Wellesley College, Wellesley, MA
September 29-30, 2007
September 29-30, 2007
Program & Schedule
All talks will take place in room 277 of the Science Center at Wellesley College.
Title: The topology of tiling spaces
Abstract: Tilings have been studied for a long time due to their fascinating beauty and symmetry properties. However it is only during the last fifteen years that topology has been introduced to investigate some of their global properties. This review talk will introduced the notion of "Hull" or "Tiling Space" and will focus on an important subclass of tilings that are "repetitive" and have "finite local complexity" (FLC). It will describe various topological invariant such as its C*-algebra, its K-group, and various equivalent definition of its cohomology. It will provide the latest results obtained to compute such invariants and give the so-called "gap labeling theorem" in full generality.
Paramita Das, Vanderbilt University
Title: The planar algebra of the group-type subfactors of Bisch and Haagerup
Abstract: We describe the planar algebra, or equivalently, the standard invariant, of a family of subfactors introduced by Bisch and Haagerup some 10 years ago. These subfactors play an important role in the theory since they provide a very simple mechanism to construct irreducible subfactors whose standard invariant has infinite depth. This is joint work with Dietmar Bisch and Shamindra Ghosh.
Ken Dykema, Texas A&M University
Title: On Horn's inequalities and Connes' embedding problem
Abstract: Connes' embedding problem asks whether every separable II1-factor can be embedded in the ultrapower of the hyperfinite II1-factor; this is equivalent to asking whether every finite set in every II1-factor has microstates. We relate this to questions concerning the possible spectral distributions of a+b, where a and b are self-adjoint elements in a II1-factor having given spectral distributions. The finite-dimensional version of the spectral distribution question was solved by Klyatchko, Totaro, Knudson and Tao, in terms of inequalities first formulated by Horn.
Søren Eilers, University of Copenhagen
Title: From substitutions to tiling spaces and C*-algebras
Abstract: A (rather complicated) computation of the K-groups of the C*-algebras associated by work of Matsumoto to substitutional dynamical systems leads to a new and computable invariant for flow equivalence of such systems. The description of this invariant, obtained in joint work with Carlsen, as a stationary inductive system involves certain so-called augmented matrices, and the fact that these lead to flow invariants is far from intuitive. However, an alternative approach by Barge and Smith gives substantial input to the understanding of this phenomenon and promises a better understanding of the equivalence relation induced by stable isomorphism of Matsumoto algebras.
Uffe Haagerup, University of Southern Denmark
Title: Solution of the Effros-Ruan conjecture for bilinear forms on C*-algebras
Abstract: In 1991 Effros and Ruan conjectured that a certain Grothendieck type inequality for a bilinear form on a pair of C*-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact, in particular they proved the Effros-Ruan conjecture for pairs of exact C*-algebras. In a recent joint work with Magdalena Musat we prove the Effros - Ruan conjecture for general C*-algebras (and with constant one), i.e. for every jointly completely bounded (jcb) bilinear form u on a pair of C*-algebras A,B there exist states f1 and f2 on A and states g1 and g2 on B, such that
|u(a,b)| ≤ ||u||jcb (f1(aa*)1/2 g1 (b*b)1/2+f2(a*a)1/2 g2(bb*) 1/2).
While the approach by Pisier and Shlyahktenko relied on free probability theory, our proof uses more classical operator algebra methods, namely Tomita Takesaki theory and special properties of the Powers factors of Type IIIλ where 0 < λ < 1.
Cyril Houdayer, Institut de Mathématiques de Jussieu and UCLA
Title: Construction of Type II1 factors with prescribed countable fundamental group
Abstract: We will present a new and rather simple construction of II1 factors with prescribed countable fundamental group. This construction is based on free products (with amalgamation) and proofs rely on Popa's deformation/rigidity techniques.
