arXiv subjects

Uffe Haagerup

Publications and source records attributed to Uffe Haagerup.

At least 19 recordsLinked to original sources

arXiv2021

Extreme Points and Factorizability for New Classes of Unital Quantum Channels

We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps. We also study a large class of maps which generalize the Werner-Holevo channel for $d = 3$ in the sense that they are defined in terms of partial isometries of rank $d-1$. Moreover, we extend this to maps whose Kraus operators have the form $t |e_j \rangle \langle e_j | \oplus V $ with $V \in M_{d-1} ({\bf C}) $ unitary and $t \in (-1,1)$. We show that almost every map in this class is extreme in both the set of unital CP maps and the set of trace-preserving CP maps. We analyze in detail a particularly interesting subclass which is extreme unless $t = -1/(d-1)$. For $d = 3$, this includes a pair of channels which have a dual factorization in the sense that they can be obtained by taking the partial trace over different subspaces after using the same unitary conjugation in $M_3({\bf C}) \otimes M_3({\bf C})$.

math.OA↗

arXiv2018

Structure of bicentralizer algebras and inclusions of type III factors

We investigate the structure of the relative bicentralizer algebra ${\rm B}(N \subset M, \varphi)$ for inclusions of von Neumann algebras with normal expectation where $N$ is a type ${\rm III_1}$ subfactor and $\varphi \in N_*$ is a faithful state. We first construct a canonical flow $\beta^\varphi : \mathbf R^*_+ \curvearrowright {\rm B}(N \subset M, \varphi)$ on the relative bicentralizer algebra and we show that the W$^*$-dynamical system $({\rm B}(N \subset M, \varphi), \beta^\varphi)$ is independent of the choice of $\varphi$ up to a canonical isomorphism. In the case when $N=M$, we deduce new results on the structure of the automorphism group of ${\rm B}(M,\varphi)$ and we relate the period of the flow $\beta^\varphi$ to the tensorial absorption of Powers factors. For general irreducible inclusions $N \subset M$, we relate the ergodicity of the flow $\beta^\varphi$ to the existence of irreducible hyperfinite subfactors in $M$ that sit with normal expectation in $N$. When the inclusion $N \subset M$ is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when $N$ is amenable.

math.OA↗

arXiv2016

Non-inner amenability of the Thompson groups T and V

In this paper we prove that the Thompson groups $T$ and $V$ are not inner amenable. In particular, their group von Neumann algebras do not have property $Γ$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of $F$.

math.OA↗

arXiv2016

A new look at C*-simplicity and the unique trace property of a group

We characterize when the reduced C*-algebra of a group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C*-algebra. We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C*-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial.

math.OA↗

arXiv2016

On the uniqueness of injective III$_1$ factor

We give a new proof of a theorem due to Alain Connes, that an injective factor $N$ of type III$_1$ with separable predual and with trivial bicentralizer is isomorphic to the Araki--Woods type III$_1$ factor $R_{\infty}$. This, combined with the author's solution to the bicentralizer problem for injective III$_1$ factors provides a new proof of the theorem that up to $*$-isomorphism, there exists a unique injective factor of type III$_1$ on a separable Hilbert space.

math.OA↗

arXiv2016

Group C*-algebras without the completely bounded approximation property

It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C*-algebra of any lattice in a non-compact simple Lie group of real rank at least 2 with finite center does not have the completely bounded approximation property. Hence, the results obtained by J. de Canniere and the author for SO(n,1), n at least 2, and by M. Cowling for SU(n,1) do not generalize to simple Lie groups of real rank at least 2.

math.OA↗

arXiv2016

A complete characterization of connected Lie groups with the Approximation Property

We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean on the space of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean is a weak* continuous functional. After proving that the groups SL(3,R), Sp(2,R), and the universal covering of Sp(2,R) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.

math.GR↗

arXiv2015

Simple Lie groups without the Approximation Property II

We prove that the universal covering group $\widetilde{\mathrm{Sp}}(2,\mathbb{R})$ of $\mathrm{Sp}(2,\mathbb{R})$ does not have the Approximation Property (AP). Together with the fact that $\mathrm{SL}(3,\mathbb{R})$ does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that $\mathrm{Sp}(2,\mathbb{R})$ does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. By an adaptation of the methods we use to study the AP, we obtain results on approximation properties for noncommutative $L^p$-spaces associated with lattices in $\widetilde{\mathrm{Sp}}(2,\mathbb{R})$. Combining this with earlier results of Lafforgue and de la Salle and results of the second named author of this article, this gives rise to results on approximation properties of noncommutative $L^p$-spaces associated with lattices in any connected simple Lie group.

math.OA↗

arXiv2014

The weak Haagerup property II: Examples

The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda. In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups the weak Haagerup constant coincides with the weak amenability constant, although this is not true for locally compact groups in general. It is also shown that the semidirect product of R^2 by SL(2,R) does not have the weak Haagerup property.

math.OA↗

arXiv2014

An asymptotic property of factorizable completely positive maps and the Connes embedding problem

