SCIENTIFIC PROGRAMS AND ACTIVITIES

October 31, 2014

Operator Algebras Seminars
July 2010 - June 2011

Seminars are generally held every Tuesday and Thursday at 2pm in Room 210. For more information about this program please contact George Elliott

UPCOMING SEMINARS

June 23, 2011
Working Seminar

 

PAST SEMINARS

June 21, 2011 - Bahen 6183
Henning Petzka
The Corona Factorization Property, III
The Corona Factorization Property (CFP) is a regularity property for C*-algebras. It arose from the study of KK-theory and extensions, but also turned out to be closely connected to stability properties of C*-algebras. I will give an overview of known results and open questions concerning the CFP.

June 16, 2011 Working Seminar

June 14, 2011
Zhiqiang Li
On his most recent work

June 9, 2011 Working Seminar

June 7, 2011 Working Seminar

June 2, 2011 Working Seminar

May 31, 2011 Working Seminar

May 26, 2011 Working Seminar

May 24, 2011 Working Seminar

May 19, 2011 -- 2:10pm
Snigdhayan Mahanta
Twisted K-theory and K-homology of some infinite dimensional spaces
In the sigma model formalism of string theory twisted K-theory is known to classify the D-brane charges in the presence of a nontrivial flux. The aim of this talk is to explain how to use the machinery of noncommutative geometry to study the twisted K-theory of some infinite dimensional spaces like SU(\infty).We are also going to discuss the dual K-homology theory, which has some physical importance, and construct the corresponding dual Chern-Connes character.

May 17 --2:10pm, Room 210,
Aaron Tikuisis
The real span of a dimension group
A dimension group is a type of ordered group which arises, for example, as the $K_0$-group of an AF algebra (and more generally, of many real rank zero $C^*$-algebras). They are well-studied and have been characterized axiomatically by Effros, Handelman, and Shen. However, it remains a difficult problem in practice to determine whether a given ordered group is a dimension group. A systematic approach to this problem taken by myself and G. Maloney centres around embedding the dimension group into a real vector space. I will discuss what properties one would want from such an embedding, and our main result: that in the finite rank case, an embedding with these desired properties exists. Some of the key steps in proving this main result can be proven using operator theory, in the case that the dimension group $G$ is the $K_0$-group of an AF algebra.

Henning Petzka
On certain multiplier projections,
We consider the stabilization of the commutative C*-algebra C(X) with compact base space X. Let X be finite dimensional and p a projection in the multiplier algebra of the stabilized algebra, such that at each point x in X, p(x) is non-compact. Then a result by Pimsner, Popa and Voiculescu implies that the projection p must be full in the multiplier algebra. I will explain why this is not always the case if we allow X to be infinite dimensional.

 

May 12, Working Seminar

May 9, 2011
Aaron Tikuisis
Strongly self-absorbing C*-algebras and the Elliott intertwining argument
An important piece of the theory of Z-stable C*-algebras is the following result characterizing when an algebra is Z-stable: a separable C*-algebra A is Z-stable if and only if there exists a *-homomorphism from A \otimes Z to the asymptotic sequence algebra of A (that is, the algebra of bounded sequences in A modulo the ideal of sequences that converge to 0), such that a \otimes 1 maps to the class of the constant sequence (a,a,a,...), for all a in A. This result is due to Toms and Winter, although it closely resembles a result by Rordam which assumes that A is unital. I will discuss this result and its proof, which relies on a particular manifestation of the Elliott intertwining argument.

May 6, 2011 -- Working Seminar

May 3, 2011
Zhiquiang Li
Calculation of K-theory with coefficients for general dimension drop algebras

April 28, 2011
Henning Petzka
The Corona Factorization Property II
The Corona Factorization Property (CFP) is a regularity property for C*-algebras. It arose from the study of KK-theory and extensions, but also turned out to be closely connected to stability properties of C*-algebras. I will give an overview of known results and open questions concerning the CFP.

