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 4dim 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 πr(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 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.
","abstract_html":"<p>Rank constrained homotopies of matrices:</p> <p>For any <em>n ≥ k ≥ l</em> ∈ N, let <em>S</em>(<em> n,k,l</em>) be the set of all non-negative definite matrices <em>a</em> ∈ <em>Mn</em>(C) with <em>l</em> ≤ rank <em>a ≤ k</em>. We investigate homotopy equivalence of continuous maps from a compact Hausdorff space <em>X</em> into sets of the form <em>S</em>(<em>n,k,l</em>). From [37] it is known that for any <em>n</em>, if 4<em>dim X</em> ≤ <em>k-l</em> where <em>dim X</em> denote the covering dimension of <em>X</em>, then there is exactly one homotopy class of maps from <em>X</em> into <em>S</em>(<em>n,k,l</em>). In Section 3.1 we improve this bound by a factor of 8 by confirming <em>C</em>(<em>X,S</em>(<em> n,k,l</em>)) to have exactly one homotopy class of maps when [floor bracket] (<em>dim X</em>/2[end floor bracket] ≤ <em>k - l</em>.) This in particular means π<em>r</em>(<em>S</em>(<em> n,k,l</em>))=0</p> <p>Blackadar-Handelman conjectures on <em>C</em>*-algebras:</p> <p>Let <em>DF</em>(<em>A</em>) denote the set of all dimension functions on a <em>C</em>*-algebra <em>A</em> and let <em>LDF</em>(<em>A</em>) be the set of all <em>s</em> ∈ <em>DF</em>(<em>A</em>) which are lower semicontinuous. It is well known that <em>DF</em>(<em>A</em>) is naturally identified with the state space of the Cuntz semigroup <em>W</em>(<em>A</em>). From [6], <em>LDF</em>(<em>A</em>) bijectively corresponds to the space of all normalized quasitraces <em>QT</em>(<em>A</em>) through a continuous affine map. [6] conjectures <em>LDF</em>(<em> A</em>) to be pointwise dense in <em>DF</em>(<em>A</em>) and <em>DF</em>(<em>A</em>) to be a Choquet simplex.</p> <p>In Theorem 5.1.1 we provide an equivalent condition for the first of these conjectures for unital <em>A</em>. Applying this condition we confirm the first conjecture for all unital <em>A</em> for which either the radius of comparison is finite or the semigroup <em>W</em>(<em> A</em>) 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 <em>LDF</em>(<em>A</em>) is dense in <em>DF</em>(<em>A</em>) for an unital <em>A</em> that has only finitely many extreme points in <em>QT</em>(<em>A</em>), through a simple application of Krein-Milman Theorem we note that <em>DF</em>(<em>A</em>)=<em>LDF</em>(<em>A</em>) and that <em>DF</em>(<em>A</em>) is affinely homeomorphic to <em>QT</em>(<em>A</em>). Together with results on the first conjecture this confirms the second conjecture for a new class of <em>C</em>*-algebras.</p> <p>Possibility of extending these results to inductive limits remain an open question.</p> <p>In general the second conjecture is true for any unital <em>A</em> for which (ordered) Grothendieck group <em>K</em>0(<em> A</em>) of <em>W</em>(<em>A</em>) has Riesz interpolation property [15] and every known confirmation of the second conjecture is achieved by showing Riesz interpolation hold for <em>K</em>0(<em> A</em>) [1,9,29]. We consider a <em>stably approximate</em> 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 <em>asymptotic interpolation</em> property considered in [28]. In Corollary 6.4.3 we confirm <em>DF</em>(<em>A</em>) to be a Choquet simplex for any unital <em>A</em> for which <em>W</em>(<em>A</em>) satisfies this weaker notion of interpolation.</p> <p>While Corollary 6.4.3 has the scope of confirming the second conjecture for a broader class of <em>C</em>*-algebras, finding a `good' class of <em>C</em>*-algebras in which <em>W</em>(<em>A</em>) exhibits stably approximate interpolation but does not satisfy Riesz interpolation remains open.</p>","abstract_has_math":true,"creators":["Silva, Kaushika De"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Thesis","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Andrew S. Toms","Lawrence G. Brown","Marius Dadarlat","David B. McReynolds"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-04-01T07:00:00Z","date_published":"2016-04-01T07:00:00Z","updated_at":"2026-07-24T03:53:47Z","subjects":["Pure sciences","Blackadar-Handelman","Rank constrained homotopy","Stably-approximate interploation","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://docs.lib.purdue.edu/open_access_dissertations/639","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Andrew S. Toms","Lawrence G. Brown","Marius Dadarlat","David B. McReynolds"]},{"key":"dc:creator","label":"Author","values":["Silva, Kaushika De"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Blackadar-Handelman","Rank constrained homotopy","Stably-approximate interploation","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://docs.lib.purdue.edu/open_access_dissertations/639"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 4dim 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 πr(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 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.
"]},{"key":"dc:title","label":"Title","values":["Rank constrained homotopies of matrices and the Blackadar-Handelman conjectures on C*-algebras"]}]}],"canonical_facts":{"dc:contributor":["Andrew S. Toms","Lawrence G. Brown","Marius Dadarlat","David B. McReynolds"],"dc:creator":["Silva, Kaushika De"],"dc:description.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 4dim 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 πr(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 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.
"],"dc:identifier":["https://docs.lib.purdue.edu/open_access_dissertations/639"],"dc:subject":["Pure sciences","Blackadar-Handelman","Rank constrained homotopy","Stably-approximate interploation","Mathematics"],"dc:title":["Rank constrained homotopies of matrices and the Blackadar-Handelman conjectures on C*-algebras"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T03:53:47Z"}