We establish a reformulation of the Connes embedding problem in terms of an asymptotic property of factorizable completely positive maps. We also prove that the Holevo-Werner channels W_n^- are factorizable, for all odd integers n different from 3. Furthermore, we investigate factorizability of convex combinations of W_3^+ and W_3^-, a family of channels studied by Mendl and Wolf, and discuss asymptotic properties for these channels.

math.OA↗

arXiv2014

Quasitraces on exact C*-algebras are traces

It is shown that all 2-quasitraces on a unital exact C*-algebra are traces. As consequences one gets: (1) Every stably finite exact unital C*-algebra has a tracial state, and (2) if an AW*-factor of type II_1 is generated (as an AW*-algebra) by an exact C*-subalgebra, then it is a von Neumann II_1-factor. This is a partial solution to a well known problem of Kaplansky. The present result was used by Blackadar, Kumjian and Rørdam to prove that RR(A)=0 for every simple non-commutative torus of any dimension.

math.OA↗

arXiv2014

Ultraproducts of von Neumann algebras

We study several notions of ultraproducts of von Neumann algebras from a unifying viewpoint. In particular, we show that for a sigma-finite von Neumann algebra $M$, the ultraproduct $M^ω$ introduced by Ocneanu is a corner of the ultraproduct $\prod^ωM$ introduced by Groh and Raynaud. Using this connection, we show that the ultraproduct action of the modular automorphism group of a normal faithful state $φ$ of $M$ on the Ocneanu ultraproduct is the modular automorphism group of the ultrapower state ($σ_t^{φ^ω}=(σ_t^φ)^ω$). Applying these results, we obtain several phenomena of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower $M^ω$ of a Type III$_0$ factor is never a factor. Moreover we settle in the affirmative a recent problem by Ueda about the connection between the relative commutant of $M$ in $M^ω$ and Connes' asymptotic centralizer algebra $M_ω$.

math.OA↗

arXiv2013

A Lévy-Khinchin formula for free groups

We find a Lévy-Khinchin formula for radial functions on free groups. As a corollary we obtain a linear bound on the growth of radial, conditionally negative definite functions on free groups of two or more generators.

math.GR↗

arXiv2013

Simple Lie groups without the Approximation Property

For a locally compact group G, let A(G) denote its Fourier algebra, and let M_0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-* topology on the space M_0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It follows that all connected simple Lie groups with finite center and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP.

math.OA↗

arXiv2013

Ultraproducts, QWEP von Neumann Algebras, and the Effros-Maréchal Topology

Based on the analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Maréchal topology on the space vN(H) of von Neumann algebras acting on a separable Hilbert space H, we show that for a von Neumann algebra M in vN(H), the following conditions are equivalent: (1) M has the Kirhcberg's quotient weak expectation property (QWEP). (2) M is in the closure of the set F_{inj of injective factors on H with respect to the Effros-Maréchal topology. (3) M admits an embedding i into the Ocneanu ultrapower R_{infty}^{omega} of the injective III_1 factor R_{\infty} with a normal faithful conditional expectation epsilon: R_{infty}^{omega} to i(M). (4) For every epsilon>0, natural number n, and xi_1,...,xi_n in P_M^{natural}, there is a natural number k and a_1,...,a_nin M_k(C)_+, such that | -tr_k(a_ia_j)|<epsilon (1<=i,j<=n) holds, where tr_k is the tracial state on M_k(C), and P_M^{natural} is the natural cone in the standard form of M.

math.OA↗

arXiv2013

The law of large numbers for the free multiplicative convolution

In classical probability the law of large numbers for the multiplicative convolution follows directly from the law for the additive convolution. In free probability this is not the case. The free additive law was proved by D. Voiculescu in 1986 for probability measures with bounded support and extended to all probability measures with first moment by J. M. Lindsay and V. Pata in 1997, while the free multiplicative law was proved only recently by G. Tucci in 2010. In this paper we extend Tucci's result to measures with unbounded support while at the same time giving a more elementary proof for the case of bounded support. In contrast to the classical multiplicative convolution case, the limit measure for the free multiplicative law of large numbers is not a Dirac measure, unless the original measure is a Dirac measure. We also show that the mean value of \ln x is additive with respect to the free multiplicative convolution while the variance of \ln x is not in general additive. Furthermore we study the two parameter family (μ_{α,β})_{α,β\ge 0} of measures on (0,\infty) for which the S-transform is given by S_{μ_{α,β}}(z) = (-z)^β(1+z)^{-α}, 0 < z < 1.

math.OA↗

arXiv2013

On the free Gamma distributions

For each positive number $α$ we study the analog $ν_alpha$ in free probability of the classical Gamma distribution with parameter $α$. We prove that $ν_α$ is absolutely continuous and establish the main properties of the density, including analyticity and unimodality. We study further the asymptotic behavior of $ν_α$ as $α\downarrow0$.

math.PR↗

arXiv2012

Inequalities for Jacobi polynomials

A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{α,β}(x)$, which is uniform for all degrees $n\ge0$, all real $α,β\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of matrix coefficients for the irreducible representations of $\mathrm{SU}(2)$ with a decay of $d^{-1/4}$ in the dimension $d$ of the representation. Moreover it complements previous results of Krasikov on a conjecture of Erdélyi, Magnus and Nevai.

math.RT↗