Mathematics – Operator Algebras
Scientific paper
2012-01-24
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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