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Evgenij Troitsky

Publications and source records attributed to Evgenij Troitsky.

25 records · Page 2Linked to original sources
arXiv2006

A C*-analogue of Kazhdan's property (T)

This paper deals with a "naive" way of generalization of the Kazhdan's property (T) to C*-algebras. This approach differs from the approach of Connes and Jones, which has already demonstrated its utility. Nevertheless it turned out that our approach is applicable to the following rather subtle question in the theory of C*-Hilbert modules. We prove that a separable unital C*-algebra A has property MI (module-infinite, i.e., any C*-Hilbert module over A is self-dual if and only if it is finitely generated projective) if and only if it has not property (T1P) (property (T) at one point}, i.e. there exists in the unitary dual of A a finite dimensional isolated point. The commutative case was studied in a previous paper.

math.OA↗
arXiv2006

Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy

The present paper consists of two parts. In the first part, we prove a noncommutative analogue of the Riesz(-Markov-Kakutani) theorem on representation of functionals on an algebra of continuous functions by regular measures on the underlying space. In the second part, using this result, we prove a weak version of Burnside type theorem for twisted conjugacy for arbitrary discrete groups.

math.OA↗
arXiv2006

A twisted Burnside theorem for countable groups and Reidemeister numbers

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one of its automorphisms, R(f) the number of f-conjugacy classes, and S(f)=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays an important role in the theory of twisted conjugacy classes and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Non-commutative Harmonic Analysis. We begin a discussion of the general case (which needs another definition of the dual object). It will be the subject of a forthcoming paper. Some applications and examples are presented.

math.RT↗
arXiv2006

Twisted Burnside theorem for type II_1 groups: an example

The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, f its automorphism, R(f) the number of f-conjugacy classes (Reidemeister number), S(f):=# Fix (f^) the number of f-invariant equivalence classes of irreducible unitary representations. If one of R(f) and S(f) is finite, then it is equal to the other. This conjecture plays a very important role in the theory of twisted conjugacy classes having a long history and has very serious consequences in Dynamics, while its proof needs rather fine results from Functional and Non-commutative Harmonic Analysis. It was proved for finitely generated groups of type I in a previous paper. In the present paper this conjecture is disproved for non-type I groups. More precisely, an example of a group and its automorphism is constructed such that the number of fixed irreducible representations is greater than the Reidemeister number. But the number of fixed finite-dimensional representations (i.e. the number of invariant finite-dimensional characters) in this example coincides with the Reidemeister number.

math.RT↗
arXiv2005

The Thom isomorphism in gauge-equivariant K-theory

In a previous paper we have introduced the gauge-equivariant K-theory group of a bundle endowed with a continuous action of a bundle of compact Lie groups. These groups are the natural range for the analytic index of a family of gauge-invariant elliptic operators (i.e. a family of elliptic operators invariant with respect to the action of a bundle of compact groups). In this paper, we continue our study of gauge-equivariant K-theory. In particular, we introduce and study products, which helps us establish the Thom isomorphism in gauge-equivariant K-theory. Then we construct push-forward maps and define the topological index of a gauge-invariant family.

math.KT↗
arXiv2002

An index for gauge-invariant operators and the Dixmier-Douady invariant

Let $\GR \to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\GR \to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\GR \to B$, which, in this approach, is an element of $K_\GR^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\GR))$, the $K$-theory group of the Banach algebra $C^*(\GR)$. We prove that $K_0(C^*(\GR)) \simeq K^0_\GR(\GR)$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\GR)$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently proved themselves useful in the study of Ramond-Ramond fields.

math.KT↗