A universal, non-commutative C*-algebra associated to the Hecke algebra of double cosets

TL;DR

This paper constructs a universal non-commutative C*-algebra linked to the Hecke algebra of double cosets, providing a new operator algebraic framework that generalizes classical Hecke operators in the modular group setting.

Contribution

It introduces a novel universal C*-algebra associated with Hecke algebras, extending the operator algebraic understanding of Hecke operators beyond classical cases.

Findings

The algebra embeds the Hecke algebra as a diagonal in a tensor product.

Representation on l^2 space yields abstract Hecke operators.

In the modular case, these operators are unitarily equivalent to classical Maass form operators.

Abstract

Let G be a discrete group and $\Gamma$ an almost normal subgroup. The operation of cosets concatanation extended by linearity gives rise to an operator system that is embeddable in a natural C* algebra. The Hecke algebra naturally embeds as a diagonal of the tensor product of this algebra with its opposite. When represented on the $l^2$ space of the group, by left and right convolution operators, this representation gives rise to abstract Hecke operators that in the modular group case, are unitarily equivalent to the classical operators on Maass wave forms