Noncommutative Geometry

Here is the schedule for the NCG seminar of the math department of UH Manoa. We meet every week on Friday 11:30-12:30 in Keller 301.

Here are some notes I wrote.

We are currently studying papers of Aaron Tikuisis, Stuart White, and Wilhelm Winter, and Christopher Schafhauser on quasi-diagonality. These are links to the preprints of the articles on the arxiv.

This paper of Jamie Gabe is also related to this circle of ideas. It contains a generalization of Tikuisis-White-Winter theorem.

Spring 2020

February 28: Clement Dell’Aiera, Rapid decay and MAP

Title: Rapid decay and MAP

Abstract: In An example of a non nuclear C*-algebra which has the Metric Approximation Property, Uffe Haagerup proved that the reduced C*-algebra of a free group has the MAP. This article contains very important ideas that became fundamental to the field in the following years. We will present why every group which has the Rapid Decay property with respect to a length that is a conditionally negative type function has the MAP. If times allows, we will also show that free groups have the RD property.

February 21: Robin Deeley (Boulder), Hilsum bordism and unbounded KK-theory

Abstract: Many cycles in Kasparov’s KK-theory are obtained from unbounded operators. Prototypical examples include the cycles associated to geometrically defined operators on a manifold (e.g., the signature operator, the spin^c Dirac operator, etc). Baaj and Julg defined the notion of an unbounded cycle in KK-theory and more recently, Hilsum defined the notion of a bordism in the context of unbounded KK-theory. Hilsum’s definition is based on operators associated to manifolds with boundary. 

In joint work with Magnus Goffeng and Bram Mesland, we defined an abelian group which is essentially unbounded KK-cycles modulo Hilsum’s notion of bordism. This group maps to the standard Kasparov group via the bounded transform and in the commutative case can be related to the geometric model for K-homology due to Baum and Douglas.

February 14: Robin Deeley (Boulder), Relative constructions in geometry K-homology

Abstract: The Baum-Douglas model for K-homology provides a geometric counterpart to the analytic construction of Kasparov. In the framework of index theory, the former is more related to the topological index, while the latter is more related to the analytic index. I will discuss both these theories and a relative construction in geometric (i.e., Baum-Douglas) K-homology. Two examples of such relative constructions are the following: 

(1) A geometric version of the analytic surgery exact sequence of Higson and Roe; 
(2) A geometric version of a recent construction of Chang, Weinbeger, and Yu. 

This is (in part) joint work with Magnus Goffeng.

February 7: Erik Guentner, Introduction to analytic K-homology

Abstract: We will explain how the differential operators introduced in last week’s talk can be organized into an abelian group, the Kasparov analytic K-homology group.

January 31: Rufus Willett, Some analysis of differential operators

Abstract: Differential operators take functions on a manifold (think the real line, or the circle) to other functions by differentiating them, and maybe multiplying by other functions. Focusing on simple examples, I’ll try to explain (some) properties of (some) differential operators, considered as unbounded operators on Hilbert space. Differential operators are the motivating example for ‘Kasparov cycles’, which Erik will speak about next week.

Fall 2019

November 15: Samantha Pilgrim, Coarse Fundamental Groups and Applications

Abstract: Algebraic topological invariants such as the fundamental group can be used to detect “holes” in spaces, i.e. obstructions to homotopy that can distinguish one topological space from another. We will construct and develop some theory for related objects known as coarse fundamental groups which also takes into account a measure of the “size” of holes and leads to a coarse-geometric analog of simple connectivity. We then show a couple of applications to coarse geometry (characterizing spaces coarsely equivalent to trees) and to geometric group theory (a new proof that finite presentability is a quasi-isometric invariant).

November 8: Rufus Willett, Quasidiagonality of group C*-algebras V

Abstract: I’ll complete the sketch of Schafhauser’s proof that group C*-algebras of amenable groups are quasi-diagonal.

November 1: Christian Bönicke (University of Glasgow), Regularity properties for ample groupoids and the type semigroup

Abstract: I will introduce the type semigroup of an ample groupoid and explain how it encodes dynamical properties of the groupoid in an algebraic framework.

In particular I will explain how the fine structure of the type semigroup relates to certain regularity properties of the groupoid, which play a prominent role in recent attempts to develop a dynamical analogue of the Toms-Winter conjecture for simple separable nuclear C*-algebras.

