Lebesgue’s criterion for the Riemann integration with respect to a state on a separable unital C*- algebra

A. I. Shtern 장전수학회 2016 Advanced Studies in Contemporary Mathematics Vol.26 No.2

In the paper “Semicontinuous envelopes, Riemann integral, and uniform distribution in C*-algebras” (Funct. Anal. Appl. 29 (4), 268– 275 (1995)(1996)), the author had constructed the Riemann integral with respect to a state on a separable unital C*-algebra A. In particular, the semicontinuous hulls of elements of the enveloping von Neumann algebra A of A were introduced and studied and, for a given state ! on A, a class of selfadjoint elements of the algebra A that are Riemann integrable with respect to the state ω were introduced. It was also proved that this class is the self-adjoint part of a C*-algebra, and relations to the uniform distribution of states of the C*-algebra A with respect to the state ω were indicated. In the present note, we supplement these results by a version of Lebesgue’s criterion for the Riemann integrability of self-adjoint elements of A.

Van der Waerden's continuity theorem for the commutator subgroups of connected Lie groups and Mishchenko's conjecture

A. I. Shtern 장전수학회 2006 Advanced Studies in Contemporary Mathematics Vol.13 No.2

As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. In this paper, with the help of an earlier result of the author claiming that the van der Waerden theorem holds for any connected semisimple Lie group, it is proved that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup of the group; moreover, it turns out that a connected Lie group satis¯es the assertion of the van der Waerden theorem (i.e., all locally bounded finite-dimensional representations of the group are continuous) if and only if the group is perfect (i.e., coincides with the commutator subgroup). Thus, for perfect connected linear Lie groups, the structure of (totally) bounded sets de¯nes the topology, and any boundedness-preserving group isomorphism of a perfect connected linear Lie group onto another perfect connected linear Lie group is automatically continuous. To study this phenomenon, the notion of discontinuity group of a locally bounded finite-dimensional representation of a topological group is introduced and studied. The notion of local boundedness of a representation is naturally related to the notion of point oscillation (at the identity element of the group) introduced by the author in 2002. According to a conjecture expressed by A. S. Mishchenko, the finite-dimensional representations of Lie groups can take only three possible values for the (reasonably defined) point oscillation, namely, 0, 2, and 1. We prove the validity of the conjecture. As a corollary, we prove that the class of connected Lie groups for which the point oscillation of a finite-dimensional representation can take only two values, 0 and 1, is the very class of perfect connected Lie groups. Related open problems are indicated.

Second preduals of tridual von Neumann algebras are geometrically unique

A. I. Shtern 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.17 No.1

The main result of the note claims that, if a von Neumann algebra Ais the third dual to some Banach space and thus, as is known, is the bidual of somevon Neumann algebra A, then the natural homomorphic projection of A onto theweakly closed two-sided ideal in A isomorphic to A (the kernel of this homomorphismcoincides with that of the canonical projection of A onto A dual to the canonicalembedding of the predual of A in the predual of A) is dened uniquely by A. Thus,the von Neumann algebra A, the image of the above projection, is not only uniquelydened (up to isomorphism) by its enveloping von Neumann algebra A but also theimage of the \canonical normal embedding" of A in A (onto the image of the aboveprojection) is uniquely dened, i.e., this weakly closed two-sided ideal in A isomorphicto A is uniquely dened geometrically. This enables us to rene some known resultson predual spaces of von Neumann algebras.

A REVISED FORMULA FOR A LOCALLY BOUNDED PSEUDOCHARACTER ON AN ALMOST CONNECTED LOCALLY COMPACT GROUP

A. I. Shtern 장전수학회 2022 Advanced Studies in Contemporary Mathematics Vol.32 No.4

As was proved in the paper Shtern A. I., Description of locally bounded pseudocharacters on almost connected locally compact groups, Russ. J. Math. Phys. 23 (2016), no. 4, 551–552, if G is an almost connected locally compact group and G0 is the connected component of the identity in G, then every locally bounded pseudocharacter of G is a uniquely defined extension to G of a locally bounded pseudocharacter on G0. We prove here that every locally bounded pseudocharacter on G0 admits an extension to a uniquely defined locally bounded pseudocharacter of G. Thus, all pseudocharacters on G are in a one-to-one correspondence with the pseudocharacters on G0 described in Theorem 1 of the aforementioned paper. We also correct the formula in the paper Shtern A. I., A formula for pseudocharacters on almost connected groups, Russ. J. Math. Phys. 25 (2018), no. 4, 531–533, connecting a locally bounded pseudocharacter of G and its restriction to G0.

