Pseudocompact C*-Algebras

Finite dimensional C*-algebras are just finite direct sums of matrix algebras, and historically limits of finite dimensional C*-algebras, such as the algebra of compact operators, UHF algebras, and AF algebras have yielded nice classification results. We study the classes of pseudocompact and pseudomatricial C*-algebras, which are the logical limits of finite-dimensional C*-algebras and matrix algebras, respectively, using the continuous model theory for metric structures. To better understand ultraproducts, we discuss when elements in ultraproducts have nice representative sequences, as well the relationship between the spectrum of an element in an ultraproduct and the spectra of the components. We introduce a notion of an ultralimit of sets, give some useful containments, and a nice way to calculate spectra for normal elements with a normal representative sequence. We study how Lin's Theorem is related to finding normal representative sequences for normal elements in an ultraproduct. We show that all the algebras of compact operators on infinite-dimensional Hilbert spaces are elementarily equivalent. We summarize work of Henson/Moore and Eagle/Vignati classifying the commutative pseudocompact C*-algebras and give an explicit axiomatization of this class. We show that the unitization of the compact operators is not pseudocompact, UHF algebras are never pseudocompact. We show that the AF algebras and pseudocompacts are distinct. We show that direct sums, matrix amplifications, corners and centers of pseudocompact C*-algebras are pseudocompact, but subalgebras and quotients of pseudocompact C*-algebras need not be pseudocompact. We show pseudocompact C*-algebras have trivial K_1 groups. We show that in pseudocompact C*-algebras every non-zero projection dominates a minimal projection, and modulo minimal projections every projection can be split into approximate fractions. We show that pseudocompact C*-algebras have the Dixmier property. We show that the pseudomatricial C*-algebras are the pseudocompact C*-algebras with trivial centers and a unique tracial state. We show there are uncountably many isomorphism classes of separable pseudomatricial C*-algebras. We show that all projections in a pseudomatricial C*-algebra are comparable, that all notions of equivalence coincide, and pseudomatricial C*-algebras have strict comparison of projections. We show that infinite-dimensional pseudomatricial C*-algebras are never simple. We give some idea of what the K_0 groups of pseudomatricial C*-algebras look like, and show that separable infinite-dimensional C*-algebras can have many Archimedean classes of projections. We also show that in pseudomatricial C*-algebras, an element is a self-commutator if and only if it is self-adjoint and has trace zero. We conclude with questions and ideas for further research.