# Journal of Operator Theory

Volume 44, Issue 1, Summer 2000 pp. 113-138.

Endomorphisms of $\cal B \cal H$, extensions of pure states, and a class of representations of ${\cal O}_n$**Authors**: Neal J. Fowler (1), and Marcelo Laca (2)

**Author institution:**(1) Department of Mathematics, University of Newcastle, Callaghan, NSW 2308, Australia

(2) Department of Mathematics, University of Newcastle, Callaghan, NSW 2308, Australia

**Summary:**We construct the pure states of ${\cal O}_n$ that extend a given pure state of the fixed point algebra ${\cal F}_n$ of the gauge action, and we show that the gauge group acts transitively on these extensions. We apply this to construct and classify the ergodic endomorphisms of ${\cal B}({\cal H})$ whose tail algebra has a minimal projection. We discuss examples arising from product states of ${\cal F}_n$ and from the trace on the Choi subalgebra of ${\cal O}_n$.

**Keywords:**Cuntz algebras, ergodic endomorphisms, shifts, pure states

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