Takeshi Katsura, University of Toronto
Title: A lifting problem of actions on K-groups of Kirchberg algebras
Abstract: A simple separable nuclear purely infinite C*-algebra satisfying the Universal Coefficient Theorem of the KK-theory is called a (UCT) Kirchberg algebra. Kirchberg and Phillips proved that Kirchberg algebras can be classified by their K-theory. In this talk, I consider the problem of lifting a given action of a finite group on the K-theory of a Kirchberg algebra to that on the Kirchberg algebra. I show that all actions lift when every Sylow subgroup of the finite group is cyclic. As a corollary, we can see that every automorphism of the K-groups of a Kirchberg algebra can be lifted to that of the Kirchberg algebra with the same order. The proof relies on the construction of Kirchberg algebras which is closely related to that of Cuntz-Krieger algebras, and a result on modules over finite groups.
Eberhard Kirchberg, Humboldt University
Title: Some tools for the classification of nonsimple C*-algebras
Magdalena Musat, University of Memphis
Title: Classification of hyperfinite factors up to completely bounded isomorphism of their predual
Abstract: In 1989 Christensen and Sinclair proved that all infinite hyperfinite (= injective) factors with separable preduals are cb-isomorphic (i.e., isomorphic as operator spaces). However, when turning to preduals, the situation is very different. We show that if M and N are hyperfinite factors (on separable Hilbert spaces) such that M is semifinite and N is of type III, then their preduals are not cb-isomorphic. Furthermore, we construct a one-parameter family of hyperfinite type III0 factors with mutually non-cb-isomorphic preduals, and we give a characterization of those hyperfinite factors M whose preduals are cb-isomorphic to the predual of the hyperfinite type III1 factor. This is joint work with Uffe Haagerup.
Jan Spakula, Vanderbilt University
Title: Uniform translation C*-algebras which are not K-exact
Abstract: Uniform translation algebras (also known as uniform Roe algebras) are C*-algebras which reflect the large-scale (coarse) geometry of a discrete metric space. For example, property A of a space is equivalent to nuclearity of its uniform translation algebra. In the talk, the other side of this correspondence is explored: we exhibit spaces (certain expanders), whose uniform translation algebra is not K-exact (hence, not exact, nor nuclear).
Andrew Toms, York University
Title: The Jiang-Su algebra
Abstract: In 1997, Jiang and Su discovered a unital simple amenable infinite-dimensional C*-algebra having the same K-theory and tracial state space as the algebra of complex numbers. Since then, it has become apparent that this algebra, denoted by Z, is connected deeply to the structure theory of separable amenable C*-algebras, and in particular to the classification program of Elliott. This talk will survey research on Z over the past decade, culminating in some recent theorems of Dadarlat, Winter, and me on the uniqueness of Z among approximately subhomogeneous C*-algebras.
Shmuel Weinberger, University of Chicago
Title: L2 Betti-less manifolds
Abstract: I will explain how to compute the cobordism classes of manifolds with free abelian fundamental group whose universal covers have vanishing L2 cohomology. An abstract torus is definable whose algebraic geometric structure is part of this story. If there is time, I will discuss some possible noncommutative generalizations and connections to the Baum-Connes philosophy.
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The following is the schedule of speakers and their lecture titles. Please scroll down to read abstracts.
