Tracial invariants, classification and II₁ factor representations of Popa algebras

2001/11/27 by Nathanial P. Brown, Brown, Nathanial P.
Mathematics · #46L #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L

paper · pdf · doi:10.48550/arxiv.math/0111286

LaTeX 2e, 37 pages, minor revisions and (hopefully) improved exposition

arxiv created 2002/03/13 · arxiv updated 2009/11/30

Abstract

Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary self adjoint operators. (2) A McDuff factor embeds into the ultrapower of the hyperfinite II₁ factor if and only if it contains a weakly dense operator system which is injective. (3) There exists a simple, quasidiagonal, real rank zero C*-algebra with non-hyperfinite II₁ factor representations and which is not tracially AF. This answers negatively questions of Sorin Popa and, respectively, Huaxin Lin.

Related

• Invariant means and finite representation theory of C*-algebras
• Tracial approximation in simple C*-algebras
• Non-unital tracially approximated \rm C^*-algebras
• Tracially Complete C*-Algebras
• Generalized Tracially Approximated C*-algebras