni-, two-or three-dimensional spatial locality. This unit is located in a different clock domain than the rest of the processor and is composed of 4 Texture Address units (TA) and 8 Texture Filtering units (TF). Spatial locality is exploited thanks to a dedicated 24 KB read-only texture cache. The 4 TAs and 8 TFs are used to access texture elements called texels and optionally apply filtering (linear, bilinear, and anisotropic). For example, a bilinear filter can be applied to eight 8-bit texels per clock or four FP16 or two FP32 per clock.

Bi-Linear filtering

In graphics languages as well as in CUDA, a texture object has several attributes. A texture can be declared as a uni-, two-or three-dimensional array that can be accessed with one, two or three texture coordinates. Fetched data can be 8-bit, 16-bit integer, 32 bit floating point numbers or a vector made of 1, 2 or 4 elements. The programmer can also choose between two addressing mode to access the texture by specifying whether texture coordinates are normalized (between 0 and 1) or not. Normalized textures are useful when the application requires texture addresses to be independent on the texture dimensions.

When a texture is configured to return floating-point data, the programmer can define the filtering mode to apply: nearest-point sampling or linear filtering. Linear filtering is a low-precision interpolation between neighboring texture data. When interpolation in enabled, texels surrounding a texture fetch location are read and used to interpolate values based on where the texture coordinates fall between the texels. Linear interpolation is performed for 1D, bilinear interpolation for 2D and trilinear for 3D texture. Let the two-dimensional texture T represent a N × M array of texels fetched using normalized floating-point texture coordinates x and y such that x ∈ [0, N ] and y ∈ [0, M ]. The filtering unit returns the value V such that:

V = (1 -α) • (1 -β) • T [i, j] +(1 -α) • β • T [i + 1, j] +α • (1 -β) • T [i, j + 1] +α • β • T [i + 1, j + 1] (1)
where i = f loor(x -0.5) and α = f rac(x -0.5) j = f loor(y -0.5) and β = f rac(y -0.5)

It should be noted that α and β are manipulated in a fixedpoint format with an 8-bit fractional part on NVIDIA GPUs.

The hardwired linear texture filter is such that internal computations are performed with application-tailored precision, which limits the accuracy to the least significant bit of the texture data format.

Logarithmic Number System

The Logarithmic Number System represents a number, X, by its base-b logarithm, x = log b |X|, rather than the exponent and mantissa used in FP systems. Normally x is manipulated in fixed-point format giving precision and range like that of a similar FP system. In this paper, x itself may be stored in FP format, making the range of the LNS so astronomical that the problems of under/overflow disappear. For algorithms involving only positive real numbers, x by itself is sufficient to represent X; algorithms needing X < 0 require a sign bit like that in FP [START_REF] Swartzlander | Sign/logarithm arithmetic for FFT implementation[END_REF]. Zero is treated specially, and often in signal processing may be ignored. Products, quotients, roots and powers are trivial in LNS; sums and differences involve the functions

s b (x) = log b (1 + b x ) and d b (x) = log b |1 -b x |. For example, since Z + Y = Y (1 + Z/Y ), LNS computes the sum as log(Z + Y ) = y + s b (z -y).
This approach has been generalized to complex values [START_REF] Arnold | A dual-purpose real/complex logarithmic number system ALU[END_REF] using logcos(θ), logsin(θ) and tan -1 (b x ) in addition to s b and d b . Complex LNS (CLNS) has been shown to be an efficient representation for the Fast Fourier Transform (FFT) [START_REF] Arnold | Fast fourier transforms using the complex logarithm number system[END_REF].

Before the widespread adoption of DSP chips, LNS offered attractive VLSI implementations for early signal processing hardware with limited-precision lookup of s b and d b from ROMs. A bibliography [4] lists around twenty papers from that era appearing in ICASSP and the Transactions on signal processing using LNS. Stouraitis [START_REF] Stouraitis | Logarithmic Number System Theory, Analysis, and Design[END_REF] gives a good summary of the state of the art in LNS for signal processing in the 1980s. Later, LNS found other niche applications, such as the massively-parallel N-body computation that won the Gordon Bell prize in 1999 [START_REF] Makino | Scientific Simulations with Special-Purpose Computers: The GRAPE Systems[END_REF]. LNS continues to be an attractive alternative number system for FPGAs [START_REF] Collange | Floating point or LNS: Choosing the right arithmetic on an application basis[END_REF] because VHDL libraries [START_REF] Detrey | A VHDL library of LNS operators[END_REF][START_REF] Vouzis | LNS subtraction using novel cotransformation and/or interpolation[END_REF] are now available that implement polynomial approximations to s b and d b , and LNS is attractive for complex signal-processing algorithms [START_REF] Albu | Pipelined implementations of the a priori error-feedback LSL algorithm using logarithmic number system[END_REF]. LNS is common in computing Hidden Markov Models (HMMs) [START_REF] Young | The HTK Book (for HTK Version 3.4[END_REF], even on conventional CPUs, because a summation of exponentials is easier to compute in LNS than FP.

With the massive FP hardware available on GPUs, implementing LNS may seem unnecessary, but like all chip sets, NVIDIA chips have fixed resources. In their original intent for accelerating graphics, the designers at NVIDIA felt it worthwhile to commit silicon to texture interpolation, hardware that typically goes unused in a GPGPU signalprocessing application. This paper considers using the texture hardware to approximate s b and related functions that implement either real or complex LNS for portions of applications, like HMM or N-body, where LNS has proven to be useful even in the presence of FP hardware, and thereby improve throughput using LNS format. The architectural complexity of GPUs make it difficult to predict when such LNS will benefit a particular application; disclosing our proposed technique gives GPGPU programmers the option to experiment with LNS.

PROPOSED METHOD

We want to evaluate f (x), for instance s b or d b . We assume x ∈ [0, 1] is in IEEE-754 single precision format. There exist several techniques to evaluate a function on a given interval. Due to the presence of filtering on GPUs, spline and B-Spline are very common. Sigg et al. described in [START_REF] Pharr | GPUGems 2 : Programming Techniques for High-Performance Graphics and General-Purpose Computation[END_REF] how to execute fast third order texture filtering while taking advantage of bilinear filtering. Another technique consists in approximating a function by one or several polynomials evaluated with SP units. The main advantage of this technique is that it is very accurate as every operation is done using floating-point arithmetic. However, it consumes computational resources to perform the higher-order interpolation.

Order-2 polynomial approximation

The proposed method consists in performing polynomial evaluation inside the texture units, whenever low input precision with high output accuracy is necessary. We propose to approximate f (x) with a piecewise degree 2 polynomial approximation on [0, 1] with k polynomials. Let P w be the polynomial used to approximate f (x) on [w.2 -k , (w + 1).2 -k ]:

P w (x) = a 0 + a 1 • x + a 2 • x 2 (2)
The novelty of this methods lies in setting the correct parameters in the texture T and the address used to fetch data such that equation 1 representing a bilinear filtering operation, is similar to the evaluation of the polynomials P w .

