scholarly journals DIRECTLY FINITE ALGEBRAS OF PSEUDOFUNCTIONS ON LOCALLY COMPACT GROUPS

2014 ◽  
Vol 57 (3) ◽  
pp. 693-707
Author(s):  
YEMON CHOI
Keyword(s):  
Invertible Element ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Complex Line ◽  
Locally Compact ◽  
The Real ◽  
C Algebra ◽  
Finite Algebras ◽  
Unimodular Group ◽  
Reduced Group

AbstractAn algebraAis said to be directly finite if each left-invertible element in the (conditional) unitization ofAis right invertible. We show that the reduced group C*-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras ofp-pseudofunctions, showing that these algebras are directly finite ifGis amenable and unimodular, or unimodular with the Kunze–Stein property. An exposition is also given of how existing results from the literature imply thatL1(G) is not directly finite whenGis the affine group of either the real or complex line.

scholarly journals On the real rank of $C^\ast$-algebras of nilpotent locally compact groups

2012 ◽  
Vol 110 (1) ◽  
pp. 99 ◽  
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Compact Group ◽  
Heisenberg Group ◽  
Identity Element ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Connected Component ◽  
Locally Compact ◽  
The Real

If $G$ is an almost connected, nilpotent, locally compact group then the real rank of the $C^\ast$-algebra $C^\ast (G)$ is given by $\operatorname {RR} (C^\ast (G)) = \operatorname {rank} (G/[G,G]) = \operatorname {rank} (G_0/[G_0,G_0])$, where $G_0$ is the connected component of the identity element. In particular, for the continuous Heisenberg group $G_3$, $\operatorname {RR} C^\ast (G_3))=2$.

scholarly journals On Godement's characterisation of amenability

1998 ◽  
Vol 57 (1) ◽  
pp. 153-158 ◽  
Author(s):  
Alain Valette
Keyword(s):  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Groups ◽  
Definite Function ◽  
Compact Groups ◽  
Locally Compact ◽  
Unimodular Group ◽  
Compactly Supported ◽  
Assembly Map

Motivated by a question related to the construction of the Baum-Connes analytical assembly map for locally compact groups, we refine a criterion of Godement for amenability: for a unimodular group G, our criterion says that G is amenable if and only if every compactly supported, positive-definite function has non-negative integral over G.

On the Lp-conjecture for locally compact groups

2007 ◽  
Vol 89 (3) ◽  
pp. 237-242 ◽  
Author(s):  
F. Abtahi ◽  
R. Nasr-Isfahani ◽  
A. Rejali
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Locally compact groups with permutable closed subgroups

2021 ◽  
Vol 390 ◽  
pp. 107894
Author(s):  
Wolfgang Herfort ◽  
Karl H. Hofmann ◽  
Francesco G. Russo
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact
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