Abstract homomorphisms from locally compact groups to discrete groups

This paper is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of this paper: for hyperbolic TDLC-groups with rational discrete cohomological dimension [Formula: see text], hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group [Formula: see text] of a negatively curved locally finite [Formula: see text]-dimensional building [Formula: see text] is a hyperbolic TDLC-group, whenever [Formula: see text] acts with finitely many orbits on [Formula: see text]. Examples where this result applies include hyperbolic Bourdon’s buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension [Formula: see text] when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

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Approximation property of C*-algebraic bundles

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005837 ◽

2002 ◽

Vol 132 (3) ◽

pp. 509-522 ◽

Cited By ~ 11

Author(s):

RUY EXEL ◽

CHI-KEUNG NG

Keyword(s):

Direct Product ◽

Approximation Property ◽

Discrete Case ◽

Locally Compact Groups ◽

Compact Groups ◽

Discrete Groups ◽

Locally Compact ◽

Cross Sectional ◽

The Cross

In this paper, we will define the reduced cross-sectional C*-algebras of C*-algebraic bundles over locally compact groups and show that if a C*-algebraic bundle has the approximation property (defined similarly as in the discrete case), then the full cross-sectional C*-algebra and the reduced one coincide. Moreover, if a semi-direct product bundle has the approximation property and the underlying C*-algebra is nuclear, then the cross-sectional C*-algebra is also nuclear. We will also compare the approximation property with the amenability of Anantharaman-Delaroche in the case of discrete groups.

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Strong and Extremely Strong Ditkin sets for the Banach Algebras Apr(G) = Ap ⋂ Lr(G)

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-077-0 ◽

2011 ◽

Vol 63 (1) ◽

pp. 123-135 ◽

Cited By ~ 1

Author(s):

Edmond E. Granirer

Keyword(s):

Banach Algebra ◽

Banach Algebras ◽

Locally Compact Groups ◽

Compact Groups ◽

Discrete Groups ◽

Fourier Algebra ◽

Locally Compact ◽

Strict Inclusion ◽

Sidon Sets ◽

Sequential Completeness

Abstract Let Ap(G) be the Figa-Talamanca, Herz Banach Algebra on G; thus A2(G) is the Fourier algebra. Strong Ditkin (SD) and Extremely Strong Ditkin (ESD) sets for the Banach algebras Apr (G) are investigated for abelian and nonabelian locally compact groups G. It is shown that SD and ESD sets for Ap(G) remain SD and ESD sets for Apr(G), with strict inclusion for ESD sets. The case for the strict inclusion of SD sets is left open.A result on the weak sequential completeness of A2(F) for ESD sets F is proved and used to show that Varopoulos, Helson, and Sidon sets are not ESD sets for A2r(G), yet they are such for A2(G) for discrete groups G, for any 1 ≤ r ≤ 2.A result is given on the equivalence of the sequential and the net definitions of SD or ESD sets for σ-compact groups.The above results are new even if G is abelian.

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New Deformations of Convolution Algebras and Fourier Algebras on Locally Compact Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-027-7 ◽

2017 ◽

Vol 69 (02) ◽

pp. 434-452 ◽

Cited By ~ 1

Author(s):

Hun Hee Lee ◽

Sang-gyun Youn

Keyword(s):

Growth Rate ◽

Lie Groups ◽

Operator Algebra ◽

Locally Compact Groups ◽

Compact Groups ◽

Discrete Groups ◽

Locally Compact ◽

Real Dimension ◽

Convolution Algebras ◽

Deformed Algebras

Abstract In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some ûnitely generated discrete groups with connection to the growth rate of the group.

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