If we perform a texture fetch at coordinates (x + 0.5, (x -f loor(x)) + 0.5) then ( 1) is equivalent to:

V = (1 -γ) 2 • T [i, 0] + (γ -γ 2 ) • T [i + 1, 0] +(γ -γ 2 ) • T [i, 1] + γ 2 • T [i + 1, 1] (3)
with γ = f rac(x), i = f loor(x) Now, let us focus on texture coefficient such that the texture T store

T [i, 0] = a 0 , T [i + 1, 0] + T [i, 1] = 2 • a 0 + a 1 and T [i + 1, 1] = a 0 + a 1 + a 2 . We have: V = (1 -γ) 2 .a 0 + (γ -γ 2 ).(2.a 0 + a 1 ) +γ 2 .a 0 + a 1 + a 2 = a 0 + a 1 .γ + a 2 .γ 2 (4) Therefore T [i, 0], T [i + 1, 0], T [i, 1] and T [i + 1, 1]
are used to store the coefficients of one polynomial. In order to store the coefficients of k polynomial approximations of f (x) on k interval, the 4.k coefficients will be stored on a 2.k × 2 texture. In that case, fetch will be performed at the coordinates illustrated by this CUDA code:

Discussion

Even though texture units have their own texture address unit, polynomial coefficients are stored in a way that requests some adjustment performed with the help of SP units. However with this technique, the execution cost of the evaluation of one function at x is almost equivalent to the evaluation cost of 4 completely different functions at the point x thanks to texture filtering units that works either on one floating point number or a vector of up to 4 floating-point numbers. In addition, one may observe that the address shifting by 0.5 can be avoided by including this bias into the polynomial approximation rather than including it in address calculations.

RESULTS AND VALIDATION

We implemented the degree-2 method described above, using the Sollya tool to generate optimized minimax polynomials with single-precision coefficients [START_REF] Brisebarre | Efficient polynomial L-∞ approximations[END_REF]. For comparison purposes, we also implemented an order-1 method that uses only linear texture filtering, a computation that takes advantage of the hardware-based exp and log in the SFU, and a direct polynomial approximation evaluated using the Horner scheme, also generated using Sollya.

Figure 3 describes the worst-case error on the s b function for the methods implemented. We consider an input precision f such that k = 2 f -8 , so the 8-bit quantized fractions of coordinates γ are exact.

We remark that storing 64 segments is necessary to provide 23 bits of output accuracy. Such storage requirements are low enough that coefficients can all fit inside the texture cache.

We evaluated the performance of each method, and combinations of them, to evaluate either the s b function or both s b and d b at the same point on a GeForce GTX 280 GPU. Results are normalized and expressed in shader clock cycles per warp, rounded to the nearest half-cycle and presented in Table 1.

When using hardware transcendentals to compute both s b and d b as log 2 (1 ± 2 x ), compiler optimizations enable the common 2 x calculation to be computed only once. Though order-2 texture interpolation provides no benefit when used by itself or to evaluate single functions, it increases the computation throughput of s b and d b when used in combination with the SFU units.

CONCLUSION

We presented a scheme that leverages the specialized texturefiltering units on GPUs to perform function evaluation, enabling a higher throughput for the functions that form the basic blocks of LNS. Other applications that require function approximation with a high output accuracy for a low input precision could also be accelerated.

The most prominent obstacle for accurate evaluation being the quantization of the inputs of the filtering unit, we make a case for the exposure of the hardware attribute interpolation unit in the CUDA environment. Direct access to the attribute interpolator, which does not suffer from the accuracy limitations of the filtering unit, would allow accurate evaluation of arbitrary functions, as already implemented in hardware for a limited set of functions [START_REF] Oberman | A highperformance area-efficient multifunction interpolator[END_REF].

Fig. 1 .

 1 Fig. 1. Block diagram of one thread processor cluster of a NVIDIA GT200.

Fig. 2 .

 2 Fig. 2. Bilinear filtering of a two dimensional texture.

d e v i c e f l o a t t e x t e v a l d e g 2 (

 2 f l o a t x , i n t k ) { x = x * k ; f l o a t xe = f l o o r ( x ) ; f l o a t x f = xxe + .5 f ; r e t u r n tex2D ( t e x R e f , ( 2 . 0 f * xe ) + xf , x f ) ; }

Fig. 3 .

 3 Fig. 3. Comparison of the worst-case error of the proposed methods on the s b function, with k = 2 f -8 .

Thread Processor Cluster (TPC)

	Multiprocessor	Multiprocessor	Multiprocessor
	Shared	Shared	Shared
	Memory	Memory	Memory
	SP SP SP SP SFU	Register File	SP SP SP SP SFU	SP SP SP SP SFU	Register File	SP SP SP SP SFU	SP SP SP SP SFU	Register File	SP SP SP SP SFU
				Texture Units

Table 1 .

 1 Performance comparison of the proposed methods and other implementations on a GeForce GTX 280, in clock cycles.

	Method	s b	s b and d b
	Order-1	42.5	42.5
	Order-2	49.0	49.0
	Order-3	49.5	49.5
	Hardware transcendentals	32.0	33.0
	Horner degree 5	35.0	44.0
	Order-2 + HW transcendentals 46.0	52.5
	2× HW transcendentals	40.0	56.0


V Jakšić

Y Pautrat

C.-A Pillet

A non-commutative Lévy-Cramér continuity theorem

Introduction

In classical probability, the Lévy-Cramér continuity theorem is a standard tool for proving convergence in distribution of a family of random variables. To be more precise let T denote either N or R and set T ≡ T ∪ {∞}. Suppose that, for each t ∈ T, A t = (A (1) t , . . . , A (n) t ) is a R n -valued random variable on the probability space (Ω t , F t , P t ) and denote by E t the expectation with respect to P t . Finally let x • y be the standard inner product of two vectors x, y ∈ R n . The Lévy-Cramér continuity theorem says that if

lim t→∞ E t (e iα•At ) = E ∞ (e iα•A∞ ), (1)
for all α ∈ R n then A t converges to A ∞ in distribution i.e., for every bounded continuous function f :

R n → R lim t→∞ E t (f (A t )) = E ∞ (f (A ∞ )). (2)
Moreover ( 2) can be shown to hold for every bounded Borel function f such that the set D(f ) of points at which f is discontinuous satisfies

P ∞ ([A ∞ ∈ D(f )]) = 0,
(see e.g., Theorem 29.2 in [Bi]).

