THEMATIC PROGRAMS

Seminar Session
Chair: Piotr M. Hajac (Poland)
Speakers: Paweł Kasprzak (University of Warsaw / Polish Academy of Sciences)
Rieffel deformation via crossed products

Adam Skalski (Polish Academy of Sciences / University of Warsaw)
Spectral triples on crossed products by equicontinuous actions

Piotr M. Sołtan (University of Warsaw)
Embeddable quantum homogeneous spaces

Seminar Session
Chair: George Elliott (Toronto)
Speakers: Alan Lai (California Institute of Technology)
Toward  understanding  the space of connections

Karen Strung (University of Muenster)
On the classification of C*-algebras of minimal dynamical systems of a product of the Cantor set and an odd dimensional sphere

Olivier Gabriel (Goettingen University) (slides)
Spectral triples on Lie groups and generalized crossed products

Qingyun Wang (Washington University in St Louis, about to join University of Toronto)
Rokhlin properties and non-commutative dimension

Exact crossed products: a counter-example revisited

The left side of BC (Baum-Connes) with coefficients “sees” any group as if the group were exact. This talk will indicate how to make a change in the right side of BC with coefficients so that the right side also “sees” any group as if the group were exact. This corrected form of BC with coefficients uses the unique minimal exact intermediate crossed-product. For exact groups (i.e., all groups except the Gromov group) there is no change in BC with coefficients. In the corrected form of BC with coefficients the Gromov group acting on the coefficient algebra obtained from an expander is not a counter-example. Thus at the present time (June, 2013) there is no known counter-example to the corrected form of BC with coefficients. The above is joint work with E. Kirchberg and R. Willett. This work is based on — and inspired by — a result of R. Willett and G. Yu.

Dimension theory for C*-algebras

We will survey the various noncommutative versions of dimension for C*-algebras and their applications, beginning with Rieffel's stable rank. Other dimension theories include real rank, tracial rank, decomposition rank, and nuclear dimension, all of which have different applications. Uses of these theories in the structure of approximately homogeneous C*-algebras will be discussed, including approximate divisibility and Z-stability. Finally, we discuss stability questions, the Cuntz semigroup, and radius of comparison.

Towards a reconstruction theorem for toric noncommutative manifolds

A toric noncommutative manifold is a spectral triple obtained from a commutative spectral triple by applying Rieffel's strict deformation quantisation to its algebra. We discuss work in progress towards extending Connes' reconstruction theorem for commutative spectral triples to a reconstruction theorem for toric noncommutative manifolds.

Alan Carey
Australian National University

Lecture Notes

Scattering theory and Noncommutative Geometry

This talk outlines work in progress on connections between index theory and scattering theory.

Heath Emerson
University of Victoria

Lecture Notes

K-theory and the Lefschetz fixedpoint formula

We describe a generalization of the Lefschetz fixed-point formula. The formula equates two invariants of a smooth, G-equivariant self-correspondence of a smooth compact manifold, where G is a compact group. As in the classical formula, one of our invariants is local and geometric and is based on a self-intersection construction, and the other is global and homological, and depends, roughly speaking, only on the R(G)-module trace of the R(G)-module map on equivariant K-theory induced by the correspondence.

Math vs. Biology

I will discuss my scientific journey from being a student of Marc Rieffel to working on genomics, and lessons learned along the way.

The Stokes Groupoids

We construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities. As a consequence, we obtain a direct, functorial method for resumming formal solutions to such equations.

Noncommutative Geometry of Parabolic Induction and Parabolic Restriction

In his influential work from the 1970s, Marc Rieffel explained how unitary induction can be neatly framed within ${C^*}$-algebras by using Hilbert modules and the concept of Morita equivalence. In the last several years, Pierre Clare has begun to study parabolic induction, which is the mainstay of Harish-Chandra-style representation theory, from the same ${C^*}$-algebraic point of view. I shall introduce Pierre’s basic construction, and then consider the problem of framing “parabolic restriction” within ${C^*}$-algebra theory. For tempered representations — roughly speaking, for the reduced rather than the full group ${C^*}$-algebra — one can expect adjunction relations between parabolic induction and restriction. The investigation of these relations leads to some interesting asymptotic geometry — for SL(2,${\mathbb{R}}$) this is the geometry of the wave equation on the hyperbolic plane. An as-yet poorly understood issue is that the constructions involve the smooth structure of the tempered dual, as captured by a smooth subalgebra, and not just the topology, as captured by the reduced ${C^*}$-algebra.

