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Bi-semi-linear mappings and commutativity in involutive Banach algebras. (Applications bi-semi-linéaires et commutativité dans les algèbres de Banach involutives.) (French) Zbl 0967.46040
Motivated by the theorem of Le Page about the commutativity of Banach algebras \(A\) such that \(\|xy|\leq C\|yx\|\) for all \(x,y\in A\), the authors prove a general result about continuous bi-semilinear maps. The proof depends on Hahn-Banach theorem and Liouville’s theorem for harmonic functions. As a consequence they obtain results about commutativity of Banach algebras with involution, and results which assure that a semilinear endomorphism is a generalized involution.
MSC:
46J05 General theory of commutative topological algebras
46K05 General theory of topological algebras with involution
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