We are interested in non-commutative analogues of these results. To formulate such extensions let us briefly introduce some basic notions of non-commutative probability theory. We refer to [Mey] or [Maa] for a more detailed introduction and to [START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 1. Second edition[END_REF] for the theory of von Neumann algebras.

We start with an algebraic reformulation of classical (i.e., commutative) probability theory: a bounded, real-valued random variable A on the probability space (Ω, F, P) can be seen as a real element of the set M ≡ L ∞ (Ω, F, P) of equivalence classes of essentially bounded F-measurable functions on Ω. An event E ∈ F (or rather, an equivalence class under the equivalence A ∼ B ⇔ P(A∆B) = 0) can be identified with the random variable 1l E ∈ M, which satisfies 1l 2 E = 1l E = 1l E ; conversely, any element A ∈ M satisfying A 2 = Ā = A is the equivalence class of the indicator function of some set E ∈ F. Denoting by E the expectation associated with P, the law of a random variable A is defined as the unique probability measure µ on R such that E(f (A)) = f (x) dµ(x) for all bounded measurable fonctions f : R → R. Now let us remark that L ∞ (Ω, F, P) is a commutative von Neumann algebra. Its elements can be interpreted as bounded multiplication operators on the Hilbert space L 2 (Ω, F, P). In non-commutative probability theory M becomes a general von Neumann algebra. For simplicity we only consider concrete von Neumann algebras, i.e., weakly (or strongly) closed * -subalgebra of B(H), the algebra of bounded linear operators on some Hilbert space H. In this extended framework, a random variable is an element A ∈ M satisfying A = A * , i.e., a selfadjoint operator of H. An event is an element A ∈ M satisfying A 2 = A * = A, i.e., the orthogonal projection on a closed subspace of H. The role of the expectation is played by a normal state ω on M, i.e., a positive linear functional on M (ω(B * B) ≥ 0 for all B ∈ M) which is continuous under monotone convergence and normalized by the condition ω(I) = 1. The law of A in the state ω is again defined as the only measure

ω A on R such that ω(f (A)) = f (x) dω A (x).
The existence of such a measure follows from the von Neumann spectral theorem (see e.g., Theorem VIII.6 in [RS]): there exists a projection valued spectral measure ξ A on R, with support on the spectrum Sp A of A, such that (u, Au)

= Sp A x d(u, ξ A (x)u) for all u ∈ H. For every bounded Borel function f and u ∈ H one has (u, f (A)u) = Sp A f (x) d(u, ξ A (x)u). In particular, ω • ξ A is a probability measure and ω(f (A)) = Sp A f (x) d(ω • ξ A )(x). (3)
The measure ω A = ω • ξ A has support in the spectrum of A, seen as an operator on H (simple examples of noncommutative probability spaces are discussed in subsection 4.1). If ω is faithful, then supp ω A = Sp A; otherwise this may not be the case. Note that the framework thus defined extends the classical one: as already remarked, the space L ∞ (Ω, F, P), acting by multiplication on the Hilbert space L 2 (Ω, F, P), is a von Neumann subalgebra of B(H).

As long as one considers the law of a single random variable, non-commutative probability reduces to classical probability; one can even discuss the convergence in distribution of a sequence of non-commutative random variables in a given state. The specificity of non-commutative probability only appears when one tries to consider two (or more) non-commuting random variables A,B ∈ M. Then, it is in general impossible to define a joint law for A and B: there is no measure µ on R 2 such that

ω(f (A)g(B)) = f (x)g(y) dµ(x, y),
for all bounded continuous functions f and g. In particular there is no measure µ such that ω(e iαA e iβB ) = e iαx e iβy dµ(x, y), for all α, β ∈ R. For this reason, quantities such as 1) . . . e iαnA (n) ), which we call quasi-characteristic functions in accordance with [CH], do not carry a simple probabilistic meaning.

ω(e iα1A (
In particular, an assumption analogous to (1):

lim t→∞ ω t (e iα1A (1) t . . . e iαnA (n) t ) = ω ∞ (e iα1A (1) ∞ . . . e iαnA (n) ∞ ), (4) where A (i)
t are non-commuting elements of some von Neumann algebra M t , has no chance of being interpreted as a convergence of measures because, in general, neither the finite t quantities, nor their limit, are characteristic functions of probability measures. Assumptions such as (4) were often considered in the non-commutative probability literature, but their rigorous implications were rarely studied (the only two exceptions we are aware of are [CH] and its extension [CGH], and [Kup]: see subsection 4.2). Instead, it was generally considered that such a convergence was a good indication of the relevance of the limiting structure (M ∞ , ω ∞ ) (another commonly used approach with similar motivations uses moments: see in particular [GvW], [AB]).

In the classical case the functional formulation of the convergence in distribution which follows from (1) is (2). This last identity defines E ∞ (f (A ∞ )) for a class of functions f , and in particular, the quantities

E ∞ (1l [a1,b1[ו••×[an,bn[ (A ∞ )),
at least for a j and b j outside some countable set. This implies that the law of A ∞ is completely determined by (1). In non commutative probability, even if no bona fide measure can be associated to the family of quantities

ω ∞ (1l [a1,b1[ (A (1) ∞ )1l [a2,b2[ (A (2) ∞ ) • • • 1l [an,bn[ (A (n) ∞ )),
this family is still relevant e.g., in the theory of repeated measurement of quantum systems (see [Dav]). It is therefore a natural question to ask how much information about quantities like

ω ∞ (f 1 (A (1) ∞ ) • • • f n (A (n) ∞ )),
can be extracted from (4). Our main result shows that (4) completely determines the values of these quantities for all bounded continuous functions. As in the classical case, an extension to discontinuous functions exists, but under assumptions stronger than those one might naively expect.

The paper is organized as follows. Section 2 contains the main results of this paper; subsection 2.1 contains the statement of our main theorem, subsection 2.2 makes more precise some objects which appear in this theorem. Section 3 describes the proof of our theorem. Section 4 comments on our theorem and its relation with previous results, gives examples and applications; more precisely, subsection 4.1 describes a simple example where it can be seen that Theorem 2 breaks down without the strenghtened continuity assumptions, subsections 4.2 discusses earlier existing results of the same type as ours and subsection 4.3 lists cases where our central assumption (A) was proven.

2 Non-commutative Lévy-Cramér continuity theorems

Recall that T denotes either N or R and T ≡ T ∪ {∞}. For any t ∈ T let (i) M t be a von Neumann algebra acting on a Hilbert space H t ;

(ii) ω t be a normal state on M t ;

(iii) A

(1) t , . . . , A

(n) t be (possibly unbounded) selfadjoint operators on H t which are affiliated to M t , i.e., such that e iαA (j) t ∈ M t for all α ∈ R.

