Open Access Dissertations

4-2016

Thesis

Degree Name

Doctor of Philosophy (PhD)

Mathematics

Andrew S. Toms

Andrew S. Toms

Committee Member 1

Lawrence G. Brown

Committee Member 3

David B. McReynolds

Abstract

Rank constrained homotopies of matrices:

For any n ≥ k ≥ l ∈ N, let S( n,k,l) be the set of all non-negative definite matrices aMn(C) with l ≤ rank a ≤ k. We investigate homotopy equivalence of continuous maps from a compact Hausdorff space X into sets of the form S(n,k,l). From [37] it is known that for any n, if 4dim Xk-l where dim X denote the covering dimension of X, then there is exactly one homotopy class of maps from X into S(n,k,l). In Section 3.1 we improve this bound by a factor of 8 by confirming C(X,S( n,k,l)) to have exactly one homotopy class of maps when [floor bracket] (dim X/2[end floor bracket] ≤ k - l.) This in particular means πr(S( n,k,l))=0

Let DF(A) denote the set of all dimension functions on a C*-algebra A and let LDF(A) be the set of all sDF(A) which are lower semicontinuous. It is well known that DF(A) is naturally identified with the state space of the Cuntz semigroup W(A). From [6], LDF(A) bijectively corresponds to the space of all normalized quasitraces QT(A) through a continuous affine map. [6] conjectures LDF( A) to be pointwise dense in DF(A) and DF(A) to be a Choquet simplex.

In Theorem 5.1.1 we provide an equivalent condition for the first of these conjectures for unital A. Applying this condition we confirm the first conjecture for all unital A for which either the radius of comparison is finite or the semigroup W( A) is almost unperforated (Theorem 5.2.5). for every $r\leq 2(k-l)+1$. Our results are achieved through applications of the techniques developed in [8] and [33]. If LDF(A) is dense in DF(A) for an unital A that has only finitely many extreme points in QT(A), through a simple application of Krein-Milman Theorem we note that DF(A)=LDF(A) and that DF(A) is affinely homeomorphic to QT(A). Together with results on the first conjecture this confirms the second conjecture for a new class of C*-algebras.

Possibility of extending these results to inductive limits remain an open question.

In general the second conjecture is true for any unital A for which (ordered) Grothendieck group K0( A) of W(A) has Riesz interpolation property [15] and every known confirmation of the second conjecture is achieved by showing Riesz interpolation hold for K0( A) [1,9,29]. We consider a stably approximate version of interpolation that is weaker than the classical Riesz interpolation. In fact it is easily seen that this property is even weaker than the asymptotic interpolation property considered in [28]. In Corollary 6.4.3 we confirm DF(A) to be a Choquet simplex for any unital A for which W(A) satisfies this weaker notion of interpolation.

While Corollary 6.4.3 has the scope of confirming the second conjecture for a broader class of C*-algebras, finding a `good' class of C*-algebras in which W(A) exhibits stably approximate interpolation but does not satisfy Riesz interpolation remains open.

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