Revista Matemática Iberoamericana

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Volume 20, Issue 1, 2004, pp. 87–105
DOI: 10.4171/RMI/381

Published online: 2004-04-30

Isometries between C*-algebras

Cho-Ho J. Chu[1] and Ngai-Ching Wong[2]

(1) Queen Mary University, London, UK
(2) National Sun Yet-sen University, Kaohsiung, Taiwan

Let $A$ and $B$ be C*-algebras and let $T$ be a linear isometry from $A$ \emph{into} $B$. We show that there is a largest projection $p$ in $B^{**}$ such that $T(\cdot)p : A \longrightarrow B^{**}$ is a Jordan triple homomorphism and $$ T(a b^* c + c b^* a) p= T(a) T(b)^* T(c) p + T(c) T(b)^* T(a) p $$ for all $a$, $b$, $c$ in $A$. When $A$ is abelian, we have $\|T(a)p\|=\|a\|$ for all $a$ in $A$. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces {\it locally} to a Jordan triple isomorphism, by a projection.

Keywords: C*-algebra, JB*-triple, isometry, Banach manifold

Chu Cho-Ho, Wong Ngai-Ching: Isometries between C*-algebras. Rev. Mat. Iberoam. 20 (2004), 87-105. doi: 10.4171/RMI/381