C denotes the set of real bounded continuous functions on R, M the set of Borel probability measures on R and B the set of real bounded Borel functions on R. For f ∈ B, D(f ) denotes the set of discontinuity points of f (D(f ) is Borel, see e.g., Theorem 25.7 in [Bi]). Finally, ω (j) t denotes the law of A (j) t in the state ω t , i.e., the unique element of M satisfying ω

(j) t (f ) = ω t (f (A (j) t )), for all f ∈ C.
Our running assumption will be the following:

Assumption (A) For all α ∈ R m , j 1 , . . . , j m ∈ {1, . . . , n} with m ≥ 1, one has lim t→∞ ω t e iα1A (j 1 ) t • • • e iαmA (jm ) t = ω ∞ e iα1A (j 1 ) ∞ • • • e iαmA (jm ) ∞ .

Statement of the results

Our first result is the following non-commutative version of the Lévy-Cramér continuity theorem.

Theorem 1 Under Assumption (A),

lim t→∞ ω t f 1 (A (1) t ) • • • f n (A (n) t ) = ω ∞ f 1 (A (1) ∞ ) • • • f n (A (n) ∞ ) , (5)
holds for any f 1 , . . . , f n ∈ C.
This result is a direct consequence of the following extension to bounded Borel functions.

Theorem 2 Under Assumption (A) there exists a family S = (S j ) j∈{1,...,n} of subsets of M such that

lim t→∞ ω t f 1 (A (1) t ) • • • f n (A (n) t ) = ω ∞ f 1 (A (1) ∞ ) • • • f n (A (n) ∞ ) , (6)
holds if, for all j ∈ {1, . . . , n}, f j ∈ B and σ(D(f j )) = 0 for every σ ∈ S j .

We shall say that a family S = (S j ) j∈{1,...,n} of subsets of M is admissible if (6) holds under the conditions of Theorem 2.

Remarks. 1. In general, the choice of S is not unique and the subsets S j ⊂ M for different j can not be chosen independently of one another. Explicit examples of admissible families are given in Subsection 2.2. 2. We will see that possible choices for S imply a strengthening of the continuity assumption with respect to the classical Lévy-Cramér theorem. This strengthening is necessary and due to the non commutativity of the problem at hand. We illustrate this in subsection 4.1.

3. The proofs of Theorem 2 is given in Section 3.

In the case where ω ∞ is faithful on the algebra M ∞ , Lemma 7 below shows that S j = {ω (j)

∞ } defines an admissible family. Theorem 2 then yields an optimal non-commutative extension of the classical Lévy-Cramér theorem.

Corollary 3 If Assumption (A) holds and ω

∞ is faithful on M ∞ then (6) holds for f 1 , . . . , f n ∈ B satisfying ω (j)
∞ (D(f j )) = 0 for every j ∈ {1, . . . , n}.

Admissible families

In this subsection we introduce possible choices of admissible families. We then discuss the special case where ω ∞ is faithful. Lemmas 6 and 7 in this section have simple proofs, which are given in Subsections 3.3 and 3.4. The combination of Theorems 2 and 5 is proved in Subsection 3.2.

Note that if ω is a normal state on the von Neumann algebra M then, for any unitary U ∈ M, the formula ω U ( • ) ≡ ω(U * ( • )U ) defines a normal state on M. In particular, for t ∈ T, j ∈ {1, . . . , n} and α 1 , . . . , α n ∈ R we can define the following normal states on M t ,

ω - jt (α 1 , . . . , α j-1 ; • ) ≡ ω t e iα1A (1) t • • • e iαj-1A (j-1) t ( • )e -iαj-1A (j-1) t • • • e -iα1A (1) t , ω + jt (α j+1 , . . . , α n ; • ) ≡ ω t e -iαnA (n) t • • • e -iαj+1A (j+1) t ( • )e iαj+1A (j+1) t • • • e iαnA (n) t .
Definition 4 By Equ.

(3), the maps

α → ω - jt α 1 , . . . , α j-1 ; e iαA (j) t , α → ω + jt α j+1 , . . . , α n ; e iαA (j) t ,
are characteristic functions of probability laws that we denote by σ - jt (α 1 , . . . , α j-1 ; • ) and σ + jt (α j+1 , . . . , α n ; • ) respectively.

Note in particular that

σ - 1t = ω (1) t and σ + nt = ω (n)
t . We define

S - j ≡ {σ - j∞ (α 1 , . . . , α j-1 ) | α 1 , . . . , α j-1 ∈ R}, S + j ≡ {σ + j∞ (α j+1 , . . . , α n ) | α j+1 , . . . , α n ∈ R}, (7)
for j ∈ {1, . . . , n}.

We can now define possible choices of admissible families.

Theorem 5 For any J ∈ {0, . . . , n} the family (S j ) j∈{1,...,n} defined by

S j ≡    S - j if j ≤ J, S + j if j > J. ( 8
)
is admissible.

The reason for the multiplicity of choices of admissible families will become clear in Subsection 3.2.

The following continuity properties of the maps t → σ ± jt are immediate consequences of the classical Lévy-Cramér continuity theorem.

Lemma 6 Fix j ∈ {1, . . . , n} and let g ∈ B. If

σ(D(g)) = 0 for every σ ∈ S - j , (9)
then lim t→∞ σ - jt (α 1 , . . . , α j-1 ; g) = σ - j∞ (α 1 , . . . , α j-1 ; g), (10)
for every α 1 , . . . , α j-1 ∈ R. Similarly, if σ(D(g)) = 0 for every σ ∈ S + j , (11)

then lim t→∞ σ + jt (α j+1 , . . . , α n ; g) = σ + j∞ (α j+1 , . . . , α n ; g), (12)
for every α j+1 , . . . ,

α n ∈ R. Last, if ω (j) ∞ (D(g)) = 0, ( 13
)
then lim t→∞ ω (j) t (g) = ω (j) ∞ (g). ( 14
)
Note that obviously ω

(j)
∞ ∈ S - j ∩ S + j , so that ( 13) is a weaker assumption than ( 9) or ( 11). The following lemma shows that they are equivalent in the case where ω ∞ is faithful.

Lemma 7 If ω ∞ is faithful on M ∞ , then for all j ∈ {1, . . . , n}, any σ ∈ S + j ∪ S - j is equivalent to ω (j)
∞ (i.e., σ and ω (j) ∞ are mutually absolutely continuous).

Proofs

We will first prove Theorem 1 for a restricted class of bounded continuous functions. The result will then be extended to bounded Borel functions using an approximation procedure and Lemma 6.

Approximation of bounded Borel functions

Let F ⊂ C denote the set of functions of the form

f (a) = R f (α) e iaα dα,
where f ∈ L 1 (R).
Lemma 8 The conclusion (5) of Theorem 1 holds for any f 1 , . . . , f n ∈ F.

