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

TL;DR

This paper explores the relationship between Toeplitz algebras of correspondences and endomorphisms of sums of Type I factors, providing criteria to identify when different representations lead to equivalent endomorphisms.

Contribution

It generalizes the connection between Toeplitz $C^*$-algebras and endomorphisms from $B(H)$ to von Neumann subalgebras, offering new criteria for representation equivalence.

Findings

Criteria for when two representations produce equal endomorphisms

Extension of the connection between graph-related $C^*$-algebras and von Neumann algebra endomorphisms

Framework for analyzing endomorphisms via Toeplitz algebra representations

Abstract

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 $C^*$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^*$-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.