Algebraically determined semidirect products

作者：Al Tameemi We' am M^*; Kallman Robert R

来源：Topology and Its Applications, 2014, 175: 43-48.

DOI：10.1016/j.topol.2014.06.016

• Polish topological groups
• Semidirect products
• Descriptive set theory
• Hilbert space
• Unitary operators
• C*-algebras
• Analytic set

Let G be a Polish (i.e., complete separable metric topological) group. Define G to be an algebraically determined Polish group if given any Polish group L and an algebraic isomorphism phi : L bar right arrow G, then phi is a topological isomorphism. The purpose of this paper is to prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This general theorem will provide a flowchart or recipe for proving that some special semidirect products are algebraically determined. For example, it may be used to prove that the natural semidirect product H (sic) g, where H is the additive group of a separable Hilbert space and g is a Polish group of unitaries on H acting transitively on the unit sphere with -I is an element of g, is algebraically determined. An example of such a g is the unitary group of a separable irreducible C*-algebra with identity on H. Not all nontrivial semidirect products of Polish groups are algebraically determined, for it is known that the Heisenberg group H-3(R) is a semidirect product of the form R-2 x (theta) R-1 and is not an algebraically determined Polish group. Published by Elsevier B.V.

• 出版日期2014-9-15