Proof. For any j ∈ {1, . . . , n}, t ∈ T and u, v ∈ H t it follows from the functional calculus that

(u, f j (A (j) t )v) = fj (α)(u, e iαA (j) t v) dα.
The σ-weak continuity of ω t thus allows us to conclude that

ω t (Bf j (A (j) t )C) = fj (α) ω t (Be iαA (j) t C) dα,
for any B, C ∈ M t . Invoking Fubini's theorem, one easily concludes that

ω t f 1 (A (1) t ) • • • f n (A (n) t ) = f1 (α 1 ) • • • fn (α n ) ω t e iα1A (1) t • • • e iαnA (n) t dα 1 • • • dα n ,
for any t ∈ T. The claim then follows from Assumption (A) and Lebesgue's dominated convergence theorem.

Lemma 9 For any f ∈ B such that sup a∈R |f (a)| ≤ R there exists a sequence

(f k ) k∈N in F such that sup k∈N,a∈R |f k (a)| ≤ R,
and

lim k f k (a) = f (a), for all a ∈ R \ D(f ). Proof. For k ∈ N set fk (α) = e -α 2 /2(k+1) +k -k f (a) e -iaα da 2π , and notice that | fk (α)| ≤ e -α 2 /2(k+1) kR/π ∈ L 1 (R).
The Fourier transform of fk can be written as

f k (a) = R 1l [-1,1] a k + b k 3/2 f a + b k dν(b),
where ν is the centered Gaussian measure of variance 1. It immediately follows that

sup a∈R |f k (a)| ≤ R. For a ∈ R \ D(f ), Lebesgue's dominated convergence theorem and the fact that lim k f (a + b/k) = f (a) for all b ∈ R imply lim k f k (a) = f (a).

Proof of Theorems 2 and 5

Let f 1 , . . . , f n ∈ B and set R ≡ max j (1 + sup a∈R |f j (a)|). Fix J ∈ {0, . . . , n} and define S j according to (8).

Denote by (f j,k ) k∈N ⊂ F the approximating sequence for f j given by Lemma 9. Writing

∆ t ≡ ω t f 1 (A (1) t ) • • • f n (A (n) t ) -ω ∞ f 1 (A (1) ∞ ) • • • f n (A (n) ∞ ) = ω t f 1 (A (1) t ) • • • f n (A (n) t ) -f 1,k1 (A (1) t ) • • • f n,kn (A (n) t ) + ω t f 1,k1 (A (1) t ) • • • f n,kn (A (n) t ) -ω ∞ f 1,k1 (A (1) ∞ ) • • • f n,kn (A (n) ∞ ) + ω ∞ f 1,k1 (A (1) ∞ ) • • • f n,kn (A (n) ∞ ) -f 1 (A (1) ∞ ) • • • f n (A (n) ∞ ) ,
and applying Lemma 8 we get

lim sup t→∞ |∆ t | ≤ lim sup t→∞ ω t f 1 (A (1) t ) • • • f n (A (n) t ) -f 1,k1 (A (1) t ) • • • f n,kn (A (n) t ) + ω ∞ f 1 (A (1) ∞ ) • • • f n (A (n) ∞ ) -f 1,k1 (A (1) ∞ ) • • • f n,kn (A (n) ∞ ) ,
for any k 1 , . . . , k n ∈ N. To study the right hand side of this inequality we fix s ∈ T, set

F j ≡ f j (A (j) s ), G j ≡ f j,kj (A (j)
s ) and proceed with the algebraic identity

F 1 • • • F n -G 1 • • • G n = J j=1 G 1 • • • G j-1 (F j -G j )F j+1 • • • F n + n j=J+1 G 1 • • • G J F J+1 • • • F j-1 (F j -G j )G j+1 • • • G n . ( 15
)
The terms of the first sum on the right hand side of this identity can be estimated as follows. Starting from the Fourier representation (see the proof of Lemma 8)

ω s (G 1 • • • G j-1 (F j -G j )F j+1 • • • F n ) = f1,k1 (α 1 ) • • • fj-1,kj-1 (α j-1 )ω s e iα1A (1) s • • • e iαj-1A (j-1) s (F j -G j )F j+1 . . . F n dα 1 • • • dα j-1 ,
and invoking the Cauchy-Schwarz inequality for ω s we can write, using Definition 4,

ω s e iα1A (1) s • • • e iαj-1A (j-1) s (F j -G j ) F j+1 . . . F n 2 ≤ ω s (F n • • • F j+1 F j+1 • • • F n ) 1/2 ω - js (α 1 , . . . , α j-1 ; (F j -G j ) 2 ) 1/2 ≤ R 2(n-j) σ - js α 1 , . . . , α j-1 ; f j -f j,kj 2 ,
from which we obtain

|ω s (G 1 • • • G j-1 (F j -G j )F j+1 • • • F n )| ≤ R n | f1,k1 (α 1 )| • • • | fj-1,kj-1 (α j-1 )| σ - js α 1 , . . . , α j-1 ; f j -f j,kj 2 1/2 dα 1 • • • dα j-1 . (16)
Furthermore, Lemma 6 and the dominated convergence theorem allow us to conclude that lim sup

t→∞ |ω t (G 1 • • • G j-1 (F j -G j )F j+1 • • • F n )| ≤ R n | f1,k1 (α 1 )| • • • | fj-1,kj-1 (α j-1 )| σ - j∞ α 1 , . . . , α j-1 ; f j -f j,kj 2 1/2 dα 1 • • • dα j-1 . ( 17
)
The terms of the second sum on the right hand side of ( 15) can be handled in a similar way, leading to the estimates

|ω s (G 1 • • • G J F J+1 • • • F j-1 (F j -G j )G j+1 • • • G n )| ≤ R n | fj+1,kj+1 (α j+1 )| • • • | fn,kn (α n )| σ + js α j+1 , . . . , α n ; f j -f j,kj 2 1/2 dα j+1 • • • dα n , (18)
and lim sup

t→∞ |ω t (G 1 • • • G J F J+1 • • • F j-1 (F j -G j )G j+1 • • • G n )| ≤ R n | fj+1,kj+1 (α j+1 )| • • • | fn,kn (α n )| σ + j∞ α j+1 , . . . , α n ; f j -f j,kj 2 1/2 dα j+1 • • • dα n . ( 19
)
Combining estimates ( 16), ( 17), ( 18) and ( 19) with identity (15) leads to

lim sup t→∞ |∆ t | ≤ 2R n D(k 1 , . . . , k n ), (20)
for any k 1 , . . . , k n ∈ N, where

