THE HAAR MEASURE ON LOCALLY COMPACT GROUPS

One motivation for studying representation theory for the unitary group of a unital C*-algebra arises from Theoretical Physics. (In the latter connection, Segal [9] and Arveson [1] have developed a representation theory for G. Their approach is in a different direction from ours.) Another motivation for studying the representation theory of G arises out of the desire to unify the theories of amenable von Neumann algebras and amenable locally compact groups.A serious problem for such a representation theory is the absence of Haar measure on G in general.In [7], the author introduced the class RepdG of contractive unitary representations of G, the strong metric condition involved compensating for the lack of Haar measure.

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Three Results for Locally Compact Groups Connected with the Haar Measure Density Theorem

Proceedings of the American Mathematical Society ◽

10.2307/2035942 ◽

1965 ◽

Vol 16 (6) ◽

pp. 1414 ◽

Cited By ~ 1

Author(s):

Bruno J. Mueller

Keyword(s):

Haar Measure ◽

Density Theorem ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact

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A Duality of Locally Compact Groups That Does Not Involve the Haar Measure

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-21162 ◽

2015 ◽

Vol 116 (2) ◽

pp. 250 ◽

Cited By ~ 1

Author(s):

Yulia Kuznetsova

Keyword(s):

Compact Group ◽

Haar Measure ◽

Von Neumann Algebras ◽

Locally Compact Group ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Von Neumann ◽

C Algebras ◽

Commutative Algebras

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf $C^*$-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the $C^*$-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.

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