April 26, 2011
Henning Petzka
The Corona Factorization Property.
The Corona Factorization Property (CFP) is a regularity property for C*-algebras. It arose from the study of KK-theory and extensions, but also turned out to be closely connected to stability properties of C*-algebras. I will give an overview of known results and open questions concerning the CFP.

April 21, 2011 Working Seminar

Moved to Bahen Centre Room 3000
April 14, 2011 -- Zhiqiang Li
K-theoretic classification for certain Z_p actions on AF algebras.
A K-theoretic classification is given of the C*-dynamical systems lim(A_n, \alpha_n, Z_p) where A_n is finite dimensional and p is a prime. Corresponding to the trivial action is the K-theoretic classification for AF algebras obtained in [Ell]

April 12, 2011 Working Seminar

April 7, 2011
Pekka Salmi (Fields and Waterloo)
Idempotent states on quantum groups
An idempotent state on a locally compact group is just a probability measure that is an idempotent with respect to the convolution. The Kawada--It\^o theorem characterizes such idempotent states as the normalised Haar measures of compact subgroups. On the dual side, idempotents states on group C*-algebras are characteristic functions of open subgroups. In this talk we consider idempotent states on coamenable locally compact \emph{quantum} groups and discuss the connections between idempotent states, quantum subgroups and invariant C*-subalgebras. This is joint work with Adam Skalski.

April 5, 2011 Working Seminar

March 31, 2011
Farzad Fathizadeh
A Noncommutative Residue for Pseudodi?erential Operators on the Noncommutative Two Torus

I will first explain a pseudodifierential calculus for the canonical dynamical system associated to the noncommutative two torus which is a special case of Connes’ pseudodifierential calculus for C*-dynamical systems. Then I will introduce a noncommutative residue for classical pseudodifierential operators on the noncommutative two torus, and prove that up to a constant multiple it is the unique trace on the algebra of classical pseudodifierential operators modulo infinitely smoothing operators. This is joint work with M. W. Wong.

March 29, 2011 Working Seminar

March 24, 2011 Working Seminar

March 22, 2011 Working Seminar

March 17, 2011 Working Seminar

March 15, 2011 Working Seminar

March 10, 2011
Adam Sierakowski
On his recent work

March 8, 2011 Working Seminar

March 3, 2011 Working Seminar

March 1, 2011 Working Seminar

February 24, 2011
Henning Petzka
A non-full extension of C(X,K)

February 22, 2011 Working Seminar

February 17, 2011
Eugene Ha
Bost-Connes systems characterize number fields
Given a number field K, one can construct a C*-algebra together with a time evolution and an action of the abelianized absolute Galois group of K, in a manner generalizing the Bost-Connes C*-dynamical system (1995). We shall try to give an overview of the proof of a theorem of Cornelissen and Marcolli (2010) which states that two number fields are isomorphic if their Bost-Connes systems are isomorphic. A theorem proved by Neukirch (1969) and Uchida (1976) states that two number fields are isomorphic if their absolute Galois groups are isomorphic. It is known, however, that neither the abelianzed Galois group nor the Dedekind zeta function nor the adele ring are strong enough to characterize number fields (up to isomorphism); therefore the theorem of Cornelissen-Marcolli is remarkable, because it is precisely these ingredients alone that go into the construction of the Bost-Connes system.

February 15, 2011 Working Seminar

February 10, 2011
Farzad FathiZadeh
The Gauss-Bonnet Theorem for Noncommutative Two Tori. cont.
Connes and Tretkoff recently gave a spectral formulation and a proof of the Gauss-Bonnet Theorem for the noncommutative two torus T^2_? endowed with its canonical conformal structure. In this talk, I will explain a recent joint work with M. Khalkhali, in which we extend this result to the noncommutative two torus equipped with an arbitrary translation invariant conformal structure. Namely, we compute the value at the origin, ?(0), of the spectral zeta function of the Laplacian attached to a complex number ? in the upper half plane representing the conformal class of a metric on T^2_?, and a Weyl factor given by a positive invertible smooth element k ? C^\infty(T^2_?). This value turns out to be independent of ? and k, thus the desired conformal invariance is shown in the general case.