D(k 1 , . . . , k n ) = J j=1 | f1,k1 (α 1 )| • • • | fj-1,kj-1 (α j-1 )| σ - j∞ α 1 , . . . , α j-1 ; f j -f j,kj 2 1/2 dα 1 • • • dα j-1 + n j=J+1 | fj+1,kj+1 (α j+1 )| • • • | fn,kn (α n )| σ + j∞ α j+1 , . . . , α n ; f j -f j,kj 2 1/2 dα j+1 • • • dα n .
By Lemma 9 and our assumptions, lim k |f j -f j,k | = 0 holds σ - j∞ (α 1 , . . . , α j-1 ; • ) almost everywhere for all j ∈ {1, . . . , J} and α 1 , . . . , α j-1 ∈ R. It follows from the dominated convergence theorem that

lim k σ - j∞ α 1 , . . . , α j-1 ; |f j -f j,k | 2 = 0.
In a similar way one shows that

lim k σ + j∞ α j+1 , . . . , α n ; |f j -f j,k | 2 = 0,
for all j ∈ {J + 1, . . . , n} and α j+1 , . . . , α n ∈ R. Applying once again the dominated convergence theorem one concludes that

lim k | f1,k1 (α 1 )| • • • | fj-1,kj-1 (α j-1 )| σ - j∞ α 1 , . . . , α j-1 ; |f j -f j,k | 2 1/2 dα 1 • • • dα j-1 = 0, (21)
for all j ∈ {1, . . . , J} and k 1 , . . . , k j-1 ∈ N, while

lim k | fj+1,kj+1 (α j+1 )| • • • | fn,kn (α n )| σ + j∞ α j+1 , . . . , α n ; |f j -f j,k | 2 1/2 dα j+1 • • • dα n = 0, (22)
for all j ∈ {J + 1, . . . , n} and k J+1 , . . . , k n ∈ N. The result now follows from ( 20) and the fact that

lim kn lim kn-1 • • • lim k J+1 lim k1 lim k2 • • • lim k J D(k 1 , . . . , k n ) = 0,
a direct consequence of ( 21) and ( 22).

Proof of Lemma 6

By Definition 4 we have σ - jt (α 1 , . . . , α j-1 ; g) = ω - jt (α 1 , . . . , α j-1 ; g(A The classical Lévy-Cramér continuity theorem readily implies that

lim t→∞ R g(x) σ - jt (α 1 , . . . , α j-1 ; dx) = R g(x) σ - j∞ (α 1 , . . . , α j-1 ; dx),
for any g ∈ B such that σ - j∞ (α 1 , . . . , α j-1 ; D(g)) = 0. This proves (10). Completely similar arguments prove ( 12) and ( 14).

Proof of Lemma 7

Let j ∈ {1, . . . , n}, σ ∈ S - j ∪ S + j and E ⊂ R a Borel set. Denoting P = 1l E (A

∞ ), there exists a unitary U ∈ M ∞ such that σ(E) = ω ∞ (U * P U ). Since both operators P and U * P U are positive, the faithfulness of ω ∞ implies that

σ(E) = 0 ⇔ U * P U = 0 ⇔ P = 0 ⇔ ω ∞ (P ) = 0 ⇔ ω (j)
∞ (E) = 0, so that σ and ω (j)

∞ are equivalent.

Applications and discussion

A simple application

We first recall standard results from non-commutative probability theory and refer the reader to [Bia] or [Mey] for proofs. We consider an orthonormal basis {Ω, X} of the Hilbert space C 2 , and the basis a × , a + , a -, a • of the algebra M (2, C) of complex, 2 × 2 matrices defined by

a × ≡ 1 0 0 0 , a + ≡ 0 0 1 0 , a -≡ 0 1 0 0 , a • ≡ 0 0 0 1 .
in the basis {Ω, X}.

For any m ∈ N * let TΦ m ≡ (C 2 ) ⊗m , the m-fold tensor product of C 2 . For i ∈ {1, . . . , m} and ε ∈ {×, +, -, •} we denote by a ε i the ampliation of a ε acting on the i-th copy of C 2 in TΦ m . The family (a ε i ) i∈{1,...,m},ε∈{×,+,-,•} is then a basis of the algebra M m ≡ M (2, C) ⊗m (this is the toy Fock space approximation of [A]). We further denote by Ω m the vector Ω ⊗m and by ω m the associated state A → (Ω m , AΩ m ). In this state, the operators

n i,m ≡ a + i + a - i √ m , p i,m ≡ a + i + a - i √ m + a • i + 1 m a × i , z i,m ≡ a • i ,
respectively follow the laws

1 2 δ -m -1/2 + 1 2 δ m -1/2 , m m + 1 δ 0 + 1 m + 1 δ 1+m -1 , δ 0 , with characteristic functions cos α √ m , e iα(1+1/m) + m 1 + m , 1.
Moreover, for b ∈ {n, p, z}, b i,m and b j,m commute if i = j. Therefore each family (b i,m ) i∈{1,...,m} , has a joint law in the state ω m and, in addition, this joint law can be seen to correspond to independent random variables. It follows that the random variables

N n ≡ m i=1 n i,m , P m ≡ m i=1 p i,m , Z m ≡ m i=1 z i,m , have characteristic functions cos α √ m m , e iα(1+1/m) + m 1 + m m , 1,
which, as m → ∞, are easily seen to converge to e -α 2 /2 , e e iα -1 , 1, the characteristic functions of the centered normal law with variance 1, the Poisson law of intensity 1 and the law δ 0 . The classical Lévy-Cramér theorem thus implies the convergence in law of the individual sequences (N m ) m∈N * , (P m ) m∈N * , (Z m ) m∈N * to the corresponding random variables. As a simple application of our quantum Lévy-Cramér theorem, we shall now consider some properties of the joint sequence (N m , P m , Z m ) m∈N * .

We start with a simple observation. Denote by Φ m ⊂ TΦ m the subspace generated by completely symmetric tensor products. An orthonormal basis of Φ m is given by the family (e k ) k∈{0,...,m} where e k is the (normalized) complete symmetrization of X ⊗k ⊗ Ω ⊗(m-k) . In particular Ω m = e 0 . The operators N m , P m , Z m clearly leave Φ m invariant. A simple calculation shows that

N m e k = 1 - k -1 m √ k e k-1 + 1 - k m √ k + 1 e k+1 , P m e k = 1 - k -1 m √ k e k-1 + 1 - k m √ k + 1 e k+1 + k + 1 - k m e k , (23)
Z m e k = k e k ,
where, by convention, e -1 = e m+1 = 0. In studying the random variables N m , P m , Z m in the state ω m we may therefore consider that these operators act on the space Φ m .

