We introduce the notion of semiprime ideals in associative triple systems and establish their fundamental equivalences. Using these characterizations, we construct several classes of semiprime ideals in TROs and in the Haagerup tensor product of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ C^* $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras. Furthermore, we examine conditions under which the Haagerup and Banach space projective norms are equivalent on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M\otimes B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>⊗</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, where M is a TRO and B is a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>-algebra. Next, we give a multiplication on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M \otimes ^h B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <msup> <mo>⊗</mo> <mi>h</mi> </msup> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> making it a Banach algebra. Finally, we analyze ideal structure of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M \otimes ^h B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <msup> <mo>⊗</mo> <mi>h</mi> </msup> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, showing that every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-ideal is an ideal and an example is given to demonstrate that the converse need not hold.

Ideal structure of Haagerup tensor product of ternary rings of operators

期刊论文
发表日期：

2026年4月20日

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Arpit Kansal,
Ajay Kumar

摘要

We introduce the notion of semiprime ideals in associative triple systems and establish their fundamental equivalences. Using these characterizations, we construct several classes of semiprime ideals in TROs and in the Haagerup tensor product of $ C^* $ $C^{*}$ -algebras. Furthermore, we examine conditions under which the Haagerup and Banach space projective norms are equivalent on $M\otimes B$ $M \otimes B$ , where M is a TRO and B is a $C^{*}$ $C^{*}$ -algebra. Next, we give a multiplication on $M \otimes ^h B$ $M \otimes^{h} B$ making it a Banach algebra. Finally, we analyze ideal structure of $M \otimes ^h B$ $M \otimes^{h} B$ , showing that every $\epsilon $ $ϵ$ -ideal is an ideal and an example is given to demonstrate that the converse need not hold.