Topological algebras and Mackey topologies

Abstract

Let <italic>E</italic> be a locally <italic>m</italic> -convex algebra with dual space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E prime"> <mml:semantics> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">E’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In a recent paper S. Warner asked if the finest locally <italic>m</italic> -convex topology on <italic>E</italic> compatible with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E prime"> <mml:semantics> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">E’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> was the mackey topology. It is shown that this is not the case. A similar result is given for this question in the <italic>A</italic> -convex algebra case. For any <italic>A</italic> -convex algebra, a construction is given of an associated locally <italic>m</italic> -convex algebra. It is shown that this associated locally <italic>m</italic> -convex topology is always the compact-open topology for the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript b Baseline left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{C_b}(S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the strict topology.