Vol. 27, No. 2, 1968

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On the Jordan structure of complex Banach algebras

Satish Shirali

Vol. 27 (1968), No. 2, 397–404

All algebras considered are complex Banach algebras with identity and continuous involution. The principal results of §1 are that for a Jordan homomorphism T of A1 into A2 where A2 is semisimple, continuity is automatic, the kernel is a closed ideal, and if A2 is commutative then the factor algebra A1kernel T is also commutative. In §2 a cone different from the usual cone is introduced and its relation to the usual cone is studied. The principal result is that if this cone coincides with the usual cone, then any Jordan representation is the sum of a representation and a antirepresentation. §3 is devoted to proving that for a semisimple algebra, the axiom xyx∥∥yfollows from the weaker axiom xy + yx2x∥∥y.

Mathematical Subject Classification
Primary: 46.60
Received: 20 September 1967
Published: 1 November 1968
Satish Shirali