To describe limiting random variables N ∞ , P ∞ , Z ∞ we denote by M ∞ the algebra of bounded operators on Φ ≡ ℓ 2 (N) with (ẽ k ) k∈N its canonical orthonormal basis, and by ω ∞ the state A → (ẽ 0 , Aẽ 0 ). The operators a + , a -, a • defined by

a + ẽk ≡ √ k + 1 ẽk+1 , a -ẽk ≡ √ k ẽk-1 , a • ẽk ≡ a + a -ẽk = k ẽk ,
(with the convention ẽ-1 = 0) are such that in the state ω ∞ , for any w ∈ C, the operators

wa + + wa -, wa + + wa -+ a • + |w| 2 I, a • , (24)
follow respectively a centered normal law with variance |w| 2 , a Poisson law of intensity |w| 2 and the law δ 0 .

Setting

N ∞ ≡ a + + a -, P ∞ ≡ a + + a -+ a • + I, Z ∞ ≡ a • , we therefore have the convergences in law N m → N ∞ , P m → P ∞ and Z m → Z ∞ .
Let us show that Assumption (A) holds with (A

m ) ≡ (N m , P m , Z m ), m ∈ N * ∪ {∞}. (1) m , A (2) m , A (3)
We first note that the partial isometry S m : Φ m → Φ induced by the map e k → ẽk , k ∈ {0, . . . , m} satisfies

ω m (e iα1A (j 1 ) m • • • e iα k A (j k ) m ) = ω ∞ (e iα1 Ã(j 1 ) m • • • e iα k Ã(j k ) m ), (25)
where

Ã(j) m ≡ S m A (j)
m S * m . Using relations (23), one easily shows that, for any k ∈ N and j ∈ {1, 2, 3}

lim m→∞ Ã(j) m ẽk = A (j) ∞ ẽk . (26)
Using the fact that the set of finite linear combinations of basis vectors ẽk is a common core for all Ã(j) m and A

∞ , it follows from (26) that the sequence Ã(j) m converges to A

∞ in strong resolvent sense (see e.g., Theorem VIII.25 in [RS]). On concludes that s-lim

m→∞ e iα Ã(j) m = e iαA (j) ∞ ,
for any α ∈ R and j ∈ {1, 2, 3}. Assumption (A) clearly follows from this relation and Equ. (25).

Computations using commutation between Weyl operators (see [Bia]) show that

e iαN∞ e iβP∞ e -iαN∞ = e iβ((1-iα)a + +(1+iα)a -+a • +|1-iα| 2 ) , so that (recall (24)) β → ω ∞ e iαN∞ e iβP∞ e -iαN∞ ,
is the characteristic function of a Poisson law of intensity |1 -iα| 2 . We therefore obtain a non-trivial consequence of Theorem 2:

lim m→∞ ω m f 1 (N m )f 2 (P m )f 3 (N m ) = ω ∞ f 1 (N ∞ )f 2 (P ∞ )f 3 (N ∞ )
for any f 1 , f 2 , f 3 ∈ B with f 2 continuous at every point of N. In particular, for any a < b in R and any

c < d in R \ N, lim m→∞ ω m 1l [a,b] (N m )1l [c,d] (P m )1l [a,b] (N m ) = ω ∞ 1l [a,b] (N ∞ )1l [c,d] (P ∞ )1l [a,b] (N ∞ ) .
In a similar way, one shows that

lim m→∞ ω m f 1 (P m )f 2 (N m )f 3 (P m ) = ω ∞ f 1 (P ∞ )f 2 (N ∞ )f 3 (P ∞ ) for any f 1 , f 2 , f 3 ∈ B.
This example also allows us to illustrate the necessity of our strengthened continuity assumptions (note that the state ω ∞ is not faithful, as for example Z ∞ is a positive operator and yet ω ∞ (Z ∞ ) = 0). For finite m it follows from ( 23) that Z m is a positive matrix with integer eigenvalues, so that 1l {1} (Z m + 1 m ) = 0 and hence

ω m e iαNm 1l {1} Z m + 1 m e -iαNm = 0.
On the other hand,

ω ∞ e iαN∞ 1l {1} (Z ∞ )e -iαN∞ = ω ∞ 1l {1} (e iαN∞ Z ∞ e -iαN∞ ) , (27)
and e iαN∞ Z ∞ e -iαN∞ = -iαa + + iαa -+ a • + |α| 2 (again by the commutation relations of Weyl operators, see [Bia]). Since this operator follows, in the state ω ∞ , a Poisson law of intensity α 2 , the right hand side of Equ. ( 27) is strictly positive provided α = 0. We thus have

ω ∞ e iαN∞ 1l {1} (Z ∞ )e -iαN∞ = lim m→∞ ω m e iαNm 1l {1} Z m + 1 m e -iαNm ,
even though assumption (A) obviously remains true if we replace Z m by Z m + 1 m . This shows that Theorem 2 is false if we only assume each f j to satisfy ω (j) ∞ (D(f j )) = 0, and illustrates why: the projection associated with the eigenvalue 1 of Z ∞ is not in the support of ω ∞ , so that the singularities of 1l {1} have measure zero under the law of Z ∞ . However, when conjugated by e iαN∞ , this projection is sent to the support of ω ∞ and the singularities of 1l {1} have non-zero measure under the law of e iαN∞ Z ∞ e -iαN∞ .

Previous results of Lévy-Cramér type

The paper [CH] was the first to study explicitly a non-commutative central limit theorem, which it proves using a result of the non-commutative Lévy-Cramér type (Theorem 2 in the cited paper). That result, in a slightly simplified framework, is the following: consider a sequence (q n , p n ) n∈N of canonical pairs on H ≡ L 2 (R), i.e. a pair of (unbounded) self-adjoint operators such that there exists a dense subspace D n ⊂ H in the domain of both q n and p n , which is stable by q n and p n , on which the canonical commutation relation (CCR)

q n p n -p n q n = iI, (28)
holds. In analytically simpler terms, this can be rewritten as the Weyl relation e i(xpn+yqn) = e ixpn e iyqn e ixy/2 , (

(see [START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 1. Second edition[END_REF] or [Pet] for more details on canonical pairs).

Assume that every (q n , p n ) is irreducible, i.e. no nontrivial subspace of H is left invariant by all operators e (ixpn+yqn) . A normal reference state ρ on B(H) is fixed; then a state ρ n on B(H) can be associated to every canonical pair (q n , p n ) by ρ n (A) ≡ ρ(U -1 n AU n ), where U n is the unitary operator mapping (q n , p n ) to the Schrödinger representation of the CCR (28). The Stonevon Neumann unicity theorem for irreducible representations of the CCR ensures its existence (see [Mey]).

Cushen and Hudson define the pseudo-characteristic function ϕ n (x, y) ≡ ρ(e i(xpn+yqn) ), this definition differs slightly from ours but it is clear from (29) that these definitions are essentially equivalent. It is then proven that there exists a state ρ ∞ such that

lim n→∞ ρ n (A) = ρ ∞ (A), (30)
for every A ∈ B(H) (a property which Cushen and Hudson call convergence in distribution) if and only if the sequence ϕ n converges pointwise on R 2 to a function which is continuous at zero.

