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 108.
If you wish to give a talk, please send you suggestions to the seminar advisor (pirkosha at gmail dot com).
Abstract. Operads are algebraic constructions which carry information on the respective algebras. We discuss the definition of operads, various types of operads and their properties, algebras over operads, and other related structures.
Abstract. Let A be a closed subalgebra of the algebra C(X) of continuous functions on a compact space X. It is known that there is a smallest closed subset Y of X such that each function in A attains its maximum at Y. Such Y is called the Shilov boundary for A. In 1970, W. Arveson introduced a noncommutative version of this notion by replacing C(X) with an arbitrary (noncommutative) C*-algebra. Later L.L.Vaksman applied Arveson's construction to define an analog of the maximum principle for "noncommutative holomorphic functions on the ball". Recently L.Turowska and coauthors extended Vaksman's results to the ball in the space of matrices. I will define boundary representations, noncommutative Shilov boundaries, some related notions, and give some examples.
Abstract. There are several ways of thinking on C*-algebra bundles. All of them give an exact meaning to the statement that a certain family of C*-algebras depend continuously on a parameter. We will concentrate on three equivalent notions: continuous fields of C*-algebras, continuous bundles of C*-algebras, and C0(X)-C*-algebras. We define the respective categories, describe equivalences between them, and show how this works in the case of the quantum torus.
Abstract. I am going to remind some noncommutative geometry in the sense of Connes and to show that a reasonable definition of a complex structure in noncommutative geometry is provided by some positive Hochschild cocycles. In particular, I am going to consider the case of 2-dimensional manifolds.
Abstract. There are several normed cohomology theories for CW-complexes, for example, bounded cohomology, summable cohomology, l2 cohomology, etc. I will talk on such theories and on Gersten's result that the vanishing of H2∞(G, l∞) is equivalent to the hyperbolicity.
Abstract. For each complete bounded Reinhardt domain D in Cn, we construct a formal Fréchet deformation Ofdef(D) of the algebra O(D) of holomorphic functions on D in such a way that the deformed coordinates satisfy the relations of the quantum affine space. In the case where D is either the polydisc or the ball, we show that Ofdef(D) admits a non-formal analog, Ohdef(D), which is an algebra over the ring O(C×) of holomorphic functions on the punctured complex plane. We show that Ofdef(D) can be obtained from Ohdef(D) via "extension of scalars". In conclusion, we prove that Ohdef(D) (in contrast to Ofdef(D)) is not topologically projective (and, a fortiori, is not topologically free) over the base ring.
Abstract. A C*-algebra is called an AF algebra if it is isomorphic to an inductive limit of finite-dimensional C*-algebras. G.A. Elliott reduced the classification problem of AF algebras to studying ordered abelian groups with some special properties. I will talk on those AF algebras whose K0 groups are isomorphic to Z2. It turns out that all such algebras (apart from two exceptional ones) can be described by an explicit construction involving continuous fractions.
Abstract. We discuss the notion of a Hilbert superspace introduced in https://arxiv.org/abs/1011.2370. We show that this notion can be used for constructing harmonic analysis on the Heisenberg supergroup.
Abstract. Super Hilbert spaces arise naturally in some parts of physics related to Quantum Field Theory. Different authors define super Hilbert spaces in a different way, according to their personal interests. We consider the classical approach of DeWitt and a generalization due to Oliver Rudolph https://arxiv.org/abs/math-ph/9910047. We also discuss another definition given by Pierre Bieliavsky https://arxiv.org/pdf/1011.2370.pdf. An interesting feature of this definition is that it gives a possible approach to the notion of a C*-superalgebra.
Abstract. I will talk on various index theorems. In particular, I will formulate the classical Atiyah-Singer index theorem (and some simplified versions of it) in terms of operator K-theory.
Abstract. We survey several known approaches to noncommutative complex analysis and to noncommutative complex analytic geometry. Some of them have a functional analytic flavour, while the others are mostly algebraic. In particular, we discuss and compare algebras of "holomorphic functions" on the quantum polydisc and on the quantum ball.
Abstract. Pirkovskii proposed to consider holomorphically finitely generated (HFG) algebras (the quotients of Taylor's algebra of free entire functions) as non-commutative analogues for Stein algebras. On the other hand, Akbarov included Arens-Michael envelopes in his duality scheme for Hopf topological algebras and proved (appropriately defined) holomorphic reflexivity for Stein groups with algebraic component of unit. The holomorphic Pontryagin duality for abelian Lie group is a special case of this construction. We combine these two approaches and consider Hopf HFG algebras and holomorphic duality in this class. Our main results are as follows:
1. The functor G ↦ O(G) (holomorphic functions on G) is an anti-equivalence of the category of Stein groups and the category of commutative Hopf HFG algebras.
2. If G is a compactly generated complex Lie group then the space Aexp(G) of exponential analytic functionals on G is a (cocommutative) Hopf HFG algebra. (The whole category of cocommutative Hopf HFG algebras is wider.)
3. We consider the holomorphic reflexivity of O(G) and Aexp(G) for a general compactly generated complex Lie group G. In particular, we show (under the assumption that G is connected) that O(G) is holomorphically reflexive iff G is linear, i.e., admits a faithful finite-dimensional holomorphic representation. Moreover, Aexp(G) is always holomorphically reflexive. The proofs of these results are based on a characterization of Aexp(G) for an arbitrary connected complex Lie group.