Connected Lie groups with many locally bounded finite-dimensional representations are linear

A. I. Shtern 장전수학회 2011 Proceedings of the Jangjeon mathematical society Vol.14 No.2

In this paper, it is proved that the following conditions are equivalent for any connected Lie group:(i) G admits sufficiently many locally bounded (not necessarily continuous) finite-dimensional representations;(ii) G admits a faithful locally bounded (not necessarily continuos) finite-dimensional representation;(iii) G admits sufficiently many continuous finite-dimensional representations;(iv) G admits a continuous faithful finite-dimensional representation, i.e., G is a linear Lie group. The equivalence (iv)⇔(i) is somewhat surprising.

Discontinuity groups of relatively compact homomorphisms of locally compact groups and their applications

A. I. Shtern 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.15 No.2

In the present paper we continue the study of the discontinuity group of a locally relatively compact homomorphism of a given topological group into another topological group (see A. I. Shtern, “Analog of the van der Waerden theorem and the validity of Mishchenko’s conjecture for relatively compact homomorphisms of arbitrary locally compact groups,” Adv. Stud. Contemp. Math. 14 (1), 1–20 (2007)). It is proved that the discontinuity group of an arbitrary locally bounded homomorphism of a connected locally compact group into another connected locally compact group is connected, and some related results are established or presented with new proofs.

A SPECTRAL CHARACTERIZATION OF ALMOST PERIODIC FUNCTIONS ON MOORE GROUPS

A. I. Shtern 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.1

It is proved that every uniformly continuous function on an [FIR] locally compact group (i.e., a group all of whose irreducible representations are finite-dimensional) is almost periodic provided that its spectrum is discrete.

TENSOR PRODUCTS OF IRREDUCIBLE FINITE-DIMENSIONAL LOCALLY BOUNDED PSEUDOREPRESENTATIONS OF CONNECTED SIMPLE LIE GROUPS

A. I. Shtern 장전수학회 2022 Advanced Studies in Contemporary Mathematics Vol.32 No.1

It is proved that the tensor product of any two irreducible finite-dimensional locally bounded pseudorepresentations of a connected simple Lie group is a quasirepresentation if and only if the group has finite center, i.e., either the group is not Hermitian symmetric or it is Hermitian symmetric and not simply connected.

A FORMULA FOR THE INVARIANT MEAN ON THE WEAK ALMOST PERIODIC FUNCTIONS ON MOORE GROUPS

A. I. Shtern 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.2

It is proved that the invariant mean on the weakly almost periodic functions on an [FIR] locally compact group (i.e., a Moore group, which means that all continuous irreducible unitary representations of the group are finite-dimensional) admits a simple formula in terms of functions on the dual space of the group.

EXTENSION OF PSEUDOCHARACTERS FROM NORMAL SUBGROUPS

A. I. Shtern 장전수학회 2015 Proceedings of the Jangjeon mathematical society Vol.18 No.4

We prove general theorems describing the conditions ensuring that a given pseudocharacter on a normal subgroup of a given group can be extended to a pseudocharacter on the group itself and the conditions under which this extension is unique. Namely, let G be a group, let N be a normal subgroup of G, let σ : G/N → G be a section (σ(gN) ∈ gN for every g ∈ G; we also assume that σ(N) = eN = eG), and let ω = ωσ : G → N be defined by the rule ω(g) = n for g = σ(gN)n, g ∈ G, n ∈ N. Let σ(giN) = si ∈ giN, i = 1, 2, and ω(gi) = ni, i = 1, 2, and thus gi = sini, i = 1, 2. This defines a mapping G/N × G/N → N such that ω(s1s2) ∈ N. We prove that an extension of a given pseudocharacter f on N to a pseudocharacter on |G exists if and only if there is a section σ for which the set {f(ω(s1s2)), si ∈ G/N; i = 1, 2} is bounded, and this extension is unique if and only if the group G/N admits no nontrivial pseudocharacters.