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Duality between the compact and discrete objects for noncommutative topological group
Alexander I. Shtern 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.16 No.2
The duality between compact and discrete objects, which is one of thebrilliant results of Pontryagin’s duality theory for locally compact Abelian groups,is partially valid for noncommutative objects. As is well known, the dual space of acompact topological group is discrete and the dual space of a discrete group is quasicompact (i.e., satises the open covering theorem but is possibly not Hausdor).The first assertion admits the converse, whereas the other does not (there is a simple example of a nondiscrete solvable group of height wo whose dual space is quasicompact and not Hausdor (a T1 space)). However, in the class of locally compact groupsall of whose irreducible representations are finite-dimensional, a group is discrete ifand only if its dual space is quasicompact (and is automatically at least a T1 space).The proof is based on a structure theorem for locally compact groups all of whoseirreducible representations are finite-dimensional.Some kind of duality in question is preserved for not necessarily locally compactgroups if these groups are reexively or unitarily representable and (in the simplestcase) their irreducible representations form a rich family and are of bounded degree.The corresponding analogs of compactness and discreteness are less recognizable butstill dual to each other.
CONNECTED LOCALLY COMPACT GROUPS HAVING SUFFICIENTLY MANY FINITE-DIMENSIONAL LINEAR REPRESENTATIONS
Alexander I. Shtern 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.1
The connected locally compact groups admitting a separating family of continuous linear finite-dimensional representations are described.
Alexander I. Shtern 장전수학회 2010 Proceedings of the Jangjeon mathematical society Vol.13 No.2
A version of the Hochschild universal kernel theorem for (not necessarily continuous) locally bounded finite-dimensional representations of connected reductive Lie groups is presented.
VON NEUMANN KERNEL OF A CONNECTED LOCALLY COMPACT GROUP, REVISITED
Alexander I. Shtern 장전수학회 2010 Advanced Studies in Contemporary Mathematics Vol.20 No.3
In the paper, a 1984 result (of two well-known mathematicians)concerning the von Neumann kernel of a connected locally compact group (the intersection of kernels of all irreducible finite-dimensional continuous complex unitary representations of the group) is corrected.
Alexander I. Shtern 장전수학회 2010 Proceedings of the Jangjeon mathematical society Vol.13 No.3
A version of the Hochschild universal kernel theorem for (not necessarily continuous) locally bounded finite-dimensional representations of connected Lie groups with simple Levi subgroup is presented.
Alexander I. Shtern 장전수학회 2010 Advanced Studies in Contemporary Mathematics Vol.20 No.2
The paper is an extended version of the text of a talk to be presented at the 22nd International Conference of the JMS. A brief survey of results that are consequences of automatic continuity theorem generalizing the van der Waerden continuity theorem is presented. These results deal with a sharpening of the Freudenthal–Weil theorem, with an analog of this theorem for amenable connected locally compact groups, and a version of the Hochschild universal kernel theorem for finite-dimensional representations of connected Lie groups.