February 8, 2011
Farzad FathiZadeh
The Gauss-Bonnet Theorem for Noncommutative Two Tori.
Connes and Tretkoff recently gave a spectral formulation and a proof of the Gauss-Bonnet Theorem for the noncommutative two torus T^2_? endowed with its canonical conformal structure. In this talk, I will explain a recent joint work with M. Khalkhali, in which we extend this result to the noncommutative two torus equipped with an arbitrary translation invariant conformal structure. Namely, we compute the value at the origin, ?(0), of the spectral zeta function of the Laplacian attached to a complex number ? in the upper half plane representing the conformal class of a metric on T^2_?, and a Weyl factor given by a positive invertible smooth element k ? C^\infty(T^2_?). This value turns out to be independent of ? and k, thus the desired conformal invariance is shown in the general case.

February 3, 2011
Henning Petzka
On his recent work

January 25, 2011 Working Seminar

January 20, 2011 Working Seminar

January 18, 2011
Adam Sierakowski
Exactness of free groups
Using the notion of amenability of C*-dynamical systems I will present a (well know) proof of the fact, that the free groups are exact.

January 13, 2011 Working Seminar

January 11, 2011
Greg Maloney
Finite rank lattice ordered groups
Recently, Aaron Tikuisis and I classified the finite rank ordered abelian groups that have Riesz interpolation. Now I will discuss a refinement of this classification scheme to the class of finite rank lattice ordered groups.

January 6, 2011
Aaron Tikuisis
Slow dimension growth (continued)
When a C*-algebra can be written as an inductive limit of subhomogeneous algebras, it is natural to expect that regularity properties of the building blocks are reflected in regularity properties of the limit algebra. To put this more concretely, note that regularity properties are ways of generalizing the notion of "low topological dimension" to noncommutative situations. Slow dimension growth for simple ASH algebras means that, among the building blocks, the topological dimension is low as compared to the matricial dimension. For unital simple ASH algebras, slow dimension growth is a well developed notion, equivalent to a number of other properties including Z-stability. I will continue to review these positive results for unital simple ASH algebras, and initiate an exploration of what the correct notion should be for nonunital simple ASH algebras.

This talk will contain new developments in light of a result by Blackadar and Cuntz pointed out by George in the first talk.

January 4, 2011
Aaron Tikuisis
Dimension growth for simple, approximately subhomogeneous algebras
When a C*-algebra can be written as an inductive limit of subhomogeneous algebras, it is natural to expect that regularity properties of the building blocks are reflected in regularity properties of the limit algebra. To put this more concretely, note that regularity properties are ways of generalizing the notion of "low topological dimension" to noncommutative situations. Slow dimension growth for simple ASH algebras means that, among the building blocks, the topological dimension is low as compared to the matricial dimension. For unital simple ASH algebras, slow dimension growth is a well developed notion, equivalent to a number of other properties including Z-stability. I will review these positive results for unital simple ASH algebras, and initiate an exploration of what the correct notion should be for nonunital simple ASH algebras.\

December 16, 2010
Working Seminar

December 14, 2010
Aaron Tikuisis
Ideals in AF algebras
In this talk, I will review the well-known relationship between AF algebras and Bratteli diagrams, with particular emphasis on ideals in AF algebras. We will have in mind the application of determining the Cuntz below relation on elements of C(X,A) where A is a (not necessarily simple) AF algebra.