October 25: Rufus Willett, Quasidiagonality of group C*-algebras IV

Abstract: Carrying on from the last few weeks, I’ll compute the K-theory of the so-called trace-kernel ideal.

October 18: Clement Dell’Aiera, Quasidiagonality of group C*-algebras III

Abstract: I will continue where Rufus left off. I’ll explain two arguments in Chris Schafhauser’s proof: why there is a trace preserving map from C*r(G) to R^w when the group is amenable, and why a lift to Q_w is enough to conclude. We will detail Choi-Effros theorem, and explain how Voiculescu’s theorem comes up in the proof. A good reference for his is Arveson’s paper, Notes on extensions of C*-algebras.

October 11: Rufus Willett, Quasidiagonality of group C*-algebras II

Abstract: I’ll continue where I left off last week, discussing Chris Schafhauser’s proof that amenable groups have quasidiagonal group C*-algebras.  In particular, I’ll explain the important C*-algebras Q and R, what their ultrapowers are, and what they have to do with the problem under discussion.  This is all needed to set up the machinery in the proof, which we discussed last time. Time permitting, I’ll also say a little about how K-theory comes in.

October 4: Rufus Willett, Quasidiagonality of group C*-algebras

Abstract: Finite-dimensional matrices are nice.  Therefore operators on infinite-dimensional Hilbert spaces that split as a block-sum of finite dimensional matrices are also nice.  Roughly, a C*-algebra is quasidiagonal if all the operators in it approximately split like this, so quasidiagonal C*-algebras are also pretty nice. 
One of the major recent results in C*-algebra theory (due to Aaron Tikuisis, Stuart White, and Wilhelm Winter) gives that the reduced C*-algebra of a group is quasi-diagonal if and only if the group is amenable.  Over a few lectures, I’ll aim to explain a different proof of this, due to Chris Schafhauser.  Today, I’ll mainly just give background and explain the problem.

September 6 and 20: Matthew Lorentz, Bounded Derivations on Uniform Roe Algebras

Abstract: In this series of talks we give conditions on a space X to give a positive answer to the question of whether or not all the derivations of the uniform Roe algebra on a space X are inner; that is, if the derivation is given by the commutator bracket [ ,b]. Specifically, if a space X has a metric d under which (X,d) is a metric space with bounded geometry having property A, then all derivations are inner. In the second talk I will state the main theorem from the paper of Spakula and Tikuisis that we will need. Then the rest of the talk will be focused on results due to Braga and Farah from their paper “On the Rigidity of Uniform Roe Algebras”. These results will allow us to consider certain families of operators simultaneously. That is, for these families, given epsilon there exists an R such that every member of this family is within epsilon of an operator of propagation at most R.

September 4: Stuart White (U. of Oxford), Amenable Operator Algebras (Keller 401)

Abstract: Operator algebras arise as suitably closed subalgebras of the bounded operators on a Hilbert space. They come in two distinct types: von Neumann algebras which have the flavour of measure theory, and C-algebras which have the flavour of topology. In the 1970’s Alain Connes obtained a deep structural theorem for amenable von Neumann algebras, leading to a complete classification of these objects. For the last 25 years the Elliott classification programme has been seeking a corresponding result for simple amenable C-algebras, and now, though the efforts of numerous researchers worldwide, we have a definitive classification theorem. In this talk, I’ll explain what this theorem says, and the analogies it makes to Connes work, using examples from groups and dynamics as motivation. I won’t assume any prior exposure to operator algebras or functional analysis.

August 30: Matthew Lorentz, Bounded derivations on uniform Roe algebras

Abstract: In this series of talks we give conditions on a space $X$ to give a positive answer to the question of whether or not all the derivations of the uniform Roe algebra on a space X are inner; that is, if the derivation is given by the commutator bracket [ ,b]. Specifically, if a space X has a metric d under which (X,d) is a metric space with bounded geometry having property A, then all derivations are inner. In the first talk I will reduce the problem to a simpler question. Then show that this new question can be partially answered using the paper of Spakula and Tikuisis that was discussed in our seminar last spring. If we have time I will give an overview of the material contained in their paper.

Spring 2019

We are currently studying papers of Spakula, Tikuisis and Zhang on quasi-locality. These are links to the preprints of the articles on the arxiv.

