Topological Algebras with Orthogonal M-bases

Abstract

<p>An M-basis in a topological vector space is a special case of the extended Markushevich basis, and a generalization of the unconditional basis. We study orthogonal bases and orthogonal M-bases in topological algebras, with emphasis on locally convex algebras. It turns out that an orthogonal basis or an orthogonal M-basis in a topological algebra is necessarily Schauder. We characterize some concrete topological algebras with orthogonal bases or orthogonal M-bases, up to a topological isomorphism. We introduce and study two classes of locally convex algebras: the class of "Φ-algebras" which includes, for example ℂʳ, c₀(r), C*₃(r) and H(D) (with the Hadamard multiplication); and the larger class of "locally convex s-algebras" which also includes - among other examples - ℓp, 1 ≤ p < ∞ and the Arens algebra L^ω[0,1]. A Φ-algebras is not necessarily locally m-convex, and a locally m-convex algebra is not necessarily a locally convex s-algebra. We give two examples of Banach algebras with orthogonal bases which are not unconditional and we prove that an orthogonal basis in a B₀ algebra is unconditional iff the algebra is a locally convex s-algebra. We also study the conversion of a Fréchet space with an unconditional basis into a Fréchet algebra with the basis under consideration as an orthogonal basis and we obtain a necessary and sufficient condition for this to be possible, revising and extending a result of Husain and Watson obtained for Banach spaces.</p>