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Duality between the compact and discrete objects for noncommutative topological group

Alexander I. Shtern
장전수학회 (Jangjeon Mathematical Society)|2008.04|Advanced Studies in Contemporary Mathematics|제16권 제2호|pp. 143-154

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topological groupcompact groupfinite-dimensional representationMAP groupBohr topologyPontryagin dualitygroup dualquasicompact spacediscrete grouppsudocompact grouplocally bounded group
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초록

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.

Abstract

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.

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