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 $$M_3({\mathbf {C}})\mapsto M_3({\mathbf {C}})$$ 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 $$V \in M_{d-1} ({\mathbf {C}}) $$ 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 family which is extreme unless $$t = \tfrac{-1}{d-1}$$ . 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 $$M_3({\mathbf {C}}) \otimes M_3({\mathbf {C}})$$ .

It is a long standing open problem whether the Thompson group F is an amenable group. In this article, we show that if A, B, C denote the standard generators of Thompson group T and D:=CBA-1 then 2+3 < 112||(I+C+C2)(I+D+D2+D3)|| <= 2+2.Moreover, the upper bound is attained if the Thompson group F is amenable. Here, the norm of an element in the group ring CT is computed in B(l2(T)) via the regular representation of T. Using the "cyclic reduced" numbers tau(((C+C2)(D+D2+D3))n),n is an element of N, and some methods from our previous article [Haagerup et al. 15] we can obtain precise lower bounds as well as good estimates of the spectral distributions of 112((I+C+C2)(I+D+D2+D3))*(I+C+C2)(I+D+D2+D3), where tau is the tracial state on the group von Neumann algebra L(T). Our extensive numerical computations suggest that 112||(I+C+C2)(I+D+D2+D3)||approximate to 3.28,and, thus that F might be non-amenable. However, we can in no way rule out that 112||(I+C+C2)(I+D+D2+D3)||= 2+2.

We investigate the structure of the relative bicentralizer algebra \(\mathrm{B}(N \subset M, \varphi )\) for inclusions of von Neumann algebras with normal expectation where N is a type \(\mathrm{III}_1\) subfactor and \(\varphi \in N_*\) is a faithful state. We first construct a canonical flow \(\beta ^\varphi : \mathbf {R}^*_+ \curvearrowright \mathrm{B}(N \subset M, \varphi )\) on the relative bicentralizer algebra and we show that the W\(^*\)-dynamical system \((\mathrm{B}(N \subset M, \varphi ), \beta ^\varphi )\) 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 \(\mathrm{B}(M,\varphi )\) 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 AFD 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.

It is shown that to every operator T in a general von Neumann factor M of type II1 and to every Borel set B in the complex plane C, one can associate a largest, closed, T -invariant subspace, K = KT (B), affiliated with M, such that the Brown measure of T |K is concentrated on B. Moreover, K is T -hyperinvariant, and the Brown measure of PK⊥T |K⊥ is concentrated on C \B. In particular, if T ∈ M has a Brown measure which is not concentrated on a singleton, then there exists a nontrivial, closed, T -hyperinvariant subspace. Furthermore, it is shown that for every T ∈ M the limit A = limn→∞[(T n)∗T ] 1 2n exists in the strong operator topology and KT (B(0, r)) = 1[0,r](A), r > 0.

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 $\Gamma$. 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 $F$.

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.

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.

We prove that the universal covering group S p ~ ( 2 , R ) \widetilde {\mathrm {Sp}}(2,\mathbb {R}) of S p ( 2 , R ) \mathrm {Sp}(2,\mathbb {R}) does not have the Approximation Property (AP). Together with the fact that S L ( 3 , R ) \mathrm {SL}(3,\mathbb {R}) does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that S p ( 2 , R ) \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 L^p -spaces associated with lattices in S p ~ ( 2 , R ) \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 L^p -spaces associated with lattices in any connected simple Lie group.

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.

Based on the analysis on the Ocneanu/Groh-Raynaud ultraproducts and the Effros-Mar\'echal 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-Mar\'echal 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)|

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.

Let F denote the Thompson group with standard generators A = x 0 , B = x 1 . It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent to [Formula: see text] and to [Formula: see text] where in both cases the norm of an element in the group ring ℂF is computed in B(ℓ 2 (F)) via the regular representation of F. By extensive numerical computations, we obtain precise lower bounds for the norms in (i) and (ii), as well as good estimates of the spectral distributions of (I+A+B)*(I+A+B) and of A+A -1 +B+B -1 with respect to the tracial state τ on the group von Neumann Algebra L(F). Our computational results suggest, that [Formula: see text] It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F.

We find a Levy-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.

All unital continuous C∗-bundles with properly infinite fibres are properly infinite C∗-algebras if and only if the full unital free product T2 ∗C T2 of two copies of the Cuntz extensions T2 generated by two isometries with orthogonal ranges is a K1-injective C ∗-algebra ([BRR08, Theorem 5.5], [Blan10, Proposition 4.2]). We show that for all integer n ≥ 3, there is a state ψn : Tn → C such that the reduced unital free product (Tn, ψn) ∗C (Tn, ψn) is a K1-injective C ∗-algebra which contains the algebraic free product Tn ⊛C Tn .

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 \({n\neq 3}\). 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.

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.

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

Since Wigner's pioneering work from 1955 random matrices have been an important tool in Mathematical Physics. After Voiculescu in 1991-95 used random matrices to solve some deep open problems about von Neumann algebras, random matrices have also played a key role in operator algebra theory. In 2005 Steen Thorbjornsen and the speaker were able to solve an old problem on C*-algebras, by making careful estimates of the largest and smallest eigenvalues in random ensembles, which can be expressed as (non-commutative) polynomials in two or more independent GUE-random matrices, [1]. Shortly after we obtained (in collaboration with Hanne Schultz) similar estimates for polynomials in GOE- and GSE-matrices [2], but the corresponding problem for polynomials in two or more non-Gaussian random matrices with independent entries was solved only recently by Greg Anderson (2011).

A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, 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 Erd\'elyi, Magnus and Nevai.

We study several notions of ultraproducts of von Neumann algebras from a unified viewpoint. In particular, we show that for a sigma-finite von Neumann algebra M, the ultraproduct Mω introduced by Ocneanu is a corner of the ultraproduct ∏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 (σtφω=(σtφ)ω). Applying these results, we obtain several properties of the Ocneanu ultraproduct of type III factors, which are not present in the tracial ultraproducts. For instance, it turns out that the ultrapower Mω of a Type III0 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 Mω and Connes' asymptotic centralizer algebra Mω.