Nonseparable approximate equivalence

Abstract

This paper extends Voiculescu’s theorem on approximate equivalence to the case of nonseparable representations of nonseparable C ∗ {C^ \ast } -algebras. The main result states that two representations f f and g g are approximately equivalent if and only if rank f ( x ) = rank g ( x ) {\text {rank}}f(x) = {\text {rank}}g(x) for every x x . For representations of separable C ∗ {C^ \ast } -algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable C ∗ {C^ \ast } -algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu’s reflexivity theorem for subalgebras of the Calkin algebra are also obtained.