Projective representations of groups using Hilbert right C*-modules

TL;DR

This paper explores projective group representations using Hilbert right C*-modules, extending classical concepts by replacing scalar functions with C*-algebras, leading to new constructions of C*-algebras and K-theories.

Contribution

It introduces a generalized framework for projective group representations with C*-algebra-valued twisting functions, broadening the scope of applications and enabling new C*-algebra constructions.

Findings

Constructed new examples of C*-algebras using this framework

Extended the concept of projective representations with C*-algebra twists

Presented specific cases including Clifford algebras

Abstract

The projective representation of groups was introduced in 1904 by Issai Schur. It differs from the normal representation of groups by a twisting factor. It starts with a group T and a scalar valued function f on T^2 satisfying the conditions: f(1,1)=1, |f(s,t)|=1, and f(r,s)f(rs,t)=f(r,st)f(s,t) for all r,s,t in T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra of operators on the Hilbert space l^2(T). This representation can be used in order to construct many examples of C*-algebras. By replacing the scalars with an arbitrary unital C*-algebra (as range of f) the field of applications is enhanced in an essential way. The projective representation of groups, which we present in this paper, has some similarities with the construction of cross products with discrete groups. It opens the way to create many K-theories. In a first section we introduce some results which are needed for this construction, which is developed in the second section. In a third section we present some examples of C*-algebras obtained by this method. Examples of a special kind (the Clifford algebras) are presented in the last section.