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Convex Sets Associated to C*-Algebras

Atkinson, Scott
Thesis/Dissertation; Online
Atkinson, Scott
Sherman, David
For a separable unital C*-algebra A and a separable McDuff II_1-factor M, we show that the space HOM_w(A,M) of weak approximate unitary equivalence classes of unital *-homomorphisms A\rightarrow 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^\mathcal{U}). When A is nuclear, HOM_w(A,M) is affinely homeomorphic to the trace space of A, but in general HOM_w(A,M) and the trace space of A do not share the same data (several examples are provided). We characterize extreme points of HOM_w(A,M) in the case where either A or M is amenable, and we give two different conditions -- one necessary and the other sufficient -- for extremality in general. The universality of C^*(F_\infty) is reflected in the fact that for any unital separable A, HOM_w(A,M) may be embedded as a face in HOM_w(C^*(F_\infty),M). We also extend Brown's construction to apply more generally to HOM(A,M^\mathcal{U}). Finally, we return to the context of HOM(N,R^\mathcal{U}) and examine the properties of finite dimensional minimal faces in that setting. The connection between algebraic and convex geometric concepts is the main theme of this thesis, and in studying this connection we uncover some new purely operator algebraic insights.
University of Virginia, Department of Mathematics, PHD, 2016
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