  • May 6th: Kenny, Specialty exam

    Abstract: A projection in A** is open if it is a strong limit on an increasing net of positive operators of A. We present examples of open projections for some C*-algebras. For instance, we show that the complement of the central support for the extension of the trivial representation 1_G to C*(G) is open. We briefly discuss a connection between open projections of A and a topology that can be placed on sets of representations of A.

  • April 22nd: Clément Dell’Aiera, Automata groups

    Abstract: I’ll finish the proof that the two-state automaton of last week generates the Lamplighter group. I will also discuss more examples, including the study of a automaton generating Thompson’s group F.

  • April 15th: Rufus Willett, Survey of Atiyah conjecture on L2-Betti numbers

    Abstract: I’ll give a survey on Atiyah’s questions on L2-Betti numbers that Clément mentioned last week.

  • April 8th: Clément Dell’Aiera, Introduction to Automata groups

    Abstract: We will define groups generated by automata with finitely many states on a finite alphabet. The reason we are interested in such objects is their importance in answering some famous group theoretic problems. We will give a non-exhaustive survey: Day’s problem on amenability, Burnside question on infinite torsion groups, Atiyah’s question on L2-Betti numbers.

    We will also show how to realize the lamplighter group as generated by an automaton, and explain how it is useful to compute the spectrum of the classical Markov operator, and how it is related to Atiyah’s conjecture.

  • April 1st: Rufus Willett, Some context for the work of Deeley-Putnam-Strung

    Abstract: I’ll try to fill in some of the ‘bigger picture’ motivating Robin Deeley’s colloquium talk last week (and based on his joint work with Ian Putnam and Karen Strung). I’ll start with homeomorphisms of odd-dimensional spheres, and discuss connections to the so-called C*-algebra classification program and the so-called Jiang-Su algebra.

    The talk will (probably) be a one-off, staying at a fairly survey-ish level.

  • March 25th: Clément Dell’Aiera,  Subgroups of RAAGs

    Abstract: We will show that 2-generated subgroups of Right-Angled Artin Groups are either free or abelian. Classical notions of Geometric Group Theory, such as Hopfian groups, residual finiteness and residually-p groups, will be introduced. The talk is scheduled for the Geometric Group Theory Seminar, be aware of the change of classroom. Here is their website.

  • March 11th: Clément Dell’Aiera,  Quasi-locality and Property A

    Abstract: We will prove that quasi-local operators on X are in the norm closure of finite propagation operators, in the case of X being a bounded geometry discrete metric space with property A. We will need several facts proven by Erik and Rufus.

  • February 25th: Rufus Willett,  Property A and its friends.

    Abstract: I’ll introduce property A, which plays the role of amenability in large scale geometry. The definition is originally due to Yu, but I’ll focus on an equivalent form due to Dadarlat and Guentner. I’ll then discuss some other properties — the metric sparsification property, and operator norm localisation — that turn out to all be equivalent (although this is not obvious). All this is with a view to generalizing the result Erik discussed during the last few lectures (although that won’t happen today).

  • February 11th: Erik Guentner,  Characterizing membership in the Roe algebra continued

    Abstract: The Roe algebra is a fundamental object in index theory
    on non-compact manifolds, and more general spaces. This series of
    talks is devoted to characterizations, some well known and others
    more recent, of membership in this important C*-algebra. We will
    focus primarily on spaces satisfying additional geometric hypotheses.

  • February 1st: Erik Guentner,  Characterizing membership in the Roe algebra

    Abstract: The Roe algebra is a fundamental object in index theory
    on non-compact manifolds, and more general spaces. This series of
    talks is devoted to characterizations, some well known and others
    more recent, of membership in this important C*-algebra. We will
    focus primarily on spaces satisfying additional geometric hypotheses.

  • January 25th: Rufus Willett, Approximation of band dominated operators

    Abstract: I’ll discuss approximation properties of so-called band-dominated operators (equivalently, operators in uniform Roe algebras) associated to a metric space, with motivation and examples coming initially from classical Fourier analysis. Erik will then take over next week with an exposition of recent results of Špakula and Tikuisis.

Fall 2018

This semester was devoted to the study of C*-simplicity for discrete groups, with a focus on the results of Breuillard, Kalantar, Kennedy and Ozawa. These are links to the preprints of the articles on the arxiv.

