<p>We present a characterization of abelian <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras among all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras in terms of strong Birkhoff-James orthogonality. As an application, it is shown that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">B</mi> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq8"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-isomorphic provided that either <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">A</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">B</mi> </math></EquationSource> </InlineEquation> is abelian and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathfrak {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathfrak {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">B</mi> </math></EquationSource> </InlineEquation> are (nonlinearly) strong Birkhoff-James isomorphic to each other.</p>

A characterization of abelian $C^*$-algebras in terms of strong Birkhoff-James orthogonality

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2025年5月16日

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Ryotaro Tanaka

摘要

We present a characterization of abelian $C^*$ $C^{*}$ -algebras among all $C^*$ $C^{*}$ -algebras in terms of strong Birkhoff-James orthogonality. As an application, it is shown that $C^*$ $C^{*}$ -algebras $\mathfrak {A}$ $A$ and $\mathfrak {B}$ $B$ are $*$ $*$ -isomorphic provided that either $\mathfrak {A}$ $A$ or $\mathfrak {B}$ $B$ is abelian and $\mathfrak {A}$ $A$ and $\mathfrak {B}$ $B$ are (nonlinearly) strong Birkhoff-James isomorphic to each other.

A characterization of abelian \(C^*\)-algebras in terms of strong Birkhoff-James orthogonality

摘要