December 9, 2010
Adam Skalski
How noncommutative is noncommutative topological entropy?
Slides from the Talk
The notion of noncommutative topological entropy for automorphisms of (nuclear) C*-algebras was introduced in 1995 by D. Voiculescu as a generalisation of the topological entropy for continuous transformations of compact spaces. Most methods of computing the Voiculescu entropy are related to finding suitable commutative subsystems of noncommutative dynamical systems, which suggests a straightforward relation between the classical and quantum case. In this talk, we will explain some of the properties of the Voiculescu entropy and present recent examples related to bitstream shifts (studied by S. Neshveyev and E. Stormer) and to endomorphisms of Cuntz algebras. They show that the connections between the commutative and noncommutative case are actually quite subtle. Finally we will discuss the general problem of finding commutative subsystems of a given quantum dynamical system. Parts of the talk are based on the joint work with Jeong Hee Hong, Wojciech Szyma?ski and Joachim Zacharias.

December 7, 2010
Adam Sierakowski
Strongly purely infinite C*-algebras

December 2, 2010
Working Seminar

November 30, 2010
Henning Petzka
On his recent work

November 25, 2010
Zhiqiang Li
On inductive limits built on splitting interval algebras with dimension drops: the ideal property case.
Ideal property is a generalization of simplicity and real rank zero. Among non-simple C*-algebras, C*-algebras with ideal property are important cases. In this talk, we are going to classify the inductive limits with ideal property built on splitting interval algebras with dimension drops.

November 23, 2010 Working Seminar

November 18, 2010
Barry Rowe Transitive Algebras

November 16, 2010 Working Seminar

November 11, 2010
Aaron Tikuisis
The Cuntz semigroup of a mapping torus over AF algebras
Interesting algebras, such as Rordam-Winter's generalized dimension drop algebras Z_{p,q}, arise using the mapping torus construction over AF algebras. I will demonstrate how to compute the Cuntz semigroup of algebras thus constructed. A key step in the computation is to show that such algebras can be written as an inductive limit of mapping tori over finite dimensional algebras (A.K.A. one-dimensional NCCW complexes), such that the connecting maps preserve the mapping torus structure. This also shows that, under suitable K_1 vanishing conditions, a mapping torus over AF algebras is in the class covered by Leonel Robert's recent classification theorem involving the invariant Cu~ (which can essentially be computed from the Cuntz semigroup of the unitization). Combining the Cuntz semigroup computations with Robert's classification results yields tools to understand homomorphisms between mapping tori over AF algebras.

November 9, 2010
Luis Santiago Moreno
The Cuntz semigrop of certain tensor products and pullbacks of C*-algebras (joint work with R. Antoine, F. Perera)

November 4, 2010 Working Seminar

November 2, 2010
Eugene Ha (Fields)
On Tate's trace
Using traces of operators on infinite-dimensional vector spaces of adeles, Tate gave a definition of residues of differentials on curves and derived the classical residue theorem from the finite dimensionality of cohomology. In this expository talk, we will give an overview of Tate's trace, following Tate (1968) and Arabarello-de Concini-Kac (1989).

October 28, 2010 Working Seminar

October 26, 2010 Working Seminar

October 21, 2010 Working Seminar

October 19, 2010 Working Seminar

October 14, 2010
Aaron Tikuisis
A stationary inductive limit construction of a stably projectionless self-absorbing algebra
In very recent work, Jacelon has brought attention to a certain stably projectionless C*-algebra R. This algebra has certain properties that suggest it may be considered a non-unital strongly self-absorbing C*-algebra (although at present, the definition of a strongly self-absorbing C*-algebra is only agreed upon in the unital case), and therefore it may play an important role in the classification of (stably projectionless) C*-algebras. I will discuss this algebra and show how to present it as a stationary inductive limit of certain 1-dimensional "ultra"-noncommutative CW complexes, closely resembling a construction by Rordam and Winter of the Jiang-Su algebra.

