Functional Analysis and Noncommutative Geometry

A. Yu. Pirkovskii

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.

Starting from April 9, the seminar takes place online on Thursdays, 17:00-18:20.

If you wish to give a talk, please send you suggestions to the seminar advisor (pirkosha at gmail dot com).

Seminar archive:


Feodor Kogan. A new look at KK-theory.

Abstract. The title of my talk is the name of an article published by Joachim Cuntz in 1986. If you don't know what KK-theory is, then this talk will hopefully be a reasonable, but non-standard introduction to this vast generalization of K-theory. Cuntz's approach requires some peculiar algebraic techniques and in fact works for arbitrary algebras; compared to other definitions, this one is accessible with little to no knowledge of C*-algebra theory. I will explain the definitions in detail, prove some basic properties, ending with the associative product in KK-theory, which can be difficult to grasp with other approaches.


Alexander Popkovich. Quantum deformations of flag varieties.

Abstract. My talk will be about a version of a quantized flag variety of SLn proposed by N. Reshetikhin and V. Lakshmibai. It is known that the homogeneous coordinate ring of the image of the usual flag variety under Plücker embedding is isomorphic to an algebra generated by certain minors of a matrix of commuting formal variables xij. The idea of deformation consists in replacing these variables with the generators of an RTT-algebra and the minors with quantum minors. Similarly to the classical case, the obtained object is closely related to the repreresentation theory of Drinfeld-Jimbo quantum group Uq(sln).


Maria Dmitrieva. A quantum analog of the Drury-Arveson space and holomorphic functions on the quantum ball.

Abstract. In my previous talk I spoke about the multiplier algebra M on the Drury-Arveson space. It is known that M is contained in A(B), the algebra of holomorphic functions on the open ball B which are continuous on the closure of B, that M is dense in A(B), and that the norm on M is strictly stronger than the norm on A(B). Today we construct a quantum analog of the Drury-Arveson space and discuss its relations to some other objects, in particular, to the quantum analog of A(B) introduced by L.L. Vaksman in 2003. We also prove the following surprising fact: in contrast to the commutative case, the quantum analog of the multiplier algebra and the quantum analog of A(B) are isomorphic.


Ananya Kaushik. Algebras of operators on a finite dimensional Hilbert space.

Abstract. We will be discussing *-algebras and their properties in the finite dimensional case. We will also discuss the double commutant theorem, factors and a theorem on decomposition of a factor.


Dmitry Golubenko. The Shilov boundary for the quantum matrix ball.

Abstract. At the previous meetings of the seminar we introduced the noncommutative Shilov boundary, following a book by D. Blecher and C. Le Merdy. Now we discuss an example of calculating the noncommutative Shilov boundary for a concrete algebra. L. Vaksman and his co-authors introduced a quantum analog of the algebra of homolorphic functions on the unit ball in the space of nxn matrices. We describe the Shilov boundary for this algebra. The talk is based on a recent preprint of O. Bershtein, O. Giselsson, and L. Turowska.


Boris Bilich. Noncommutative holomorphic functions in elements of a Lie algebra.

Abstract. Despite the undeniable importance of holomorphic functions in geometry, their applications in the noncommutative case are quite restricted. The generalizations are usually constructed as suitable completions of a "base" skew polynomial ring. Following A. Dosiev's papers, we will discuss the case where the base ring is the universal enveloping algebra of a Lie algebra.


Dmitry Golubenko. C*-envelopes and boundary representations (Part 2).


Dmitry Golubenko. C*-envelopes and boundary representations.

Abstract. Following a monograph of Blecher and Le Merdy, we introduce Shilov boundaries for subalgebras of C*-algebras, prove some elementary properties of C*-envelopes, and discuss applications to operator algebra theory.


Maria Dmitrieva. Abstract operator algebras.

Abstract. The talk will be devoted to operator algebras, that is, to algebras of operators on a Hilbert space. Obviously, each such algebra has the structure of a normed algebra, but this structure is insufficient for many purposes.

It is easy to see that, if A is an operator algebra, then a natural norm is defined not only on A itself, but also on all matrix algebras Mn(A). Thus we have the following natural question. Suppose that A is an abstract algebra, and that each Mn(A) is equipped with a norm. Under what conditions is A an operator algebra?

The answer to this question is given by the Blecher-Ruan-Sinclair theorem. The conditions which imply that A is an operator algebra are rather mild. For example, each quotient of an operator algebra satisfies these conditions.

I will also define the Haagerup tensor product, which is closely related to the Blecher-Ruan-Sinclair theorem, and I will explain how these results can be applied to a problem that I am currently working on.

No previous knowledge of operator algebra theory is required.


Alexei Pirkovskii. Bornological vector spaces.

Abstract. A bornological vector space is a vector space in which bounded subsets are defined axiomatically. Such spaces were introduced by L. Waelbroeck (1960), who was motivated by some problems of functional calculus in topological algebras. In the beginning of this century, bornological spaces attracted attention of specialists in noncommutative geometry, starting from R. Meyer's thesis (1999). The key observation was that bornological spaces provide a convenient setting for "analytic" versions of cyclic homology and for bivariant K-theory. Bornological spaces are closely related to topological vector spaces, but there is no 1-1 correspondence between them. One of the nice features of bornological spaces is that they form a closed symmetric monoidal category, in contrast to standard classes of topological vector spaces. In this introductory talk, we define bornological spaces, discuss their relashionship with topological vector spaces (given by an adjoint pair of functors), define some constructions of bornological spaces, and give a number of examples.


Alexander Kalmynin. Graph C*-algebras.

Abstract. We will discuss a way to construct a C*-algebra C*(E) from a given directed graph E and study basic properties of the resulting algebras. It turns out that many questions about C*(E) can be answered in terms of properties of the graph E. Also, some interesting algebras, such as the algebra of functions on the noncommutative sphere, the Toeplitz algebra, and AF-algebras (up to Morita equivalence), are isomorphic to graph C*-algebras. This allows, for example, to explicitly compute the K-theory of these algebras.


Vladislav Balakirev. Quantum balls and quantum spheres.

Abstract. The talk is intended to give an overview of several ways to define quantum spheres and balls. All of them are generalizations of classical topological constructions such as glueing, suspensions, homogeneous spaces, etc.


S. S. Akbarov. Stereotype approximation property for the algebras C(M) of continuous functions on metric spaces.

Abstract. The category Ste of stereotype spaces possesses a natural analog of the approximation property called "stereotype approximation property". Its study is hindered by the fact that the operator space in Ste is defined in a more complicated way than in the category of locally convex spaces. For this reason, little is known about spaces with stereotype approximation, and each example is of interest. In this talk, I will explain why the space C(M) of continuous functions on a complete metric space M has the stereotype approximation.


Stepan Kazanin. Quantum automorphism groups of finite spaces.

Abstract. Since most natural and important examples of groups arising in commutative mathematics are automorphism groups of simple mathematical structures (finite sets, linear spaces etc), one would expect noncommutative spaces to also carry quantum automorphism groups. Nevertheless, the notion of quantum automorphism group is yet to be completely clarified. I will give a talk on a few specific examples, worked out by S. Wang, in particular, on his construction of quantum automorphism group of a finite set.

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