I'll give an overview of recent progress in the structure and classification of simple amenable C*-algebras, making parallels to the Connes-Haagerup classification of amenable von Neumann algebras and drawing examples from group actions.
Tag - Functional analysis
This is a 35-lecture course, with each lecture being an hour, given by George Elliott. Note that the 32nd lecture was not recorded. The first 31 lectures are still of great interest, but this needs to be known.
The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.
Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.
Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)
The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)
The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classication of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras, both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras - what Bratteli called AF algebras - is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).
This is a 24-lecture course, with each lecture being around 80 minutes, given by Robert McCann. It gives an introduction to optimal transport.
This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, nonlinear partial differential equations, and machine learning. The basic problem is to find the most efficient structure linking two or more continuous distributions of mass; think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.
Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), geometric inequalities with sharp constants, image processing, optimal decision making, long time asymptotics of dissipative systems, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, fluid mechanics, physics, economics, and geometry. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provide a framework for adapting powerful ideas from Riemannian and Lorentzian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.
In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting "nicely" inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence).
In recent years, Matsumoto and Matui exploited this to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type using graph C*-algebras and their commutative subalgebras of continuous functions on the shift space (which form a Cartan pair under mild assumptions on the graph). The key point was translating these dynamical conditions into groupoid language. The ring theoretic analogue of graph C*-algebras are Leavitt path algebras. Leavitt path algebras are also connected to Thompson's group V and some related simple groups considered by Matui and others. Since the Leavitt path algebra associated to a graph is the "Steinberg" algebra of the same groupoid (a ring theoretic version of groupoid C*-algebras whose study was initiated by the speaker), this led people to wonder whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the shift space. The answer is yes, and it turns out in the algebraic setting one doesn’t even need any conditions on the graph. Initially work was focused on recovering a groupoid from the pair consisting of its "Steinberg" algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture.
It turns out that a twist on a groupoid gives rise to a Cartan pair when the algebra satisfies a groupoid analogue of the Kaplansky unit conjecture. In particular, if the groupoid has a dense set of objects whose isotropy groups satisfy the Kaplansky unit conjecture (e.g., are unique product property groups or left orderable), then the groupoid gives rise to a Cartan pair. This is what happens in the case of Leavitt path algebras where the isotropy groups are either trivial or infinite cyclic and hence left orderable.
A conjecture of George Elliott dating back to the early 1990s asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely infinite algebras, this is the famous Kirchberg-Phillips Theorem. The stably finite setting proved to be much more subtle and has been a driving force in research in C*-algebras over the last 30 years. A series of breakthroughs were made in 2015 through the classification results of Elliott, Gong, Lin, and Niu and the quasidiagonality theorem of Tikuisis, White, and Winter. Today, the classification conjecture is now a theorem under two additional regularity assumptions: Z -stability and the UCT. In my recent joint work with José Carrión, Jamie Gabe, Aaron Tikuisis, and Stuart White a much shorter and more conceptual proof of the classification theorem in the stably finite setting was provided. I hope to give an overview of the classification problem for C*-algebras and discuss some of the new techniques that led to the new proof.
An online lecture course by the University of Münster in Von Neumann algebras.
Lecture 1: We'll briskly review basic properties of semi-finite von Neumann algebras. The standard representation, completely positive maps, group von Neumann algebras, the group-measure space construction, and some characterizations of the hyperfinite II1 factor.
Lecture 2: We discuss some approximation properties that are common in "rank 1" groups: Weak amenability and biexactness.
Lecture 3: We discuss properly proximal groups as defined by Boutonnet, Ioana, and myself, and give some applications to group von Neumann algebras associated to higher-rank groups.
Lecture 4: We’ll introduce measure equivalence (ME), W*-equivalence (W*E), and von Neumann equivalence (VNE). We’ll give examples and discuss invariants.
This is a 22-lecture course, with each lecture being around 90 minutes, given by Ilijas Farah.
The route to understanding separable C*-algebras frequently involves a detour via non-separable C*-algebras, such as the Calkin algebra, the asymptotic sequence algebras, ultrapowers, and ultraproducts. Some basic ideas from logic can be used to analyse these massive C*-algebras. Among other things, we will see that the existence of outer automorphisms of the Calkin algebra depends on the set-theoretic axioms.
An LMS online lecture course in functional analysis.
We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.
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