Inner product modules over 𝐵*-algebras

Abstract

This paper is an investigation of right modules over a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{B^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <italic>B</italic> which posses a <italic>B</italic>-valued “inner product” respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which <italic>B</italic> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>W</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{W^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>W</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{W^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras. The extension of an inner product module over <italic>B</italic> by a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{B^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <italic>A</italic> containing <italic>B</italic> as a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.