On enveloping C*-algebras of Hecke algebras

Details On enveloping C*-algebras of Hecke algebras On enveloping C*-algebras of Hecke algebras

2012-01-03

arxiv.org/abs/1201.0709v2

Mathematics

Operator Algebras

31 pages; minor corrections

Scientific paper

We give a sufficient condition for a $^*$-algebra with a specified basis to have an enveloping $C^*$-algebra. Particularizing to the setting of a Hecke algebra $\h(G,\gm)$, we show that under a suitable assumption not only we can assure that an enveloping $C^*$-algebra $C^*(G, \gm)$ exists, but also that it coincides with $C^*(L^1(G,\gm))$, the enveloping $C^*$-algebra of the $L^1$-Hecke algebra. Our methods are used to show the existence of $C^*(G, \gm)$ and isomorphism with $C^*(L^1(G,\gm))$ for several classes of Hecke algebras. Most of the classes which are known to satisfy these properties are covered by this approach, and we also describe some new ones.

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