Foundations of the Theory of Murray-von Neumann Algebras

F. J. Murray and J. von Neumann introduced the family of unbounded, closed, densely-defined operators that are closely associated (“affiliated” as they termed them) with a finite von Neumann algebra. They proved that they have remarkable common, dense domain properties and described surprising addition and multiplication operations for them. Zhe Liu and the speaker have defined algebras based on these operations, Murray-von Neumann algebras, and studied their basic structure. This will be discussed during the lecture with special emphasis on the nature of the derivations of these algebras.

Algebraic and Hermitian K-theory of stable algebras

This is joint work with Mariusz Wodzicki. It was conjectured in 1978 that algebraic K-theory and topological K-theory coincide on the category of stable complex ${C^*}$-algebras. This conjecture was proved by Suslin and Wodzicki about 20 years ago. In this lecture we give another proof of the conjecture which may be applied to stable real ${C^*}$-algebras or Banach algebras like the algebra of compact operators in a separable real Hilbert space. We apply our new method to compute also Hermitian K-theory of a large class of operator algebras by a comparison theorem involving the topological analog.

Rieffel deformation via crossed products

The aim of this talk is to present a description of the Rieffel deformation in the crossed product terms. The starting point of our construction is an action of an abelian group G on a C*-algebra A. The crossed product algebra of A by the action of G is canonically equipped with the dual action of the dual group of G. The algebra A can be embedded into this crossed product and it is characterized by the Landstad conditions, one of which is the invariance with respect to the dual action. Using a 2-cocycle on the dual group one can deform the dual action and the Rieffel deformation of A is defined as the Landstad algebra for this deformed dual action. A particular benefits of our approach are immediate proofs of invariance of K-groups and preservation of nuclearity under the Rieffel deformation.

Toward understanding the space of connections

What do people mean when they integrate over a large space like the connection space? Inspired by the formal definition of a path integral, I attempt to give an interpretation on a measure of the connection space in loop quantum gravity literature. I will end with a natural way of interpreting a connection as an operator on a Hilbert space.

Franz Luef
UC Berkeley

Lectures Notes

Noncommutative geometry and time-frequency analysis

In this talk I describe a link between projective modules over noncommutative tori and time-frequency analysis, where they are known as Gabor frames and are of relevance in wireless communication. The main focus will be on properties of Gabor frames that follow from the existence of Hermitian connections on projective modules over noncommutative tori.

Zhiqiang Li
University of Toronto

Lecture Notes

KK-lifting problem and order structure on K-groups

We investigate the KK-lifting problem for ${C^*}$-algebras, namely, the problem which KK-class is representable by a *-homomorphism between the algebras (allowing the tensor product with a matrix algebra for the codomain algebra). This problem not only makes sense in its own right, but also has application to the classification of ${C^*}$-algebras. To be more precise, we look at this problem for dimension drop interval algebras (with possibly different dimension drops at the endpoints). It turns out that there exist KK-elements between two such algebras which preserve the Dadarlat-Loring order on K-theory with coefficients, but can not be lifted to a *-homomorphism between the algebras. This is different from the equal dimension drop case, as shown by S. Eilers.This is a joint work with George A. Elliott.

Bivariant theories for C*-algebras

Bivariant theories are two variable theories which provide an axiomatic framework to study E-theory and (local) cyclic homology theory amongst others. One fundamental example of such a theory is noncommutative stable homotopy, which has received much less attention in the literature. It is a sharper invariant than E-theory (or KK-theory for nuclear C*-algebras) and hence deserves a closer look. Standard constructions view them as Hom-groups of certain triangulated categories. I will demonstrate that they have a higher categorical origin and the noncommutative stable homotopy groups are simply the ordinary homotopy groups of this higher category. This construction will be applied to address some questions on the global aspects of noncommutative stable homotopy.

Classification of actions of finite abelian groups on AI-algebras

In this talk, I will introduce an equivariant version of the Cuntz semigroup. Then I will discuss some of its properties. I will show that if G is a compact group then the equivariant Cuntz semigroup of a G-${C^*}$-algebra is naturally isomorphic to the Cuntz semigroup of the associated crossed product ${C^*}$-algebra. I will also explain how this semigroup can be used to classify actions of finite abelian groups on AI-algebras with the Rokhlin property. This is a joint work with Eusebio Gardella.

