Polynomial Representations of marklogic.com/appservices/search">\(C^*\) -Algebras and their Applications

详细信息    查看全文

• 作者：Daniel Beltiţă ; Karl-Hermann Neeb
• 关键词：\(C^*\) ; algebra ; \(*\) ; semigroup ; completely positive map
• 刊名：Integral Equations and Operator Theory
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：86
• 期：4
• 页码：545-578
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-8989
• 卷排序：86

文摘

This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all classification problems to the passage from a \(C^*\)-algebra \(\mathcal {A}\) to its symmetric powers \(S^n(\mathcal {A})\), resp., to holomorphic representations of the multiplicative \(*\)-semigroup \((\mathcal {A},\cdot )\). Here we study the correspondence between representations of \(\mathcal {A}\) and of \(S^n(\mathcal {A})\) in detail. As \(S^n(\mathcal {A})\) is the fixed point algebra for the natural action of the symmetric group \(S_n\) on \(\mathcal {A}^{\otimes n}\), this is done by relating representations of \(S^n(\mathcal {A})\) to those of the crossed product \(\mathcal {A}^{\otimes n} \rtimes S_n\) in which it is a hereditary subalgebra. For \(C^*\)-algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of \(S^n(\mathcal {A})\) and we relate this to the Schur–Weyl theory for \(C^*\)-algebras. Finally we show that if \(\mathcal {A}\subseteq B(\mathcal {H})\) is a factor of type II or III, then its corresponding multiplicative representation on \(\mathcal {H}^{\otimes n}\) is a factor representation of the same type, unlike the classical case \(\mathcal {A}=B(\mathcal {H})\).