Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

On the similarity problem for locally compact quantum groups

A well-known theorem of Day and Dixmier states that any uniformly bounded representation of an amenable locally compact group G on a Hilbert space is similar to a unitary representation. Within the category of locally compact quantum groups, the conjectural analogue of the Day–Dixmier theorem is that every completely bounded Hilbert space representation of the convolution algebra of an amenable locally compact quantum group should be similar to a ?-representation. We prove that this conjecture is false for a large class of non-Kac type compact quantum groups, including all q-deformations of compact simply connected semisimple Lie groups. On the other hand, within the Kac framework, we prove that the Day–Dixmier theorem does indeed hold for several new classes of examples, including amenable discrete quantum groups of Kac-type.     

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.     

Canonical extension of actions of locally compact quantum groups

In this paper, we generalize the notion of the canonical extension of automorphisms of von Neumann algebras to the case of actions of locally compact quantum groups (in the sense of Kustermans and Vaes). Various expected properties will be shown to hold for this new canonical extension. As an application, we describe the flow of weights of the crossed product of a type III factor by some special action of a discrete Kac algebra.     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

The central kernel of the Peirce-one space

Further investigation into the properties of the Peirce-one space J₁ corresponding to a weak^*-closed inner ideal J in a JBW^*-triple A is carried out, and, in particular, it is shown that J₁ contains no non-trivial weak^*-closed ideals.Received: 12 June 2002     

Drinfeld–Sokolov reduction for quantum groups and deformations of W-algebras

We define deformations of W-algebras associated to comple semisimple Lie algebras by means of quantum Drinfeld-Sokolov reduction procedure for affine quantum groups. We also introduce Wakimoto modules for arbitrary affine quantum groups and construct free field resolutions and screening operators for the deformed W-algebras. We compare our results with earlier definitions of q-W-algebras and of the deformed screening operators due to Awata, Kubo, Odake, Shiraishi [60],[6], [7], Feigin, E. Frenkel [22] and E. Frenkel, Reshetikhin [34]. The screening operator and the free field resolution for the deformed W-algebra associated to the simple Lie algebra sl₂ coincide with those for the deformed Virasoro algebra introduced in [60]. The author is supported by the Swiss National Science Foundation.     

Compact quantum groups and group duality

In this paper, we consider the *-representations of compact quantum groups and group duality. The main results in the paper are: (1) there is a one-to-one correspondence between the *-representations of compact quantum groups and *-representations of the dual Banach *-algebra; (2) the category of commutative compact quantum groups (semigroups) is a dual category to the category of compact groups (semigroups); (3) the dual category of the category of locally compact groups (semigroups) is the category of commutative Hopf C^*-algebras with a particular property. Our group duality has the flavor of a Gelfand-Naimark type theorem for compact quantum groups, and for Hopf C^*-algebras.     

Fixed point property and normal structure for Banach spaces associated to locally compact groups

In this paper we investigate when various Banach spaces associated to a locally compact group have the fixed point property for nonexpansive mappings or normal structure. We give sufficient conditions and some necessary conditions about for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a -algebra has the fixed point property then for any normal element of , the spectrum is countable and that the group -algebra has weak normal structure if and only if is finite.

     

The linearization of the central limit operator in free probability theory

We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory [VDN92], we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT. In the classical case the analysis involves dilation operators; in the free case more general composition operators appear. Received: 3 December 1997     

On cocycle twisting of compact quantum groups

We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C^∗-algebra of the quantum group changes.     

Jordan operator algebras revisited

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.     

On admissible topologies in JB*‐triples

Given a complex JB*‐triple X, we define and study admissible topologies on X, i.e., locally convex topologies τ on X coarser than the norm topology, invariant under the group of surjective linear isometries of X, and such that the triple product is jointly ‐continuous on bounded subsets of X. As a consequence of the joint ‐continuity of the triple product, all holomorphic automorphisms of the open unit ball are homeomorphisms of and the natural action is jointly ‐continuous on .     

Noncommutative topology and Jordan operator algebras

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.     

Jordan operator algebras: basic theory

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.     

Fixed point property for Banach algebras associated to locally compact groups

In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak^∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier-Stieltjes algebra of G has the weak^∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T.