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

It is a well-known fact that endomorphisms of $B(H)$ are intimately connected with families of mutually orthogonal isometries, i.e. 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 to graphs and endomorphisms of certain von Neumann subalgebras of $B(H)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.