Jean Renault
Given a locally compact group bundle, we show that the system of the Plancherel weights of their C*-algebras is lower semi-continuous. As a corollary, we obtain that the dual Haar sytem of a continuous Haar system of a locally compact abelian group bundle is also continuous.
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We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. In particular, we obtain a generalisation of Ellis's joint continuity theorem. We also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a "noncommutative" version of the converse Haar's theorem.
We study states on the universal noncommutative *-algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. F…
We develop the twisting construction for locally compact quantum groups. A new feature, in contrast to the previous work of M. Enock and the second author, is a non-trivial deformation of the Haar measure. Then we construct Rieffel's deformation of locally compact quantum groups and show that it is dual to the twisting. This allows to give new interesting concrete examples of locally compact quantum groups, in particular, deformations of the classical $az+b$ group and of the Woronowicz' quantum $az+b$ group.
We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the \cst-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorph…
We consider groupoids on $\{1,2,..,d\}^\mathbb{N}$, cocycles and the counting measure as transverse function. We generalize results relating quasi-invariant probabilities with eigenprobabilities for the dual of the Ruelle operator. We assume a mild compatibility of the groupoid with the symbolic structure. We present a generalization of the Ruelle operator - the Haar-Ruelle operator - taking into account the Haar structure. We consider continuous and also Hölder cocycles. IFS with weights appears in our reasoning in the Hölder case.
In this note we study asymptotic properties of the *-distribution of traces of some matrices, with respect to the free Haar trace on the unitary dual group. The considered matrices are powers of the unitary matrix generating the Brown algebra. We proceed in two steps, first computing the free cumulants of any R-cyclic family, then characterizing the asymptotic *-distributions of the traces of powers of the generating matrix, thanks to these free cumulants. In particular, we obtain that these traces are asymptotic *-free circular variables.
The continuity of the core inverse and the dual core inverse is studied in the setting of C*-algebras. Later, this study is specialized to the case of bounded Hilbert space operators and to complex matrices. In addition, the differentiability of these generalized inverses is studied in the context of C*-algebras.
We construct a one parameter deformation of the group of $2\times 2$ upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual $\cs$-algebra and the dual comultiplication.
An approximate equivalence for the GNS representation of the Haar state of $SU_{q}(2)$
本研究利用q=0时的结晶化$C(SU_{q}(2))$代数构造酉算子，建立量子SU(2)群Haar态GNS表示与非零q下该代数无限维不可约表示直积分的近似等价，并推导对应的KK类与对偶量子群的Fredholm表示。
A notion of L^2-homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its L^2-Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these L^2-Betti numbers vanish for the Gelfand dual of a compact Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of the underlying Lie group. Finally, we relate our approach to the approach of A. Connes and D. Shlyakhtenko by proving that the L^2-Betti numbers of a c…
We prove that (a) discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and (b) the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of $C^*$-algebras are again free via a Fell-topology characterization for $C^*$-field continuity, recovering a result of Blanchard's in a somewhat more general setting.
The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding
本文提出了一种基于II型C*-代数的规范群oid构造，通过规范对角嵌入将代数嵌入到群oidC*-代数中，展示了其在K-理论和拓扑学中的应用。
We introduce the notion of a Jordan triple module and determine the precise conditions under which every derivation from a JB*-triple E into a Banach (Jordan) triple E-module is continuous. In particular, every derivation from a real or complex JB*-triple into its dual space is automatically continuous. Among the consequences, we prove that every triple derivation from a C*-algebra A to a Banach triple A-module is continuous. In particular, every Jordan derivation from A to a Banach A-bimodule is a derivation, a result which complements a classical theorem due to B.E. Johnson and solves a prob…
Automatic continuity of biorthogonality preservers between compact C$^*$-algebras and von Neumann algebras
该研究证明了对偶/紧C$^*$-代数及冯·诺依曼代数间的保双正交线性满射具有自动连续性，推进了算子代数映射性质研究。
Free wreath products by finite group duals
研究经典群的广义自由圈积，通过显式计算Haar态，证明其von Neumann代数为全II_1因子，约化C*-代数为具有唯一迹的简单代数。
Let $\mathbb{G}$ be a locally compact quantum group with dual $\widehat{\mathbb{G}}$. Suppose that the left Haar weight $φ$ and the dual left Haar weight $\widehatφ$ are tracial, e.g. $\mathbb{G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb{G}},\widehatφ)$ to $L_q(\widehat{\mathbb{G}},\widehatφ)$ whenever the symbol $x$ lies in $L_{r,\infty}(\mathbb{G},φ)$, where $1/r=1/p-1/q$. Moreover, we have \begin{equation*} \|m_{x}:L_p(\widehat{\mathbb{G}},\widehatφ)\to L_q(\widehat{\mathbb{G}},\widehatφ)\|\l…
In this paper, we give a construction of a (C*-algebraic) quantum Heisenberg group. This is done by viewing it as the dual quantum group of the specific non-compact quantum group (A,Δ) constructed earlier by the author. Our definition of the quantum Heisenberg group is different from the one considered earlier by Van Daele. To establish our object of study as a locally compact quantum group, we also give a discussion on its Haar weight, which is no longer a trace. In the latter part of the paper, we give some additional discussion on the duality mentioned above.
Scott continuity is a concept from domain theory that had an unexpected previous life in the theory of von Neumann algebras. Scott-continuous states are known as normal states, and normal states are exactly the states coming from density matrices. Given this, and the usefulness of Scott continuity in domain theory, it is natural to ask whether this carries over to generalized probabilistic theories. We show that the answer is no - there are infinite-dimensional convex sets for which the set of Scott-continuous states on the corresponding set of 2-valued POVMs does not recover the original conv…
Absolutely continuous commuting row contractions admit a weak-$*$ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury-Arveson space, we give a complete characterization of these commuting row contractions in measure theoretic terms. We also establish that completely non-unitary row contractions are necessarily absolutely continuous, in direct parallel with the case of a single contraction. Finally, we consider refinements of this question for row contractions that are annihilated by a give…
The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the Fourier transform: Motivated by some recent works by Van Daele in the multiplier Hopf algebra framework, and by using the Haar weights, we define here the (generalized) Fourier transform and the inverse Fourier transform, at the level of locally compact quantum groups. We will then consider the analogues of the Fourier inversion theorem, Plancherel theorem, and t…
In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra $\mathcal{T}^*$ is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between $\mathcal{T}$ and the structure of weak Hopf algebra.
Almost Representations of Groupoids on Banach Bundles
本研究将紧群几乎表示的经典结果推广到带连续Haar测度系统的真群胚，证明其在有限秩Hilbert丛上局部有足够多连续表示，进而得到纯拓扑下群胚的Tannaka对偶新推广。
In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a C*-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper "Locally compact quantum groups". We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its du…
In this paper, we carry out the ``quantum double construction'' of the specific quantum groups we constructed earlier, namely, the ``quantum Heisenberg group algebra'' (A,Δ) and its dual, the ``quantum Heisenberg group'' (\hat{A},\hatΔ). Our approach will be by constructing a suitable multiplicative unitary operator. In this way, we are able to retain the C*-algebra framework, and thus able to carry out our construction within the category of locally compact quantum groups. This construction is a kind of a generalized crossed product. To establish that the quantum double we obtain is indeed a …
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