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Isometries of Banach algebras satisfying the von Neumann inequality. (English) Zbl 0818.46064
A well-known result of R. Kadison asserts that every surjective isometry between two \(C^*\)-algebras is the superposition of a Jordan isomorphism and the multiplication by a unitary element. The aim of this paper is to extend this result to the class of those algebras that satisfy the von Neumann inequality. A (complex, unital) Banach algebra \(B\) is said to satisfy the von Neumann inequality if for every element \(b\in B\) with \(\| b\|\leq 1\) one has \(\| p(b)\|\leq \max\{| p(z)|; | z|\leq 1\}\), \(p\) a polynomial. Of course, the author has to give a meaning to the concepts of Jordan automorphism and unitary element in this context.

46L05 General theory of \(C^*\)-algebras
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