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 - C*-algebras
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).
In this talk I will introduce the notion of self-similarity for compact quantum groups. I will start by looking at the quantum automorphism group of an infinite homogeneous rooted tree. Self-similar quantum groups are then certain quantum subgroups of these quantum automorphisms. I will then look at a class of examples called finitely-constrained self-similar quantum groups, and I will describe a subclass as quantum wreath products by subgroups of the quantum permutation group.
We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.
Many combinatorial objects (or similar) give rise to a C-algebra with a distinguished C-subalgebra. Examples of such which have been studied a lot in the past decade are étale groupoids which give rise to a so-called Cartan pair, consisting of the groupoid C-algebra and a commutative C-subalgebra. Other examples include discrete groups acting on compact Hausdorff spaces, for which the reduced crossed product C-algebra contains the reduced group C-algebra as a C- subalgebra. I will talk about invariants and properties for such inclusions of C- algebras. This talk will be somewhat non-standard (for me, at least) and will only contain few (if even any) theorems, but should more be thought of as a conversation starter/brainstorm for what invariants one can associate to certain mathematical objects.
The category of Hilbert spaces and bounded linear functions forms the mathematical basis for quantum theory. But why? What physical principles enforce this mathematical structure? This category is also where C*-algebra theory lives, as it's the universal C*-category. But why? What properties does it have that accommodate this mathematical structure? As a first answer, this talk provides axioms that guarantee a category is equivalent to that of Hilbert spaces and bounded linear functions. The axioms are purely categorical and do not presuppose any analytical structure such as complex numbers, continuity, dimension, convexity, probabilities, etc. We will also discuss variations, such as linear contractions, finite-dimensional Hilbert spaces, and Hilbert C*-modules.
The dynamics of iterating a rational function exhibits complicated and interesting behaviour when restricted to points in its Julia set. Kajiwara and Watatani constructed a C*-algebra from a rational function restricted to its Julia set in order to study its dynamics from an operator algebra perspective. They showed the C*-algebras are Kirchberg algebras that satisfy the UCT, and are therefore classified by K-theory. The K-theory groups of these algebras have been computed in some special cases, for instance by Nekrashevych in the case of a hyperbolic and post-critically finite rational function. We compute the K-theory groups for a general rational function using methods different to those used before. In this talk, we discuss our methods and results.
Given a C*-algebra B, and a regular, abelian, sub-C*-algebra A ⊆ B, we will discuss the opaque ideal Δ ⊴ B, which is a somewhat mysterious ideal that tends to vanish most of the time, but not always. In the last part of the talk I will give an example of a non-vanishing opaque ideal based on an idea of Rufus Willett, and related to the celebrated action of the free group on the 2-sphere used by Banach and Tarski to produce their paradox. This is based on joint work with David Pitts and Vrej Zarikian.
The C∗-algebra generated by the left-regular representation of ℕn twisted by a 2-cocycle is a Toeplitz extension of an n-dimensional non-commutative torus, on which each vector r ∈ [0,∞)n determines a one-parameter subgroup of the gauge action. I will report on joint work with Z. Afsar, J. Ramagge and M. Laca, in which we show that the equilibrium states of the resulting C∗-dynamical system are parametrized by tracial states of the non-commutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of r. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Our results generalize the phase transition on the Toeplitz non-commutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims.
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