Conference Schedule:
| Saturday, September 29, 2007 | ||
| 8:00-9:00 | Registration | |
| 9:00-9:55 | Uffe Haagerup | Solution of the Effros-Ruan conjecture for bilinear forms on C*-algebras |
| 10:00-10:25 | Paramita Das | The planar algebra of the group-type subfactors of Bisch and Haagerup |
| 10:30-11:00 | Break | |
| 11:00-11:55 | Ken Dykema | On Horn's inequalities and Connes' embedding problem |
| 12:00-1:30 | Lunch | |
| 1:30-2:25 | Eberhard Kirchberg | Some tools for the classification of nonsimple C*-algebras |
| 2:30-2:55 | Andrew Toms | The Jiang-Su algebra |
| 3:00-3:30 | Break | |
| 3:30-4:25 | Søren Eilers | From substitutions to tiling spaces and C*-algebras |
| 4:30-4:55 | Takeshi Katsura | A lifting problem of actions on K-groups of Kirchberg algebras |
| 5:00-5:25 | Jan Spakula | Uniform translation C*-algebras which are not K-exact |
| Sunday, September 30, 2007 | ||
| 9:00-9:55 | Jean Bellissard | The topology of tiling spaces |
| 10:00-10:25 | Magdalena Musat | Classification of hyperfinite factors up to completely bounded isomorphism of their predual |
| 10:30-11:00 | Break | |
| 11:00-11:55 | Shmuel Weinberger | L2 Betti-less manifolds |
| 12:00-12:25 | Cyril Houdayer | Construction of Type II1 factors with prescribed countable fundamental group |
Available Abstracts:
Jean Bellissard, Georgia Institute of TechnologyTitle: The topology of tiling spaces
Abstract: Tilings have been studied for a long time due to their fascinating beauty and symmetry properties. However it is only during the last fifteen years that topology has been introduced to investigate some of their global properties. This review talk will introduced the notion of "Hull" or "Tiling Space" and will focus on an important subclass of tilings that are "repetitive" and have "finite local complexity" (FLC). It will describe various topological invariant such as its C*-algebra, its K-group, and various equivalent definition of its cohomology. It will provide the latest results obtained to compute such invariants and give the so-called "gap labeling theorem" in full generality.
Paramita Das, Vanderbilt University
Title: The planar algebra of the group-type subfactors of Bisch and Haagerup
Abstract: We describe the planar algebra, or equivalently, the standard invariant, of a family of subfactors introduced by Bisch and Haagerup some 10 years ago. These subfactors play an important role in the theory since they provide a very simple mechanism to construct irreducible subfactors whose standard invariant has infinite depth. This is joint work with Dietmar Bisch and Shamindra Ghosh.
Ken Dykema, Texas A&M University
Title: On Horn's inequalities and Connes' embedding problem
Abstract: Connes' embedding problem asks whether every separable II1-factor can be embedded in the ultrapower of the hyperfinite II1-factor; this is equivalent to asking whether every finite set in every II1-factor has microstates. We relate this to questions concerning the possible spectral distributions of a+b, where a and b are self-adjoint elements in a II1-factor having given spectral distributions. The finite-dimensional version of the spectral distribution question was solved by Klyatchko, Totaro, Knudson and Tao, in terms of inequalities first formulated by Horn.
Søren Eilers, University of Copenhagen
Title: From substitutions to tiling spaces and C*-algebras
Abstract: A (rather complicated) computation of the K-groups of the C*-algebras associated by work of Matsumoto to substitutional dynamical systems leads to a new and computable invariant for flow equivalence of such systems. The description of this invariant, obtained in joint work with Carlsen, as a stationary inductive system involves certain so-called augmented matrices, and the fact that these lead to flow invariants is far from intuitive. However, an alternative approach by Barge and Smith gives substantial input to the understanding of this phenomenon and promises a better understanding of the equivalence relation induced by stable isomorphism of Matsumoto algebras.
Uffe Haagerup, University of Southern Denmark
Title: Solution of the Effros-Ruan conjecture for bilinear forms on C*-algebras
Abstract: In 1991 Effros and Ruan conjectured that a certain Grothendieck type inequality for a bilinear form on a pair of C*-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact, in particular they proved the Effros-Ruan conjecture for pairs of exact C*-algebras. In a recent joint work with Magdalena Musat we prove the Effros - Ruan conjecture for general C*-algebras (and with constant one), i.e. for every jointly completely bounded (jcb) bilinear form u on a pair of C*-algebras A,B there exist states f1 and f2 on A and states g1 and g2 on B, such that
|u(a,b)| ≤ ||u||jcb (f1(aa*)1/2 g1 (b*b)1/2+f2(a*a)1/2 g2(bb*) 1/2).
