Morita isomorphism for Cuntz algebras

Abstract

Let Rep On denote the category of all nondegenerate * representations of the Cuntz algebra On. For any 2 ≤ n, m < ∞, we construct an isomorphism functor Fn,m from Rep Om to Rep On such that (i) Fn,m commutes with infinite direct sum, (ii) Fn,m • F m,l = F n,l and Fm,n = F −1 n,m for any 2 ≤ n, m, l < ∞, (iii) for the von Neumann algebra Nπ generated by the image of a representation π, N Fn,m(π) and Nπ are isomorphic for any π in Rep Om, and (iv) there exists a functor F∞,n from Rep On to Rep O∞ with a right inverse such that F∞,n • Fn,m = F∞,m for any 2 ≤ n, m < ∞.