, Volume 61, Issue 3, pp 297-314

Jordan axioms for C*-algebras

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Abstract

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a 2)*=(a *)2, ∥aa * a∥=∥a3 and ∥ab+ba∥≦2a∥∥b∥ for alla, b inA is shown to be a C*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C*-algebra. Thus we obtain a purely Jordan characterization of C*-algebras.