\(C^\) -代数正锥上映射的Jordan \(\) -同构的非线性表征

IF 0.6 Q3 MATHEMATICS

ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-05-27 DOI:10.1007/s44146-024-00140-y

Osamu Hatori, Shiho Oi

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引用次数: 0

摘要

研究了一元\(C^*\) -代数的正定锥与正半定锥之间的映射。我们描述了保持(1)元素的商或积的范数的满射映射；（二）元素的商或积的谱；(3)元素的商或积的谱半模。这些映射与指定的\(C^*\) -代数之间的约旦\(*\) -同构有关。当正定锥之间保留元素商的范数的射不能被扩展到底层的\(C^*\) -代数之间的线性映射时，其他类型的射可以被扩展到Jordan \(*\) -同构或Jordan \(*\) -同构后与一个正可逆元素的2边乘法。我们还研究了正可逆元中心性的条件。我们推广了关于酉\(C^*\) -代数的正半定锥之间的射的“推论”。应用它，我们给出了一般一元\(C^*\) -代数问题Molnár的正解。

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Non-linear characterization of Jordan \(*\)-isomorphisms via maps on positive cones of \(C^*\)-algebras

We study maps between positive definite or positive semidefinite cones of unital \(C^*\)-algebras. We describe surjective maps that preserve

(1)
the norm of the quotient or product of elements;
(2)
the spectrum of the quotient or product of elements;
(3)
the spectral seminorm of the quotient or product of elements.

These maps relate to the Jordan \(*\)-isomorphisms between the specified \(C^*\)-algebras. While a surjection between positive definite cones that preserves the norm of the quotient of elements may not be extended to a linear map between the underlying \(C^*\)-algebras, the other types of surjections can be extended to a Jordan \(*\)-isomorphism or a Jordan \(*\)-isomorphism followed by 2-sided multiplication by a positive invertible element. We also study conditions for the centrality of positive invertible elements. We generalize “the corollary” regarding surjections between positive semidefinite cones of unital \(C^*\)-algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital \(C^*\)-algebras.

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