  • December 5: Clément Dell’Aiera, Thompson’s group V is C*-simple

    Abstract: We realize Thompson’s group V as the topological full group of the Renault-Deaconnu or Cuntz groupoid, and use a result of Le Boudec and Matte Bon to prove that it is C*-simple.

  • November 28: Rufus Willett, Some examples of C*-simple groups

    Abstract: I’ll prove that any group that has no amenable normal subgroups, and contains only countably many amenable subgroups is C*-simple. This includes many interesting examples, starting with free groups but including many other natural geometric examples (e.g. torsion free hyperbolic groups), algebraic examples (e.g. SL(3,Z)), and other more pathological things (e.g. so-called Tarski monsters).

  • November 21: Matthew Lorentz, A Generalized Dimension Function for K_0 of C*-Algebras

    Abstract: One way to classify projections in the matrices over the complex numbers is to consider their rank. Since the complex numbers are the prototypical C*-algebra we would like to generalize this to C*-algebras to help us better understand their structure and perhaps assist with their classification. Note that matrices of different sizes can have the same rank. Thus, to consider matrices of all sizes we use K-theory, denoted K_0(A), the K-theory of the C*-algebra A. Recall that we can compute the rank of a square idempotent matrix over the complex numbers by taking its trace. We can generalize this to many C*-algebras; however, sometimes we cannot. To this end, we will define and use an unbounded trace. Then using our unbounded trace we will create a group homomorphism from K0(A) to the complex numbers viewed as an additive group.

  • November 14: Rufus Willett, Recurrent subgroups

    Abstract: We prove that a group is C*-simple iff it contains no amenable recurrent subgroup.

  • October 31: Clément Dell’Aiera, Dynamical characterization of C*-simplicity again

    Abstract: We prove the same result than last week, this time with a proof of Ozawa, avoiding representation theoretic techniques.

  • October 24: Clément Dell’Aiera, Dynamical characterization of C*-simplicity

    Abstract: We prove that a discrete group is C*-simple iff it acts freely on its Furstenburg boundary. The talk will use several properties of the boundary that were covered in the previous talks. The proof is the one given in the paper by Breuillard, Kalantar, Kennedy and Ozawa.

  • October 3: Rufus Willett, The Furstenburg boundary and injectivity

    Abstract: Two weeks ago Matt introduced the notion of a G-boundary. Today I’ll prove that there is a maximal such G-boundary called the ‘Furstenburg boundary’, and that C*-algebra of continuous functions on the Fursternburg boundary is an injective C*-algebra in the sense of last week’s talk.

  • September 26: Rufus Willett, Injective C*-algebras

    Abstract: Injective C*-algebras are analogues of injective modules from pure algebra. Although not immediately obvious why, they turn out to be very important in various aspects of the theory (particularly von Neumann algebras). I’ll aim to introduce injective C*-algebras and some of the underlying theory, and deduce some consequences.

  • September 19: Matthew Lorentz, G-C*-algebras

    Abstract: I’m going to talk about groups acting on compact Hausdorff spaces X by homeomorphism and C*-algebras by automorphism. This will lead to showing that P(X) is weak* closed and G-invariant. Then an introduction to a G-boundary.

  • September 12: Clément Dell’Aiera, C*-simplicty continued

    Abstract: We give two proofs that if a group has a non trivial amenable normal subgroup, then it is not C*-simple. One uses induction for unitary representations, the other is direct but restricts to the discrete countable case.

  • September 5: Rufus Willett, C*-simplicty of discrete groups

    Abstract: A discrete group is called C*-simple if its reduced group C*-algebra C*_r(G) is simple. This roughly says (in stark contrast to the case e.g. of finite groups) that all subrepresentations of the regular representation of G on l^2(G) are equivalent to the regular representation itself. The inputs for proving that a particular group has this property come, however, not from representation theory but from geometric group theory and dynamics.

    I’ll discuss some background to this, and the (non-)connection with amenability, and a little on the classic (‘Powers’) technique for proving it. In the next few weeks, we’ll discuss the recent work of Kennedy et al that completely solves the problem of characterizing C*-simple groups.

  • August 29: Benedikt Hunger (Ausburg University), Almost flat bundle