October 12, 2010 Working Seminar

October 7, 2010
Zhiqiang Li
A classification of certain simple ASH-algebra over an interval with two singular endpoints cont.
Let $A(\phi_0,\phi_1)$ be K.Thomsen's building block of ASH-algebra, which contains splitting interval algebra and dimension drop algerbra. In the talk, under some assumption of the two homomorphisms, we are going to classify the simple inductive limit of finite direct sums of this building block.

October 5, 2010
Zhiqiang Li
A classification of certain simple ASH-algebra over an interval with two singular endpoints.
Let $A(\phi_0,\phi_1)$ be K.Thomsen's building block of ASH-algebra, which contains splitting interval algebra and dimension drop algerbra. In the talk, under some assumption of the two homomorphisms, we are going to classify the simple inductive limit of finite direct sums of this building block.

September 30, 2010
Volker Runde
Amenability of operator algebras on Banach spaces

September 28, 2010
Adam Sierakowski (Fields)
Purely infinite C*-algebras arising from crossed products (part IV)
This will be the fourth of a series of talks reporting new progress made in the study of purely infinite C*-algebras arising from crossed products by M. Roerdam and A. Sierakowski. We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions for example can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

September 16, 2010
Working Seminar

September 14, 2010
Adam Sierakowski (Fields)
Purely infinite C*-algebras arising from crossed products (part III)
This will be the third of a series of talks reporting new progress made in the study of purely infinite C*-algebras arising from crossed products by M. Roerdam and A. Sierakowski. We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions for example can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

September 9, 2010
Working Seminar

September 7, 2010
Working Seminar

September 2, 2010
Working Seminar

August 31, 2010
Working Seminar

August 26, 2010
Working Seminar

August 24, 2010
Aaron Tikuisis
Cu(C(T, . )) as an invariant for C*-algebras
Brown, Perera and Toms (and Elliott, Robert and Santiago in the non-unital case) demonstrated that for simple, Z-stable, finite C*-algebras, the Cuntz semigroup contains the same information as the Elliott invariant minus the K_1 group. I will show that the Cuntz semigroup of C(T,A) (where T is the circle) contains the same information as the Elliott invariant of A, when A is a simple unital ASH algebra with slow dimension growth. This is a fairly general result, as there are no known examples of simple nuclear unital C*-algebras that are not ASH, and conjecturally, every simple, nuclear, finite, Z-stable unital C*-algebra is an ASH algebra with slow dimension growth. I hope to discuss some elements of the proof, including the role played by Riesz interpolation in the Cuntz semigroup of A.

August 19, 2010
Pinar Colak
Leavitt path algebras and their interplay with graph C*-algebras
Leavitt path algebras are a natural generalization of the Leavitt algebras, which are a class of algebras introduced by Leavitt in 1962. For a directed graph E, the Leavitt path algebra L_K(E) of E with coefficients in K has re- ceived much recent attention both from algebraists and analysts over the last decade, due to the fact that they have some immediate structural connections with graph C*-algebras. So far, some of the algebraic properties of Leavitt path algebras have been investigated, including primitivity, simplicity and being Noetherian. We explicitly describe two-sided ideals in Leavitt path algebras associated with an arbitrary graph. Our main result is that any two-sided ideal I of a Leavitt path algebra associated with an arbitrary directed graph is generated by elements of the form (v+\sum_{i=1}^n\lambda_i g_i)(1-\sum_{e\in S ee^*) where g is a cycle based at vertex v, and S is a finite subset of s^{-1}(v). We show that this result can be used to unify and simplify many known results for Leavitt path algebras some of which have been proven by utilizing established methodologies from C*-algebras. Moreover, we discuss the methods to carry this theorem into the graph C*-algebra setting.

August 17, 2010
Eugene Ha
C*-Algebras for Arakelov line bundles cont.
The cohomology groups of a line bundle on a curve are modules over the field of constants. In the arithmetic case, that is, for Arakelov line bundles, the situation is less satisfactory: it is not known what kind of algebraic structure should play the role of "cohomology groups," though it is nonetheless possible to define of their "absolute" dimension in a manner compatible with the arithmetic Riemann-Roch theorem. In this talk, I will discuss a natural construction of C*-algebras that may be regarded as the "group" C*-algebras for the cohomology of an Arakelov line bundle.

