Amenability and compact type for Hopf-von Neumann algebras from the homological point of view

Oleg, Aristov,

doi:10.1090/conm/363/06638

Cited by 14 publications

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“…We remark that in the setting of locally compact groups G, the above type of multiplicative structure on T (L 2 (G)) has been studied in [1,[39][40][41][42]. As defined in Section 3, the *-antiautomorphism R : B(L 2 (G)) → B(L 2 (G)) extends the unitary antipode R of G, mapping L ∞ (G) and L ∞ (Ĝ) onto L ∞ (G) and L ∞ (Ĝ ), respectively.…”

Section: Luc(g) and Ruc(g)mentioning

confidence: 99%

Completely bounded multipliers over locally compact quantum groups

Neufang

Ruan

2011

Proceedings of the London Mathematical Society

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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 algebra M r cb (L1(G)). We show that M r cb (L1(G)) is a dual Banach algebra with a natural operator predual Q r cb (L1(G)), and the completely isometric representation of M r cb (L1(G)) on B(L2(G)), studied recently by Junge, Neufang and Ruan, is actually weak*-weak* continuous if the quantum group G has the right co-approximation property. Secondly, we study the space LUC(G) of left uniformly continuous functionals on L1(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 semi-regular. 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 continuous complete quotient space UCQ r (G) and its Banach algebra dual UCQ r (G) * . For quantum groups G with the right coapproximation property, we establish a completely contractive injection Q r cb (L1(G)) → UCQ r (G) which is compatible with the relation C0(G) ⊆ LUC(G). For co-amenable quantum groups G, we obtain the weak*-weak* homeomorphic and completely isometric algebra isomorphism M r cb (L1(G)) ∼ = M (G) and the completely isometric isomorphism UCQ r (G) ∼ = LUC(G).

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Section: Luc(g) and Ruc(g)mentioning

confidence: 99%

Completely bounded multipliers over locally compact quantum groups

Neufang

Ruan

2011

Proceedings of the London Mathematical Society

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“…Since the span of the set {ω η : η ∈ L 2 (G)} is norm dense in T (L 2 (G)), it follows that (ω ξi ) is a right bounded approximate identity for T ⋆ (G). Conversely, if T ⋆ (G) has a right bounded approximate identity, then a similar argument to the proof of part (1) shows that G is co-amenable. Proposition 3.11.…”

Section: Convolution and Pointwise Products For Locally Compact Quantmentioning

confidence: 77%

Duality, cohomology, and geometry of locally compact quantum groups

Kalantar

Neufang

2013

Journal of Mathematical Analysis and Applications

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Abstract. In this paper we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally compact quantum group with two products which are operator versions of convolution and pointwise multiplication, respectively; we investigate the relation between these two products, and derive a formula linking them. Furthermore, we define some canonical module structures on these convolution algebras, and prove that certain topological properties of a quantum group, can be completely characterized in terms of cohomological properties of these modules. We also prove a quantum group version of a theorem of Hulanicki characterizing group amenability. Finally, we study the Radon-Nikodym property of the L 1 -algebra of locally compact quantum groups. In particular, we obtain a criterion that distinguishes discreteness from the Radon-Nikodym property in this setting.

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“…The relative biprojectivity of L 1 (G), that is, relative projectivity of L 1 (G) as an operator bimodule over itself, has been completely characterized: L 1 (G) is relatively C-biprojective if and only if L 1 (G) is relatively 1-biprojective if and only if G is a compact Kac algebra [2,20,12]. The corresponding characterization for (relative) C-biflatness remains an interesting open question.…”

Section: Proposition 52 Let G Be a Locally Compact Group Then L 1 mentioning

confidence: 99%

Inner amenability and approximation properties of locally compact quantum groups

Crann¹

2019

Indiana Univ. Math. J.

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We introduce an appropriate notion of inner amenability for locally compact quantum groups, study its basic properties, related notions, and examples arising from the bicrossed product construction. We relate these notions to homological properties of the dual quantum group, which allow us to generalize a well-known result of Lau-Paterson [46], resolve a recent conjecture of Ng-Viselter [51], and prove that, for inner amenable quantum groups G, approximation properties of the dual operator algebras can be averaged to approximation properties of G. Similar homological techniques are used to prove that ℓ 1 (G) is not relatively operator biflat for any non-unimodular discrete quantum group G; a unimodular discrete quantum group G with Kirchberg's factorization property is weakly amenable if and only if L 1 cb ( G) is operator amenable, and amenability of a locally compact quantum group G implies Cu( G) ∼ = L 1 ( G) ⊗ L 1 ( G) C0( G) completely isometrically. The latter result allows us to partially answer a conjecture of Voiculescu [67] when G has the approximation property.2010 Mathematics Subject Classification. Primary 22D35; Secondary 46M10, 46L89.

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Amenability and compact type for Hopf-von Neumann algebras from the homological point of view

Cited by 14 publications

References 0 publications

Completely bounded multipliers over locally compact quantum groups

Completely bounded multipliers over locally compact quantum groups

Duality, cohomology, and geometry of locally compact quantum groups

Inner amenability and approximation properties of locally compact quantum groups

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