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# 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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