Journal of Operator Theory

Volume 40, Issue 1, Summer 1998 pp. 71-85.

Isometries and Jordan-isomorphisms onto

$\scriptstyle C^*$ -algebras

Authors: Angel Rodriguez Palacios
Author institution: Departamento de An\'{a}lisis Matem\'{a}tico, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain

Summary: Let

$A$ be a

$C^*$ -algebra, and

$B$ a complex normed non-associative algebra. We prove that, if

$B$ has an approximate unit bounded by one, then, for every linear isometry

$F$ from

$B$ onto

$A$ , there exists a Jordan-isomorphism

$G:B \to A$ and a unitary element

$u$ in the multiplier algebra of

$A$ such that

$F(x)=u G(x)$ for all

$x$ in

$B$ . We also prove that, if

$G$ is an isometric Jordan-isomorphism from

$B$ onto

$A$ , then there exists a self-adjoint element

$\varphi$ in the centre of the multiplier algebra of the closed ideal of

$A$ generated by the commutators satisfying

$\|\varphi\|\leq 1$ and

$G(xy) = {1\over 2} (G(x)G(y) + G(y)G(x) + \varphi (G(x)G(y)-G(y)G(x)))$ for all

$x,y$ in

$B$ .

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