Bi-semi-linear mappings and commutativity in involutive Banach algebras (Q1969336)

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.