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COMPLETELY BOUNDED MULTIPLIERS OVER LOCALLY COMPACT QUANTUM GROUPS
"... Abstract. In this paper, we consider several interesting multiplier algebras associated with a locally compact quantum group G. Firstly, we study the completely bounded right multiplier algebraMrcb(L1(G)). We show that Mrcb(L1(G)) is a dual Banach algebra with a natural operator predual Qr(L1(G)), a ..."
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Abstract. In this paper, we consider several interesting multiplier algebras associated with a locally compact quantum group G. Firstly, we study the completely bounded right multiplier algebraMrcb(L1(G)). We show that Mrcb(L1(G)) is a dual Banach algebra with a natural operator predual Qr(L1(G)), and the completely isometric representation ofMrcb(L1(G)) on B(L2(G)), studied recently by Junge, Neufang and Ruan, is actually weak*weak * continuous. Secondly, we study the left uniformly continuous space LUC(G) and its Banach algebra dual LUC(G)∗. We prove that LUC(G) is a unital C*subalgebra of L∞(G) if the quantum group G is semiregular. We show the connection between LUC(G) ∗ and the quantum measure algebra M(G), as well as their representations on L∞(G) and B(L2(G)). Finally, we study the right uniformly complete qotient space UCQr(L1(G)) and its Banach algebra dual UCQr(L1(G))∗. For coamenable quanum groups G, we obtain the weak*homeomorphic completely isometric algebra isomorphism Mrcb(L1(G)) ∼=M(G) and the completely isometric isomorphism UCQr(L1(G)) ∼ = LUC(G). 1.
THE CHOQUETDENY EQUATION IN A BANACH SPACE
, 2006
"... Abstract. Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure µ on G we study the ChoquetDeny equation π(µ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection ..."
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Abstract. Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure µ on G we study the ChoquetDeny equation π(µ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula ” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law µ. The relation between the space of solutions of the ChoquetDeny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling the W ∗crossed product and coinciding precisely with the crossed product in the special case of the ChoquetDeny equation in the space E = B(L 2 (G)) of bounded linear operators on L 2 (G). Other general properties of the ChoquetDeny equation in a Banach space are also discussed.
Quantum group amenability, injectivity, and a question of Bédos– Tuset
, 2012
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Amenability and coamenability in nonabelian group duality
, 2006
"... Leptin’s theorem asserts that a locally compact group is amenable if and only if its Fourier algebra has a bounded (by one) approximate identity. In the language of locally compact quantum groups—in the sense of J. Kustermans and S. Vaes—, it states that a locally compact group is amenable if and on ..."
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Leptin’s theorem asserts that a locally compact group is amenable if and only if its Fourier algebra has a bounded (by one) approximate identity. In the language of locally compact quantum groups—in the sense of J. Kustermans and S. Vaes—, it states that a locally compact group is amenable if and only if its quantum group dual is coamenable. It is an open problem whether this is true for general locally compact quantum groups. We approach this problem focussing on the rôle of multiplicative unitaries. For a Hilbert space H, a multiplicative unitary W ∈ B(H˜⊗2H) defines a comultiplication ΓW on B(H), so that (B(H), ΓW) is a Hopf–von Neumann algebra. We introduce the notion of an admissible, multiplicative unitary. With an admissible, multiplicative unitary W, we associate another Hopf–von Neumann algebra (M W, ΓW). We show that (B(H), ΓW) is left amenable (coamenable) if and only if this is true for (M W, ΓW). Setting ˆ W: = σW ∗ σ, where σ is the flip map on H˜⊗2H, we prove that the left coamenability of (B(H), ΓW) implies the left amenability of (B(H), Γ ˆ W), and—for infinitedimensional H and under an additional technical hypothesis—also establish the converse. Applying these results to locally compact quantum groups—and, in particular, to Kac algebras—, we obtain that a Kac algebra is amenable if and only if its dual is coamenable. This extends Leptin’s theorem to Kac algebras and answers a problem left open by D. Voiculescu. Keywords: amenability; coamenability; convolution of trace class operators; Kac algebras; Leptin’s theorem; locally compact quantum groups; multiplicative unitaries.