#### Title

Rank constrained homotopies of matrices and the Blackadar-Handelman conjectures on C*-algebras

#### Date of Award

4-2016

#### Degree Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Andrew S. Toms

#### Committee Chair

Andrew S. Toms

#### Committee Member 1

Lawrence G. Brown

#### Committee Member 2

Marius Dadarlat

#### 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 *a* ∈ *Mn*(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 4*dim X* ≤ *k-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 π*r*(*S*(* n,k,l*))=0

Blackadar-Handelman conjectures on *C**-algebras:

Let *DF*(*A*) denote the set of all dimension functions on a *C**-algebra *A* and let *LDF*(*A*) be the set of all *s* ∈ *DF*(*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 *K*0(* 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 *K*0(* 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.

#### Recommended Citation

Silva, Kaushika De, "Rank constrained homotopies of matrices and the Blackadar-Handelman conjectures on C*-algebras" (2016). *Open Access Dissertations*. 639.

https://docs.lib.purdue.edu/open_access_dissertations/639