Convex sets associated to C^⁎-algebras

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• 作者：Scott Atkinson ^{saa6uy@virginia.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：C⁎C⁎-algebra ; II1-factor ; Convex ; Approximate unitary equivalence
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 September 2016
• 年：2016
• 卷：271
• 期：6
• 页码：1604-1651
• 全文大小：673 K

文摘

For a separable unital C^⁎$C^{⁎}$-algebra A$\mathfrak{A}$ and a separable McDuff II₁-factor M  , we show that the space Hom_w(A,M)$\mathbb{H}{\text{om}}_w(\mathfrak{A},M)$ of weak approximate unitary equivalence classes of unital ⁎-homomorphisms a683745de71a13bfcc4129" title="Click to view the MathML source">A→M$\mathfrak{A}\to M$ may be considered as a closed, bounded, convex subset of a separable Banach space – a variation on N. Brown's convex structure Hom(N,R^U)$\mathbb{H}\text{om}(N,R^{\mathcal{U}})$. Many separable unital C^⁎$C^{⁎}$-algebras, including all (separable unital) nuclear C^⁎$C^{⁎}$-algebras, have the property that for any McDuff II₁-factor M  , Hom_w(A,M)$\mathbb{H}{\text{om}}_w(\mathfrak{A},M)$ is affinely homeomorphic to the trace space of A$\mathfrak{A}$. In general Hom_w(A,M)$\mathbb{H}{\text{om}}_w(\mathfrak{A},M)$ and the trace space of A$\mathfrak{A}$ do not share the same data (several examples are provided). We characterize extreme points of Hom_w(A,M)$\mathbb{H}{\text{om}}_w(\mathfrak{A},M)$ in many cases, and we give two different conditions – one necessary and the other sufficient – for extremality in general. The universality of C^⁎(F_∞)$C^{⁎}({\mathbb{F}}_{\infty })$ is reflected in the fact that for any unital separable A,Hom_w(A,M)$\mathfrak{A},\mathbb{H}{\text{om}}_w(\mathfrak{A},M)$ may be embedded as a face in Hom_w(C^⁎(F_∞),M)$\mathbb{H}{\text{om}}_w(C^{⁎}({\mathbb{F}}_{\infty }),M)$. We also extend Brown's construction to apply more generally to Hom(A,M^U)$\mathbb{H}\text{om}(\mathfrak{A},M^{\mathcal{U}})$.