August 12, 2010
Greg Maloney
Ordered vector spaces with interpolation and their subgroups
Aaron Tikuisis and I have been working jointly on a project to find a canonical description of the finite dimensional ordered real vector spaces that have Riesz interpolation. This project is almost finished, and we are now applying these results to the study of countable dimension groups. I will outline some progress that we have made on the following question: If V is an ordered real vector space with interpolation that is expressed in canonical form and G is a subgroup of V, what conditions are necessary and sufficient for G to have interpolation? My talk will only cover the case where V is separated by its states, although we hope to extend these results to the general case.

August 10, 2010
Eugene Ha
C*-Algebras for Arakelov line bundles cont.
The cohomology groups of a line bundle on a curve are modules over the field of constants. In the arithmetic case, that is, for Arakelov line bundles, the situation is less satisfactory: it is not known what kind of algebraic structure should play the role of "cohomology groups," though it is nonetheless possible to define of their "absolute" dimension in a manner compatible with the arithmetic Riemann-Roch theorem. In this talk, I will discuss a natural construction of C*-algebras that may be regarded as the "group" C*-algebras for the cohomology of an Arakelov line bundle.

August 6, 2010
Eugene Ha
C*-Algebras for Arakelov line bundles
The cohomology groups of a line bundle on a curve are modules over the field of constants. In the arithmetic case, that is, for Arakelov line bundles, the situation is less satisfactory: it is not known what kind of algebraic structure should play the role of "cohomology groups," though it is nonetheless possible to define of their "absolute" dimension in a manner compatible with the arithmetic Riemann-Roch theorem. In this talk, I will discuss a natural construction of C*-algebras that may be regarded as the "group" C*-algebras for the cohomology of an Arakelov line bundle.

August 3, 2010
Abhijnan Rej
Periods, geometry and arithmetic in quantum fields and strings
I shall report on my doctoral dissertation work on the appearance of periods (in the sense of Kontsevich--Zagier) in calculations of Feynman integrals in quantum field theory through the study of certain classes of algebraic varieties naturally associated to finite graphs. I shall also talk about the Landscape of string vacua, certian moduli spaces appearing in their study and some computation--theoretic results related to them.

July 27, 2010
Ping Wong Ng
On his most recent work

July 22, 2010
Aaron Tikuisis
Describing the Cuntz semigroup of C(X,Z)
In a previous talk, I discussed a characterization of Cuntz comparison for positive elements of C(X,A) where A is a simple unital ASH algebra with no dimension growth. We find that the Cuntz class of such an element is determined by certain codified data. In order to obtain a description of the Cuntz semigroup of C(X,A), what's missing is a description of what Cuntz classes arise (in terms of the codified data). In this talk, we shall look at how to prove such a data attainment result for the situation where A is the Jiang-Su algebra Z. Part of the reason that we are able to prove this for the Jiang-Su algebra, but not for the same generality of algebras as in the Cuntz comparison result, is that the Cuntz semigroup of Z is totally ordered, which is a rarely occurring property of the Cuntz semigroup.

July 20, 2010
Working Seminar

July 15, 2010
Adam Sierakowski
Purely infinite C*-algebras arising from crossed products (part II)
This will be the second talk of a series of talks reporting new progress made in the study of purely infinite C*-algebras arising from crossed products by M. Roerdam and A. Sierakowski. We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions for example can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

July 13, 2010
Working Seminar

July 8, 2010
Aaron Tikuisis
Cuntz comparison in C_0(X,ASH)
For certain simple ASH algebras A, we can find explicitly when two positive elements of C_0(X,A \otimes K) are Cuntz comparable. I will explain this result in detail and discuss the proof.

July 6, 2010
Working Seminar

 

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