Unitary Corepresentations Research Articles

Amenability properties of unitary co-representations of locally compact quantum groups

For locally compact quantum groups [Formula: see text], we initiate an investigation of stable states with respect to unitary co-representations [Formula: see text] of [Formula: see text] on Hilbert spaces [Formula: see text]; in particular, we study the subject on the multiplicative unitary operator [Formula: see text] of [Formula: see text] with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of [Formula: see text] associated to a suitable vector [Formula: see text], to present an operator theoretic characterization of stable states with respect to a related unitary co-representation [Formula: see text]. This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.

Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups

In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group G is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary WG is Bekka-amenable, and that G is amenable if and only if its dual quantum group’s fundamental multiplicative unitary WGˆ is weakly Bekka-amenable.

Amenability of locally compact quantum groups and their unitary co-representations

We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of B\'edos, Conti and Tuset. As a corollary, we extend to locally compact quantum groups a result of the first-named author, which characterizes amenability of a locally compact group $G$ by nuclearity of the reduced group $C^{*}$-algebra $C_{r}^{*}(G)$ and an additional condition.

Property T for locally compact quantum groups

In this short paper, we obtained some equivalent formulations of property T for a general locally compact quantum group 𝔾, in terms of the full quantum group C*-algebras [Formula: see text] and the *-representation of [Formula: see text] associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if 𝔾 is of Kac type, we show that 𝔾 has property T if and only if every finite-dimensional irreducible *-representation of [Formula: see text] is an isolated point in the spectrum of [Formula: see text] (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property T discrete quantum groups using bicrossed products.

COMPLETELY POSITIVE MULTIPLIERS OF QUANTUM GROUPS

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group 𝔾 (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of 𝔾. It follows that there is an order bijection between the completely positive multipliers of L1(𝔾) and the positive functionals on the universal quantum group [Formula: see text]. We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*–weak*-continuous.

The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Let G be a locally compact group, and let A ( G ) and VN ( G ) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A ( G ) : Is every bounded representation of A ( G ) on a Hilbert space H similar to a *-representation? We show that the similarity problem for A ( G ) has a negative answer if and only if there is a bounded representation of A ( G ) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π : A ( G ) → B ( H ) is similar to a *-representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN ( G ) associated to non-degenerate completely bounded representations of A ( G ) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A ( G ) is a *-representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].

On corepresentations of multiplier Hopf algebras

We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.

A new approach to induction and imprimitivity results

Given a closed quantum subgroup of a locally compact quantum group, we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group. More generally, we study induction given a coaction of the quantum subgroup on a C * -algebra. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. This means that we define quantum homogeneous spaces as C * -algebras and that we prove Morita equivalence of crossed products and homogeneous spaces. We essentially use von Neumann algebraic techniques to prove these Morita equivalences between C * -algebras.

On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations

AbstractWe introduce and study several notions of amenability for unitary corepresentations and *-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantumgroups. As a background for this study, we investigate the associated tensor C*-categories.

COVARIANT REPRESENTATIONS FOR COACTIONS OF HOPF ${\rm C}^*$-ALGEBRAS

Given a coaction α of a Hopf C*-algebra A on a C*-algebra B with an α-invariant C*-subalgebra C, and a conditional expectation E: B → C commuting with α, it is shown that if (π, u) is a covariant representation of the system (C, A, α∣C), then there is an associated covariant representation ( π ˜ , u ˜ ) of the system (B, A, α), where π ˜ is the representation induced from π up to B via E, and u ˜ is a unitary corepresentation of A naturally associated with u. Some applications are also discussed, including a lifting of ergodic coactions to von Neumann algebras, and a characterization of the amenability of multiplicative unitary operators via infinite tensor product covariant representations. 2000 Mathematics Subject Classification 46L55, 46L60, 46L87, 46L89, 81R15, 81R50, 81T05.

Amenable Representations and Reiter's Property for Kac Algebras

In this paper, we study amenable unitary corepresentations of Kac algebras. We also study some sorts of “noncommutative” Reiter's properties. As an application, we find some new equivalent conditions for the amenability of Kac algebras.

LOCALLY COMPACT QUANTUM GROUPS IN THE UNIVERSAL SETTING

In this paper we associate to every reduced C *-algebraic quantum group (A, Δ) (as defined in [11]) a universal C *-algebraic quantum group (Au, Δu). We fine tune a proof of Kirchberg to show that every *-representation of a modified L 1-space is generated by a unitary corepresentation. By taking the universal enveloping C *-algebra of a dense sub *-algebra of A we arrive at the C *-algebra Au. We show that this C *-algebra Au carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.

COREPRESENTATION THEORY OF MULTIPLIER HOPF ALGEBRAS II

We continue our development of the corepresentation theory of multiplier Hopf algebras. In this paper, we consider the corepresentations of a multiplier Hopf algebra A in a nondegenerate algebra B rather than on a vector space (cf. [25]). We concentrate ourself on those corepresentations of A in B which are invertible elements of the multiplier algebra M(B⊗A). They are called the unitary corepresentations of A. In particular, the generalized R-matrices or quasi-triangular structures of a regular multiplier Hopf algebra are unitary (bi)corepresentations. As an application the quantum double of an algebraic quantum group can be constructed by means of the universal unitary corepresentation. Moreover, a unitary corepresentation of A in B can implement an inner coaction of A on B which allows us to study the covariant theory and crossed products.

Compact quantum groups associated with monoidal functors

We provide a C ∗ {C}^\ast -algebra structure on the bialgebra associated with a monoidal linear ∗ {}^\ast -functor. The C ∗ {C}^\ast -algebra obtained in this way is a compact quantum group in the sense of Baaj and Skandalis. We show that the category of finite dimensional unitary corepresentations of this C ∗ {C}^\ast -algebra is equivalent to the given category.

Spectral analysis and the Haar functional on the quantumSU (2) group

The Haar functional on the quantumSU(2) group is the analogue of invariant integration on the groupSU(2). If restricted to a subalgebra generated by a self-adjoint element the Haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both. These results by Woronowicz and Koornwinder have been proved by using the corepresentation theory of the quantumSU(2) group and Schur's orthogonality relations for matrix elements of irreducible unitary corepresentations. These results are proved here by using a spectral analysis of the generator of the subalgebra. The spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of Jacobi matrices.

A unified theory of point groups. VI. The projective corepresentations of the magnetic point groups of infinite order

This paper provides all the p-inequivalent projective irreducible unitary corepresentations of all the magnetic point groups of infinite order with full use of their isomorphisms.

Magnetic Groups and Their Corepresentations

A review is given of the theory of magnetic groups and of their unitary corepresentations. Particular application is made to magnetic space groups, this part of the work being set in the framework of little-group theory. The symmetry problems in physics which lead to magnetic groups are analyzed and various applications of the theory to such problems are pointed out. Finally a method is given for obtaining the Kronecker products of corepresentations of magnetic groups, and an example is presented in which the unitary subgroup of the magnetic group is the point group ${\mathrm{C}}_{2v}$.