The enveloping algebra of a covariant system

Abstract

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a “covariant system”. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.