The left Hilbert algebra associated to a semi-direct product

1977 ◽  
Vol 82 (3) ◽  
pp. 411-418 ◽  
Author(s):  
R. Rousseau
Keyword(s):  
Direct Product ◽  
Compact Support ◽  
Continuous Functions ◽  
Crossed Product ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Hilbert Algebra ◽  
Locally Compact ◽  
Continuous Action ◽  
Canonical Weight

AbstractLet A and G be locally compact groups and α a continuous action of G on A, and let denote the semi-direct product of A and G. Then we prove that the left Hilbert algebra of continuous functions with compact support, has the same achieved left Hilbert algebra, as the crossed product of K(A)" by the associated action α̃ of G on . As a consequence we obtain that the canonical weight on is the dual weight of the canonical weight on K(A)".

scholarly journals Ends of locally compact groups and their coset spaces

1974 ◽  
Vol 17 (3) ◽  
pp. 274-284 ◽  
Author(s):  
C. H. Houghton
Keyword(s):  
Compact Group ◽  
Direct Product ◽  
Compact Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Countable Base ◽  
Locally Compact ◽  
Coset Spaces ◽  
Compact Boundary

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

Exponential Boundedness and Amenability of Open Subsemigroups of Locally Compact Groups

1994 ◽  
Vol 46 (06) ◽  
pp. 1263-1274 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Open Subset ◽  
Compact Support ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Complete Proof ◽  
Absolutely Continuous ◽  
Crucial Part

Abstract Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L ∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

Compactifications of locally compact groups and quotients

1994 ◽  
Vol 116 (3) ◽  
pp. 451-463 ◽  
Author(s):  
A. T. Lau ◽  
P. Milnes ◽  
J. S. Pym
Keyword(s):  
Normal Subgroup ◽  
Compact Group ◽  
Semidirect Product ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Positive Answer ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Uniformly Continuous

AbstractLet N be a compact normal subgroup of a locally compact group G. One of our goals here is to determine when and how a given compactification Y of G/N can be realized as a quotient of the analogous compactification (ψ, X) of G by Nψ = ψ(N) ⊂ X; this is achieved in a number of cases for which we can establish that μNψ ⊂ Nψ μ for all μ ∈ X A question arises naturally, ‘Can the latter containment be proper?’ With an example, we give a positive answer to this question.The group G is an extension of N by GN and can be identified algebraically with Nx GN when this product is given the Schreier multiplication, and for our further results we assume that we can also identify G topologically with N x GN. When GN is discrete and X is the compactification of G coming from the left uniformly continuous functions, we are able to show that X is an extension of N by (GN)(X≅N x (G/N)) even when G is not a semidirect product. Examples are given to illustrate the theory, and also to show its limitations.

scholarly journals Landstad–Vaes theory for locally compact quantum groups

2018 ◽  
Vol 29 (04) ◽  
pp. 1850028
Author(s):  
Sutanu Roy ◽  
Stanisław Lech Woronowicz
Keyword(s):  
Dynamical System ◽  
Quantum Groups ◽  
Compact Quantum Group ◽  
Crossed Product ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups

Landstad–Vaes theory deals with the structure of the crossed product of a [Formula: see text]-algebra by an action of locally compact (quantum) group. In particular, it describes the position of original algebra inside crossed product. The problem was solved in 1979 by Landstad for locally compact groups and in 2005 by Vaes for regular locally compact quantum groups. To extend the result to non-regular groups we modify the notion of [Formula: see text]-dynamical system introducing the concept of weak action of quantum groups on [Formula: see text]-algebras. It is still possible to define crossed product (by weak action) and characterize the position of original algebra inside the crossed product. The crossed product is unique up to an isomorphism. At the end we discuss a few applications.

scholarly journals Approximation property of C*-algebraic bundles

2002 ◽  
Vol 132 (3) ◽  
pp. 509-522 ◽  
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.

LOCALLY COMPACT GROUPS ADMITTING FAITHFUL STRONGLY CHAOTIC ACTIONS ON HAUSDORFF SPACES

2013 ◽  
Vol 23 (09) ◽  
pp. 1350158 ◽  
Author(s):  
FRIEDRICH MARTIN SCHNEIDER ◽  
SEBASTIAN KERKHOFF ◽  
MIKE BEHRISCH ◽  
STEFAN SIEGMUND
Keyword(s):  
Hausdorff Space ◽  
Geometric Characterization ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Action

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

The reduced dual of a direct product of groups

1966 ◽  
Vol 62 (1) ◽  
pp. 5-6 ◽  
Author(s):  
Aubrey Wulfsohn
Keyword(s):  
Direct Product ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Type I ◽  
Locally Compact ◽  
Product Of Groups

AbstractIf one of the locally compact groups H and K is of type I, the reduced dual of their product is shown to be homeomorphic to the product of the reduced duals.

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