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.

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