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FREUDENTHAL-WEIL THEOREM FOR ARBITRARY EMBEDDINGS OF CONNECTED LIE GROUPS IN COMPACT GROUPS
Shtern 장전수학회 2009 Advanced Studies in Contemporary Mathematics Vol.19 No.2
It is proved that, if the underlying abstract group of a connected Lie group admits an embedding in a compact topological group, then the connected Lie group necessarily admits a continuous embedding in a compact topological group, and thus, by the famous Freudenthal-Weil theorem, the group is a direct product of a compact connected Lie group and a vector group. Some applications are indicated and some generalizations and related results are mentioned.
HOMOMORPHIC IMAGES OF CONNECTED LIE GROUPS IN COMPACT GROUPS
Shtern 장전수학회 2010 Advanced Studies in Contemporary Mathematics Vol.20 No.1
It is proved that the image under a homomorphism (which need not be continuous) of a connected Lie group in a compact topological group is the image (in the same compact group) under a homomorphism (which need not be continuous) of a direct product of a compact Lie group and a vector group.
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.
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. I. Shtern 장전수학회 2009 Advanced Studies in Contemporary Mathematics Vol.19 No.1
In the paper, which is an extended version of the text of a talk to be presented at the 20th International Congress of the JMS, the main results obtained till now in the theory of finite-dimensional locally bounded quasirepresentations of connected locally compact groups are summarized.
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.
A. I. Shtern 장전수학회 2018 Proceedings of the Jangjeon mathematical society Vol.21 No.4
We obtain a criterion for the continuity of the restriction of a locally bounded finite-dimensional representation of an almost connected locally compact group to the commutator subgroup of the group.
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.
A. I. Shtern 장전수학회 2018 Advanced Studies in Contemporary Mathematics Vol.28 No.4
We obtain a criterion for the continuity of a locally bounded finite-dimensional representation of a totally disconnected locally compact group.
A. I. Shtern 장전수학회 2019 Proceedings of the Jangjeon mathematical society Vol.22 No.2
We describe the structure of indecomposable locally bounded finite-dimensional pseudorepresentations of connected Lie groups.