Abstract:We study Property (T) for locally compact quantum groups, providing several new characterisations, especially related to operator algebraic ergodic theory. Quantum Property (T) is described in terms of the existence of various Kazhdan type pairs, and some earlier structural results of Kyed, Chen and Ng are strengthened and generalised. For second countable discrete unimodular quantum groups with low duals, Property (T) is shown to be equivalent to Property (T) 1,1 of Bekka and Valette. This is used to extend to this class of quantum groups classical theorems on 'typical' representations (due to Kerr and Pichot), and on connections of Property (T) with spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a particular von Neumann algebra (due to Connes and Weiss). Finally, we discuss in the Appendix equivalent characterisations of the notion of a quantum group morphism with dense image.
In this paper we generalize the notion of Cuntz-Pimsner algebras of C * -correspondences to the setting of subproduct systems. The construction is jus-Definition 1.1 ([17, Definition 2.1]). Let M , N be C * -algebras. A (right) Hilbert C * -module E over M is called an N −M (C * -) correspondence if it is also equipped with a left N -module structure, implemented by a * -homomorphism ϕ : N → L(E); that is, a · ζ := ϕ(a)ζ for a ∈ N , ζ ∈ E. We say that E is faithful if ϕ is faithful and essential if ϕ(N )E is total in E. If N = M , we say that E is a (C * -) correspondence over M . Example 1.2. Every Hilbert space is a C * -correspondence over C. Example 1.3. If M is a C * -algebra and α is an endomorphism of M , we write α M for the C * -correspondence that is equal to M as sets, with the obvious right M -module structure, and left M -action given by ϕ(a)b := α(a)b for a, b ∈ M .Example 1.4. Every quiver (directed graph) possesses an associated C * -correspondence. See [21, p. 193, (2)] or [17, Example 2.9] for details.The original definition of subproduct systems ([23, Definitions 1.1, 6.2]) is for the context of von Neumann algebras. We require its adaptation to the C * -setting of [26]. Definition 1.5. A family X = (X(n)) n∈Z + of C * -correspondences over a C * -algebra M is called a (standard) subproduct system if X(0) = M and for all n, m ∈ Z + , X(n+m) is an orthogonally-complementable sub-correspondence of X(n)⊗X(m). This implies, in particular, that X(n) is essential for each n ∈ N.Example 1.6. If E is an essential C * -correspondence over M , the product system X E , defined by X E (n) := E ⊗n for each n ∈ Z + , is trivially a subproduct system. Example 1.7 ([23, Example 1.3]). Fix a Hilbert space H, and let X(n) := H n (the n-fold symmetric tensor product of H) for every n. The resulting family X satisfies the requirements of Definition 1.5. It is called the symmetric subproduct system over H, and denoted by SSP H . Specifically, we put SSP d := SSP C d for d ∈ N and SSP ∞ := SSP ℓ 2 (N) .The reader is urged to consult [23] for many other interesting examples of subproduct systems. Given a subproduct system X, we shall use the following notation throughout the paper. Set E := X(1). The X-Fock space is the sub-correspondence F X := n∈Z + X(n)
A celebrated theorem of Pimsner states that a covariant representation T of a C * -correspondence E extends to a C * -representation of the Toeplitz algebra of E if and only if T is isometric. This paper is mainly concerned with finding conditions for a covariant representation of a subproduct system to extend to a C * -representation of the Toeplitz algebra. This framework is much more general than the former. We are able to find sufficient conditions, and show that in important special cases, they are also necessary. Further results include the universality of the tensor algebra, dilations of completely contractive covariant representations, Wold decompositions and von Neumann inequalities.
Recent results of Zsidó, based on his previous work with Niculescu and Ströh, on actions of topological semigroups on von Neumann algebras, give a Jacobs–de Leeuw–Glicksberg splitting theorem at the von Neumann algebra (rather than Hilbert space) level. We generalize this to the framework of actions of quantum semigroups, namely Hopf–von Neumann algebras. To this end, we introduce and study a notion of almost periodic vectors and operators that is suitable for our setting.
Abstract. In this short note we introduce a notion called "quantum injectivity" of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups.
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