C+ - Algebras and the Uncountable: A Systematic Study of the Combinatorics of the Uncountable in the Noncommutative Framework
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In this dissertation we investigate nonseparable C-algebras using methods coming from logic, specifically from set theory. The material is divided into three main parts. In the first part we study algebras known as counterexamples to Naimarks problem, namely C-algebras that are not isomorphic to the algebra of compact operators on some Hilbert space, yet still have only one irreducible representation up to unitary equivalence. Such algebras have to be simple, nonseparable and non-type I, and they are known to exist if the diamond principle (a strengthening of the continuum hypothesis) is assumed. With the motivation of finding further characterizations for these counterexamples, we undertake the study of their trace spaces, led by some elementary observations about the unitary action on the state space of these algebras, which seem to suggest that a counterexample to Naimarks problem could have at most one trace. We show that this is not the case and, assuming diamond, we prove that every Choquet simplex with countably many extreme points occurs as the trace space of a counterexample to Naimarks problem and that, moreover, there exists a counterexample whose tracial simplex is nonseparable. The second part of this dissertation revolves around the Calkin algebra (H) and the general problem of what nonseparable C-algebras embed into it. We prove that, under Martins axiom, all C-algebras of density character less than 20 embed into the Calkin algebra. Moving to larger C-algebras, we show that (within ZFC alone) Cred(F20 ) and Cm ax(F20 ), where F20 is the free group on 20 generators, and every nonseparable UHF algebra with density character at most 20 , embed into the Calkin algebra. On the other hand, we prove that it is consistent with ZFC + 20 , for every ordinal 2, that the abelian C-algebra generated by an increasing chain of 2 projections does not embed into Q(H). Hence, the statement Every C-algebra of density character strictly less than 20 embeds into the Calkin algebra is independent from ZFC+ 20 , for every ordinal > 2. Finally, we show that the proof of Voiculescus noncommutative version of the Weyl- von Neumann theorem consists, when looked from the right perspective, of a sequence of applications of the Baire category theorem to certain ccc posets. This allows us, assuming Martins axiom, to generalize Voiculescus results to nonseparable C-algebras of density character less than 20 . The last part of this manuscript concerns lifting of abelian subalgebras of coronas of non-unital C-algebras. Given a subset of commuting elements in a corona algebra, we study what could prevent the existence of a commutative lifting of such subset to the multiplier algebra. While for finite and countable families the only issues arising are of K-theoretic nature, for larger families the size itself becomes an obstruction. We prove in fact, for a primitive, non-unital, -unital C-algebra A, that there exists a set of 1 orthogonal positive elements in the corona of A which cannot be lifted to a collection of commuting elements in the multiplier algebra of A.