Cocycle deformation of operator algebras

Given a ${C^*}$-algebra A with an action of a locally compact quantum group G on it and a unitary 2-cocycle ${\Omega}$ on ${\hat{G}}$, we define a deformation ${A_\Omega}$ of A. We will be particularly interested in the cases when G is either a genuine group or a group dual. The construction behaves well under the regularity assumption on ${\Omega}$, meaning that ${C_0 (G)_\Omega \rtimes G}$ is isomorphic to the algebra of compact operators on some Hilbert space. In particular, then ${A_\Omega}$ is stably isomorphic to the iterated twisted crossed product ${G^{op}\rtimes_\Omega G \rtimes A}$. Also, in good situations, the ${C^*}$-algebra ${A_\Omega}$ carries a left action of the deformed quantum group ${G_\Omega}$ and we have an isomorphism ${G_\Omega \rtimes A_\Omega \cong G \rtimes A}$. As examples we consider Rieffel’s deformation and deformations by cocycles on the duals of some solvable Lie groups recently constructed by Bieliavsky and Gayral. (Joint work with J. Bhowmick, L. Tuset and A. Sangha.)

Complex Vector Bundles over Higher-Dimensional Connes-Landi Spheres

We classify and construct (up to isomorphism) all finitely-generated projective modules over higher-dimensional Connes-Landi spheres for totally irrational values of the deformation parameter.

In January 1997, Marc Rieffel gave a talk at a special session of the Joint Annual Meetings entitled “Multiwavelets and operator algebras”, which related wavelet theory to the K-theory of the (commutative) torus. Rieffel’s talk related the multiresolution analysis theory of wavelets due to S. Mallat and Y.Meyer to a nested sequence of Hilbert modules over continuous functions on the torus, and the theory of projective multi-resolution analyses had its origins here. The talk today will relate some of this theory, as well as discussing some recent developments due to B. Purkis of the University of Colorado, Boulder.

The twisted local index formula is primary

In this talk we introduce a new Hopf algebra with a characteristic map that captures the twisted local index formula on the groupoid action algebra. In contrast with the Connes-Moscovici Hopf algebra the cohomology of this new
Hopf algebra is comprised of all universal Chern classes. This proves that the cyclic cohomology class of the twisted index cocycle is primary. The talk is based on the collaboration with Henri Moscovici.

Noncommutative resistance networks

To avoid the technicalities of unbounded operators and their dense domains, in this talk I will deal only with finite-dimensional C*-algebras. I will introduce what I am calling a Riemannian metric over such an algebra A. When A is commutative I will indicate how we essentially obtain a (finite) resistance network. I will describe interesting non-commutative examples. In particular, in our setting every spectral triple determines a Riemannian metric. I will sketch how from a Riemannian metric we obtain further interesting structures, such as Laplace operators, seminorms equipping A with the structure of a quantum metric space, and corresponding metrics on the state space. These seminorms have surprisingly strong properties. I will also mention how this setup is closely related to Dirichlet forms and quantum semigroups.

The structure of quantum line bundles over quantum teardrops

Over the quantum weighted 1-dimensional complex projective spaces, called quantum teardrops, the quantum line bundles associated with the quantum principal U(1)-bundles introduced and studied by Brzezinski and Fairfax are explicitly identified among the finitely generated projective modules which are classified up to isomorphism.

Spectral triples on crossed products by equicontinuous actions

I will discuss a method of constructing spectral triples on crossed products by actions of discrete groups, inspired by the Kasparov product. A sufficient condition for the method to work, introduced by Jean Belissard, Mathilde Marcolli and Kamran Reihani for actions of Z, turns out to be closely related with the topological equicontinuity of the action, if only the original triple is Lipschitz regular (in the sense of Rieffel). I will also present certain examples and further related problems. (Joint work with Andrew Hawkins, Stuart White and Joachim Zacharias.)

Embeddable quantum homogeneous spaces

I will review some aspect of the theory of noncommutative (or quantum) homogeneous spaces and describe a natural class of such objects which in joint work with P. Kasprzak we called "embeddable" following the original use of this term by Podleś. Along the way I will devote some attention to a von Neumann algebraic version of this theory which exhibits an interesting duality. As an example I will shed some light on the concept of the diagonal subgroup of the direct product of a quantum group with itself.

On the classification of C*-algebras of minimal dynamical systems of a product of the Cantor set and an odd dimensional sphere

Let : ${\beta: S^n \rightarrow S^n}$ be one of the known examples of minimal dynamical systems of n dimensional spheres, n ${\geq}$ 3 odd. For every such (${\beta; S^n}$), there is a Cantor minimal system (X; ${\alpha}$) such that the product system (${X \times S^n; \alpha \times \beta}$) is minimal and such that tracial state space of ${C(S^n) \rtimes_ \beta \mathbb{Z}}$ is preserved in ${C(X \times S^n) \rtimes_{\alpha \times \beta}\mathbb{Z}}$.
I show that ${C(X \times S^n) \rtimes_{\alpha\times\beta}\mathbb{Z}}$ is a tracially approximately interval (TAI) algebra and hence classifiable. Moreover, with forthcoming work of Wilhelm Winter this implies that ${C(Y ) \rtimes_\beta\mathbb{Z}}$ is TAI after tensoring with the universal UHF algebra, showing that such crossed products are classified by their tracial state spaces, as conjectured by N. Christopher Phillips.

