The students who participate in the seminar give talks on functional analytic aspects of noncommutative geometry. Talks devoted to noncommutative algebraic geometry and to "pure" functional analysis (preferably with an algebraic or geometric flavour) are also welcome. The topics of talks are usually taken from the literature, but sometimes the participants present their own results. Occasionally, talks are given by the seminar advisor or by an invited speaker.
The seminar takes place on Fridays, 14:00-15:30, Faculty of Mathematics, HSE (Usacheva, 6), room 306.
If you wish to give a talk, please send you suggestions to the seminar advisor (pirkosha at gmail dot com).
Abstract. I will explain the construction of the operator K0 and K1 functors and prove some of their properties (half-exactness, continuity w.r.t. inductive limits). If time permits, I will explain the construction of the index map and of the Bott map and will show that they preserve exactness.
Abstract. The talk will be devoted to an interesting operator algebra which was introduced by G.Popescu in 1991 and which is being intensively studied since then by many specialists in operator theory and operator algebras. The algebra is called "noncommutative disc algebra" and is defined to be the completion of the free algebra in several generators by a certain special norm. For a number of reasons, this algebra is usually interpreted as a free analog of the classical disc algebra, which consists of those analytic functions on the open unit disc which are continuous on the closed disc. We will give several equivalent definitions of the noncommutative disc algebra and characterize its quotient modulo the commutator ideal as a multiplier algebra on a classical Hilbert function space (the Drury-Arveson space, a.k.a. the symmetric Fock space).
Abstract. Despite the achievements of formal deformation quantization, in practice one often wants power series to converge. For example to make sense in physical applications. Following a recent preprint of Stefan Waldmann I am going to talk about different approaches to the problem of convergence of star products.
Abstract. Let X be a topological space and R be some equivalence relation on X. It often happens that X/R is indiscrete. However, in some cases it is possible to construct a C*-algebra C*(R) and hence to recover some information about X and R. We will discuss a particular example of this construction in which X is the unit circle and R corresponds to a subgroup of rank one. The resulting C*-algebra turns out to be connected with one-dimensional almost periodic tilings, AF-algebras with rk K0=2 and continued fractions.
Abstract. The stereotype approximation property is an analogue of the classical approximation property transferred into the category Ste of stereotype spaces. Formally this is a stronger condition than the classical approximation (although it is not known up to now if these conditions coincide or not in the class Ste). As a corollary, the question which spaces in the standard list of functional analysis have the stereotype approximation is quite difficult (the only exception is the situation when the space has a topological basis in a reasonable sense). At the talk of 22/02/2019 we were discussing the category Ste of stereotype spaces and the equivalent definitions of the stereotype approximation in it. This time we shall give the proof of the fact that the stereotype group algebra of measures C*(G) on an arbitrary locally compact group G has the stereotype approximation property.
Abstract. The stereotype approximation property is an analogue of the classical approximation property transferred into the category Ste of stereotype spaces. Formally this is a stronger condition than the classical approximation (although it is not known up to now if these conditions coincide or not in the class Ste). As a corollary, the question which spaces in the standard list of functional analysis have the stereotype approximation is quite difficult (the only exception is the situation when the space has a topological basis in a reasonable sense). In this talk I will show that the group algebra of measures C*(G) on an arbitrary locally compact group G has the stereotype approximation property.
Abstract. Following a recent paper by Nemirovski and Shafikov, I will show that the unit ball in Cn is the only simply connected complex manifold that can cover both Stein and non-Stein strictly pseudoconvex domains. I will remind some basic notions needed (strictly pseudoconvex domains, plurisubharmonic functions, Stein manifolds).
Abstract. The Gelfand-Naimark theorem identifies a commutative unital C*-algebra A with C(Spec A). This leads to a natural conjecture that each noncommutative C*-algebra A corresponds to an algebra of operator-valued functions on Prim A. This was the original motivation for the development of C*-bundle theory. The results of this program are not completely satisfactory. Noncommutative generalizations of the Gelfand-Naimark theorem were proved only for rather narrow classes of C*-algebras. On the other hand, some progress has been made, and the Dauns-Hofmann theorem is a good illustration. The theorem states that each C*-algebra is a module over the algebra of continuous bounded functions on the primitive ideal space. In our talk, we discuss the structure of Prim A and prove the Dauns-Hofmann theorem.
Abstract. A set H of natural numbers is called a van der Corput set if for any sequence x1,x2,... of points on the unit circle the uniform distribution of the sequences xn+hxn-1 for all h in H implies that the sequence xn is also uniformly distributed. In my talk I will discuss van der Corput sets, generalizations of this notion to the case of other compact groups, and applications of Banach limits to the study of these sets.
Abstract. Deformation quantization is a natural way of constructing quantum systems out of a classical one. Following M.Rieffel's survey papers, we discuss the definition of strict deformation quantization, give basic examples, and consider some questions about deformation quantization of Poisson manifolds.
Abstract. We discuss (following mostly M. Putinar and J. Eschmeier) some applications of sheaves in the spectral theory of linear operators on Banach spaces. In particular, we discuss the dictionary "operator with Bishop's property (β) = quasicoherent analytic Fréchet sheaf", "Foias decomposable operator = soft analytic Fréchet sheaf", "spectral subspace of an operator = section space of a sheaf"...
Abstract. In the early 1990s enough results have been accumulated to suggest a strong link between C*-algebras and their K-theory. It was conjectured that certain classes of C*-algebras are completely determined by K-theoretic invariants. This conjecture was disproven multiple times by tricky counterexamples and then modified by widening the class of invariants and shortening the class of algebras by considering certain "regularity" properties. I will give a historical overview of this classification program, with a goal to introduce some of the important definitions and constructions that appear along the way.
