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A correspondence between algebra endomorphisms of a finite sum of copies of the algebra of all bounded operators on a Hilbert space and representations of certain norm closed ∗-subalgebras of bounded operators generated by a finite collection of partial isometries is introduced. Basic properties of this correspondence are investigated after developing some operations on bipartite graphs that usefully describe aspects of this relationship.

Cuntz-Krieger algebras and endomorphisms of finite direct sums of type I$_{\infty }$ factors