Convex sets associated to \(C^{\ast}\)-algebras. (English) Zbl 1358.46049
Summary: For a separable unital \(C^\ast\)-algebra \(\mathfrak{A}\) and a separable McDuff \(\mathrm{II}_{1}\)-factor \(M\), we show that the space \(\mathbb{H} \text{om}_w(\mathfrak{A}, M)\) of weak approximate unitary equivalence classes of unital homomorphisms \(\mathfrak{A} \to M\) may be considered as a closed, bounded, convex subset of a separable Banach space – a variation on N. Brown’s convex structure \(\mathbb{H} \text{om}(N, R^{\mathcal{U}})\). Many separable unital \(C^\ast\)-algebras, including all (separable unital) nuclear \(C^\ast\)-algebras, have the property that for any McDuff \(\mathrm{II}_{1}\)-factor \(M\), \(\mathbb{H} \text{om}_w(\mathfrak{A},M)\) is affinely homeomorphic to the trace space of \(\mathfrak{A}\). In general \(\mathbb{H} \text{om}_w(\mathfrak{A},M)\) and the trace space of \(\mathfrak{A}\) do not share the same data (several examples are provided). We characterize extreme points of \(\mathbb{H} \text{om}_w(\mathfrak{A}, M)\) in many cases, and we give two different conditions – one necessary and the other sufficient – for extremality in general. The universality of \(C^\ast(\mathbb{F}_\infty)\) is reflected in the fact that for any unital separable \(\mathfrak{A}, \mathbb{H} \text{om}_w(\mathfrak{A},M)\) may be embedded as a face in \(\mathbb{H} \text{om}_w(C^\ast(\mathbb{F}_\infty),M)\). We also extend Brown’s construction to apply more generally to \(\mathbb{H}\text{om}(\mathfrak{A},M^{\mathcal{U}})\).
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