Vol. 15, No. 3, 1965

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Lie and Jordan structures in Banach algebras

Paul Civin and Bertram Yood

Vol. 15 (1965), No. 3, 775–797

We first consider the theory of Jordan homomorphisms and Jordan ideals in Banach algebras. If B is a B-algebra or a semi-simple annihilator algebra, any closed Jordan ideal in B is a two-sided ideal. Any Jordan homomorphism of a Banach algebra onto B is automatically continuous. That Jordan homomorphisms are continuous and Jordan ideals are ideals is shown to hold in a number of other situations. We also study the Lie ideals in a semi-simple Banach algebra A. If the center of A is zero and proper closed Lie ideals do not contain their Lie annihilators, then A is direct topological sum of its minimal closed ideals. An H-algebra with zero center is an example of such an algebra.

Mathematical Subject Classification
Primary: 46.50
Secondary: 17.30
Received: 23 June 1964
Published: 1 September 1965
Paul Civin
Bertram Yood