A duality of locally compact groups which does not involve the Haar measure

Details A duality of locally compact groups which does not involve the Haar measure A duality of locally compact groups which does not involve the Haar measure

2012-01-24

arxiv.org/abs/1201.5023v2

Mathematics

Operator Algebras

19 pages

Scientific paper

We present a duality construction for locally compacts groups that is simpler than the theory of Kac algebras and does not involve the Haar measure in the definition of the duality functor. On the category of coinvolutive Hopf-von Neumann algebras (roughly speaking, these are Kac algebras without weight), we define a functor $M\mapsto \hatM$ such that for every locally compact group $G$, the algebra $C_0(G)^{**}$ is reflexive. Here $C_0(G)^{**}$ is the enveloping von Neumann algebra of $C_0(G)$, identified with its second dual. Conversely, every commutative reflexive algebra is of this form. The dual algebra $\hat{C_0(G)^{**}}$ is the big group algebra $W^*(G)$ of J.Ernest.

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