Toeplitz Algebras of Correspondences and Endomorphisms of Sums of Type I Factors

Abstract

It is a well-known fact that endomorphisms of $B\left(H\right)$ are intimately connected with families of mutually orthogonal isometries, that is, with representations of the so-called Toeplitz $C^{\ast }$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^{\ast }$-algebras associated with graphs and endomorphisms of certain von Neumann subalgebras of $B\left(H\right)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.