C*-Multipliers, crossed product algebras, and canonical commutation relations

TL;DR

This paper generalizes the concept of group multipliers to C*-multipliers and explores their relationship with projective actions, leading to a unified framework for twisted group algebras and crossed product algebras with applications in mathematical physics.

Contribution

It introduces a one-to-one correspondence between C*-multipliers and projective actions, unifying twisted group algebra and crossed product constructions in a novel framework.

Findings

C*-multipliers correspond to projective actions.

The framework unifies twisted group algebras and crossed products.

Application to quantum spacetime and Weyl operators.

Abstract

The notion of a multiplier of a group X is generalized to that of a C*-multiplier by allowing it to have values in an arbitrary C*-algebra A. On the other hand, the notion of the action of X in A is generalized to that of a projective action of X as linear transformations of the space of continuous functions with compact support in X and with values in A. It is shown that there exists a one-to-one correspondence between C*-multipliers and projective actions. C*-multipliers have been used to define twisted group algebras. On the other hand, the projective action tau can be used to construct the crossed product algebra A x_tau X. Both constructions are unified in the present approach. The results are applicable in mathematical physics. The multiplier algebra of the crossed product algebra A x_tau X contains Weyl operators {W(x),x in X}. They satisfy canonical commutation relations w.r.t. the C*-multiplier. Quantum spacetime is discussed as an example.