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

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 τ can be used to construct the crossed product algebra A ×τ 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 ×τ X contains Weyl operators {W (x), x ∈ X}. They satisfy canonical commutation relations w.r.t. the C-multiplier. Quantum spacetime is discussed as an example.