While the approach by Pisier and Shlyahktenko relied on free probability theory, our proof uses more classical operator algebra methods, namely Tomita Takesaki theory and special properties of the Powers factors of Type IIIλ where 0 < λ < 1.
Cyril Houdayer, Institut de Mathématiques de Jussieu and UCLA
Title: Construction of Type II1 factors with prescribed countable fundamental group
Abstract: We will present a new and rather simple construction of II1 factors with prescribed countable fundamental group. This construction is based on free products (with amalgamation) and proofs rely on Popa's deformation/rigidity techniques.
Takeshi Katsura, University of Toronto
Title: A lifting problem of actions on K-groups of Kirchberg algebras
Abstract: A simple separable nuclear purely infinite C*-algebra satisfying the Universal Coefficient Theorem of the KK-theory is called a (UCT) Kirchberg algebra. Kirchberg and Phillips proved that Kirchberg algebras can be classified by their K-theory. In this talk, I consider the problem of lifting a given action of a finite group on the K-theory of a Kirchberg algebra to that on the Kirchberg algebra. I show that all actions lift when every Sylow subgroup of the finite group is cyclic. As a corollary, we can see that every automorphism of the K-groups of a Kirchberg algebra can be lifted to that of the Kirchberg algebra with the same order. The proof relies on the construction of Kirchberg algebras which is closely related to that of Cuntz-Krieger algebras, and a result on modules over finite groups.
Eberhard Kirchberg, Humboldt University
Title: Some tools for the classification of nonsimple C*-algebras
Magdalena Musat, University of Memphis
Title: Classification of hyperfinite factors up to completely bounded isomorphism of their predual
Abstract: In 1989 Christensen and Sinclair proved that all infinite hyperfinite (= injective) factors with separable preduals are cb-isomorphic (i.e., isomorphic as operator spaces). However, when turning to preduals, the situation is very different. We show that if M and N are hyperfinite factors (on separable Hilbert spaces) such that M is semifinite and N is of type III, then their preduals are not cb-isomorphic. Furthermore, we construct a one-parameter family of hyperfinite type III0 factors with mutually non-cb-isomorphic preduals, and we give a characterization of those hyperfinite factors M whose preduals are cb-isomorphic to the predual of the hyperfinite type III1 factor. This is joint work with Uffe Haagerup.
Jan Spakula, Vanderbilt University
Title: Uniform translation C*-algebras which are not K-exact
Abstract: Uniform translation algebras (also known as uniform Roe algebras) are C*-algebras which reflect the large-scale (coarse) geometry of a discrete metric space. For example, property A of a space is equivalent to nuclearity of its uniform translation algebra. In the talk, the other side of this correspondence is explored: we exhibit spaces (certain expanders), whose uniform translation algebra is not K-exact (hence, not exact, nor nuclear).
Andrew Toms, York University
Title: The Jiang-Su algebra
Abstract: In 1997, Jiang and Su discovered a unital simple amenable infinite-dimensional C*-algebra having the same K-theory and tracial state space as the algebra of complex numbers. Since then, it has become apparent that this algebra, denoted by Z, is connected deeply to the structure theory of separable amenable C*-algebras, and in particular to the classification program of Elliott. This talk will survey research on Z over the past decade, culminating in some recent theorems of Dadarlat, Winter, and me on the uniqueness of Z among approximately subhomogeneous C*-algebras.
Shmuel Weinberger, University of Chicago
Title: L2 Betti-less manifolds
Abstract: I will explain how to compute the cobordism classes of manifolds with free abelian fundamental group whose universal covers have vanishing L2 cohomology. An abstract torus is definable whose algebraic geometric structure is part of this story. If there is time, I will discuss some possible noncommutative generalizations and connections to the Baum-Connes philosophy.