On enveloping C*-algebras of Hecke algebras

TL;DR

This paper establishes conditions under which Hecke algebras have enveloping C*-algebras and demonstrates their equivalence with certain L^1-Hecke algebra C*-algebras, covering known and new classes.

Contribution

It provides a sufficient condition for the existence of enveloping C*-algebras of Hecke algebras and proves their isomorphism with L^1-Hecke algebra C*-algebras, extending previous results.

Findings

Existence of enveloping C*-algebras for several classes of Hecke algebras.

Isomorphism between the enveloping C*-algebra and L^1-Hecke algebra C*-algebra.

Extension of known classes satisfying these properties to new classes.

Abstract

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.