Equivariant quantization and K-theory

In the early 90s, Marc Rieffel proved that K-groups of ${C^*}$-algebras are invariant under strict deformations. In this talk, we will explain a generalization of this theorem to the equivariant setting. As an application, this result allows us to compute K-groups of some noncommutative orbifold algebras. (Joint work with Yijun Yao).

Entropy, Determinants, and ${L^2}$-Torsion

This talk is about the entropy of group actions of amenable groups. I will present recent progress on questions asked by Christopher Deninger about the entropy of certain principal algebraic dynamical systems. I will show that the entropy of an algebraic dynamical system agrees with the ${L^2}$-torsion of the dual module over the integral group ring of the group acting. As a by-product we prove vanishing of the ${L^2}$-torsion of amenable groups, which was conjectured by Wolfgang Luck. This is joint work with Hanfeng Li.

Alfons Van Daele
University of Leuven, Belgium

Lecture Slides

The Larson-Sweedler theorem and the operator algebra approach to quantum groups

The Larson-Sweedler theorem says that a bialgebra is a Hopf algebra if there exist a left and a right integral. More precisely, let A be a unital algebra (say over the field of complex numbers) with a coproduct ${\Delta : A \rightarrow A \bigotimes A}$ and a counit ${\varepsilon : A \rightarrow \mathbb{C}}$. If there exist non-zero linear functionals ${\varphi}$ and ${\psi}$ on ${A}$ satisfying ${(\iota \bigotimes \varphi) \Delta (a) = \varphi (a) 1}$ and ${(\psi \bigotimes \iota) \Delta (a)= \psi(a)1}$ for all ${a \in A}$ (where ${\iota}$ is the identity map on ${A}$), then there is an antipode on ${A}$ and (${A, \Delta}$) is a Hopf algebra. Compare this result with the notion of a locally compact quantum group (in the von Neumann algebra setting). Given is a pair ${(M,\Delta)}$ of a von Neumann algebra M and a coproduct ${\Delta: M \rightarrow M \bigotimes M}$ (where now the von Neumann algebraic tensor product is considered). If there exist a left and a right Haar weight ${\varphi}$ and ${\psi}$ on M, then ${(M,\Delta)}$ is a locally compact quantum group. The key result in the theory of locally compact quantum groups is the construction of the antipode from these axioms. Then the similarity between this and the Larson-Sweedler theorem for Hopf algebras is clear. We will mainly talk about this connection. But at the end of the talk, we will briefly indicate how the same link pops up in the more recent work on quantum groupoids (joint work with B.-J. Kahng).

Some fundamental problems in the index theory of non-elliptic Fredholm operators.

In the investigation of index formulas for non-elliptic Fredholm operators, interesting new phenomena appear that are not apparent in elliptic theory. In testing the boundaries of known approaches to index problems, fundamental problems arise when we try to extend the theory to more general Fredholm operators. I want to discuss some "negative results" that may be of philosophical interest, partly in the hope that a clear exposition of some of these issues might elicit new ideas from the audience.

Rokhlin properties and non-commutative dimension

Given a C*-dynamic system ${\alpha: G \rightarrow Aut(A)}$, we could form the crossed product ${C*(G,A,\alpha)}$, which is again a C*-algebra, in a way similar to semi-product of groups. ${A}$ naturally embeds into ${C*(G,A,\alpha)}$, hence it's interesting to see what properties of ${A}$ can be inherited by the crossed product. In this talk, I will present several definitions of dimension for C*-algebras, which are non-commutative versions of Lebesgue's covering dimension for topological spaces. Then I'll show that when the action ${\alpha}$ is nice, namely having the Rokhlin properties, the property on ${A}$ of having dimension ${<k}$ will be inherited by the crossed products.

Wave operators and quantum groups in the C*-setting

Wave operators appeared first in the quantum scattering theory. From the mathematical point of view they belong to the perturbation theory of self-adjoint operators. We shall show, that after a suitable modification they are useful when we construct examples of locally compact quantum groups. The set of presented examples will include the (unpublished as of yet) quantum ‘ax + b’-group with Schmudgen commutation relations.

K-theory of C*- algebras associated to Hilbert manifolds and applications

I will introduce a ${C^*}$-algebra associated to Hilbert manifolds and discuss its applications to the Novikov conjecture for a certain diffeomorphism group. This is joint work with Jianchao Wu and Erik Guentner.