It is the easy part of the theorem to show that, if the sequence (ϕ n ) n∈N has a pointwise limit ϕ ∞ which is continuous at zero, then ϕ ∞ is of the form

ϕ ∞ (x, y) = ρ(e i(xp∞+yq∞) ),
for some canonical pair (q ∞ , p ∞ ). This and the Weyl relation ( 29) imply that pointwise convergence of ϕ n to a function which is continuous at zero is equivalent to our assumption (A) for ω = ρ and A

(1)

n = q n , A (2)
n = p n . Moreover, the Weyl relation ( 29) implies that e ixpn e iyqn e -ixpn = e -ixy e iyqn , e iyqn e ixpn e -iyqn = e +ixy e ixpn , and the law of both p ∞ and q ∞ in the state ω ∞ is Gaussian, so that Theorem 2 implies (6) for all bounded Borel functions. The conclusion of [CH], i.e. the convergence (30), is stronger than ours at first sight, but it is a consequence of the properties of the Weyl correspondence (as described in the proofs of Proposition 6 and Theorem 2 of that paper) that both conclusions are actually equivalent. Our results therefore extend the results of Cushen and Hudson, which rely heavily on the particular properties of canonical pairs.

The other occurrence of a non-commutative Lévy-Cramér type result we are aware of is [Kup]. In this paper, Kuperberg proves implications of pointwise convergence of pseudo-characteristic functions, of the same type as (6): for M a von Neumann algebra equipped with a normal state ρ, he considers for any A ∈ M the elements

A N = 1 √ N N k=1 I ⊗k-1 ⊗ A ⊗ I ⊗N -k ,
(we use different notations from that in [Kup] to stay as close as possible to our own) and shows that, for fixed self adjoint elements A (1) , . . . , A (k) ∈ M, for any self adjoint non commutative polynomial p, p(A

N , . . . , A

N ) converges in distribution to p(X (1) , . . . , X (k) ) in any tracial state ρ, where (X (1) , . . . , X (k) ) is a (classical) centered Gaussian vector with covariance matrix C ij ≡ ρ(A (i) A (j) ) -ρ(A (i) )ρ (A (j) ). One of the steps of that proof is to show that the convergence of ρ e iα1A

(1) • • • e iα k A (k) to E e iα1X (1) • • • e iα k X (k) for every α 1 , . . . , α k implies the convergence of any quantity ρ f (A (1) ) • • • f (A (k) ) to E f (X (1) ) • • • f (X (k) ) for every f 1 , . . . , f k ∈ C.
The same method could easily be extended to include bounded Borel functions with the standard continuity assumptions (here the limiting quantities are purely commutative) but the proof here uses the fact that the GNS norm associated with the reference state ρ is spectral, which is only true if ρ is tracial. Our result therefore improves the scope of application of this part of Kuperberg's results.

Applications

In this subsection, we simply cite results of the type (A) found in the literature. Our theorem therefore applies to the frameworks of these results, but describing each of these different frameworks would make this paper too long. We beg the reader not yet familiar with the cited papers to apologize for our possibly cryptic comments.

The first series of results of this type originates in the paper [AFL] (later extended in more than one direction, see e.g. [Gou] and references therein) for the weak coupling limit and [AL] for the low density limit. The results of the form of (A) in the cited papers exist at two different levels. First there is the kinematical results: Theorem 3.4 in [AFL], and Lemma 2.1 in [AL]. Note that, at this level, the von Neumann algebras M t and the states ω t are the same (the algebra being of the form B(H), the state being the pure state associated with the vacuum vector) for all t in the case of the weak coupling limit, but not in the case of the low density limit, where the parameter z enters the definition of the considered scalar product. The second level at which these papers prove results of the form (A) is the dynamical one: Theorem (II) in [AFL], Theorem 5.1 in [AL] where this time the structure depends on t in all cases. These theorems consider only one unitary U t at a time; we can, however, make a connection with a non-trivial form of (A) by noting that they can be easily extended to the case where the single unitary operator U t is replaced with a product of different operators corresponding to different couplings V in the Hamiltonian.

Another possible application comes from [AP] (and its extension in [AJ]); this time the considered limit is that of "repeated to continuous" interactions. The whole picture, that is both the h > 0 systems and the limiting case can be described within a single algebra. Here again the main result of the paper, Theorem 13, shows a priori a result of type (A) for the case of a single operator, but Corollary 18 implies that one has, in the common Hilbert space for all operators, strong convergence of the operators e iαiA (i) n to e iαiA (i) ∞ . Therefore, (A) will also hold for a product of more that one operator, corresponding to possibly different operators L (when using Theorem 17) or different Hamiltonians H (when using Theorem 19) -it is Theorem 19 that we used in subsection 4.1, in the simple case where H 0 = C. Note that standard results on strong resolvent convergence of operators (see [RS]) imply the strong convergence of f i (A

(i) n ) to f i (A (i)
∞ ), hence the convergence (6) for bounded continous functions f i . It is a non-trivial improvement to obtain this for non-continuous f i .

Last, we mention the study of "fluctuation algebras" in the papers [GV], [GVV] (and subsequent papers by the same authors), [Mat], [AJPP] and [JPP]. These papers consider operators A

A (i) n = 1 √ n |x|≤n (τ x (A i ) -ω(A i )), or
A (i) t = 1 √ t t 0 (τ s (A i ) -ω(A i )) ds,
where τ x is a translation operator (as in [GV], [GVV], [Mat], which study spatial fluctuations) or τ s is a dynamical group (as in [AJPP] and [JPP], which study time fluctuations) and every A i is an observable of the considered system (belonging to some subalgebra M 1 ). In both cases, a result of the type (A) is proven in which the limiting quantities ω ∞ (e iα1A (1) ∞ • • • e iαpA (p) ∞ ) are of the form ρ(W (A (1) ) . . . W (A (p) )), where the W are elements of a Weyl algebra over M 1 for an explicit symplectic form, and ρ is a quasi-free state on this Weyl algebra (see [START_REF] Bratteli | Operator Algebras and Quantum Statistical Mechanics 1. Second edition[END_REF] or [Pet]). A case of particular interest is when this symplectic form is found to be null, so that the Weyl algebra is abelian; in this case, the law of the operators A

(1) ∞ , . . . , A (n) ∞ in the state ω ∞ is that of a (classical) Gaussian vector.

  for g ∈ B and Assumption (A) translates into lim t→∞ R e iαx σ - jt (α 1 , . . . , α j-1 ; dx) = R e iαx σ - j∞ (α 1 , . . . , α j-1 ; dx).