Journal of Operator Theory

Volume 37, Issue 2, Spring 1997 pp. 357-369.

Full projections, equivalence bimodules and automorphisms of stable algebras of unital C*-algebras

Authors: Kazunori Kodaka
Author institution:Department of Mathematical Sciences, College of Science, Ryukyu University, Nishihara-cho, Okinawa, 903-01, JAPAN

Summary: Let A be a unital C*-algebra and $\mathbb K$ the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. Let $K_0(A)$ and $K_0 (A \otimes \mathbb K)$ be the $K_0$-groups of A and $A \otimes \mathbb K$ respectively. Let $\beta_*$ be an automorphism of $K_0 (A \otimes \mathbb K)$ induced by an automorphism $\beta$ of $A \otimes \mathbb K$. Since $K_0 (A) \cong K_0 (A \otimes \mathbb K)$, we regard $\beta_*$ as an automorphism of $K_0(A)$. In the present note we will show that there is a bijection between equivalence classes of automorphisms of $A \otimes \mathbb K$ and equivalence classes of full projections p of $A \otimes \mathbb K$ with $p(A \otimes \mathbb K)p \cong A$. Furthermore, using this bijection, we give a sufficient and necessary condition that there is an automorphism $\beta$ of $A \otimes \mathbb K$ such that $\beta_* \ne \alpha_*$ on $K_0(A)$ for any automorphism $\alpha$ of A if A has cancellation or A is a purely infinite simple C*-algebra.

Keywords: Automorphisms, cancellation, equivalence bimodules, full projections, $K_0$-groups.

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