<p>We study the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-norms on the twisted tensor product between two graded <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>. Such an analysis extends the analogous one concerning the usual (i.e. untwisted) tensor product <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${{\mathfrak {A}}}\otimes {{\mathfrak {B}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">A</mi> <mo>⊗</mo> <mi mathvariant="fraktur">B</mi> </mrow> </math></EquationSource> </InlineEquation>, and includes the so-called Fermi tensor product as the simplest nontrivial twisted case. After a detailed study of representations of the involutive algebra under consideration on pre-Hilbert spaces (which, in principle, might be made by unbounded operators), we show that such <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-norms are associated with classes of positive functionals and their Gelfand-Naimark-Segal representations. We then prove the equivalence between nuclearity and the uniqueness of a compatible <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-norm, as it happens for the usual tensor product. The result concerning nuclearity is new also for the relevant case of Fermi systems.</p>

$C^*$-norms on the twisted tensor product and nuclearity

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发表日期：

2025年7月11日

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Francesco Fidaleo,
Elia Vincenzi

摘要

We study the $C^*$ $C^{*}$ -norms on the twisted tensor product between two graded $C^*$ $C^{*}$ -algebras ${{\mathfrak {A}}}$ $A$ and ${{\mathfrak {B}}}$ $B$ . Such an analysis extends the analogous one concerning the usual (i.e. untwisted) tensor product ${{\mathfrak {A}}}\otimes {{\mathfrak {B}}}$ $A \otimes B$ , and includes the so-called Fermi tensor product as the simplest nontrivial twisted case. After a detailed study of representations of the involutive algebra under consideration on pre-Hilbert spaces (which, in principle, might be made by unbounded operators), we show that such $C^*$ $C^{*}$ -norms are associated with classes of positive functionals and their Gelfand-Naimark-Segal representations. We then prove the equivalence between nuclearity and the uniqueness of a compatible $C^*$ $C^{*}$ -norm, as it happens for the usual tensor product. The result concerning nuclearity is new also for the relevant case of Fermi systems.

\(C^*\)-norms on the twisted tensor product and nuclearity

摘要