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GEOMETRIC CHARACTERIZATION OF HERMITIAN ALGEBRAS WITH CONTINUOUS INVERSION

Part of: Infinite-dimensional manifolds Topological (rings and) algebras with an involution Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  02 October 2009

DANIEL BELTIŢĂ  and
KARL-HERMANN NEEB
DANIEL BELTIŢĂ*
Affiliation:
Institute of Mathematics, ‘Simion Stoilow’ of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania (email: Daniel.Beltita@imar.ro)
KARL-HERMANN NEEB
Affiliation:
Department of Mathematics, Darmstadt University of Technology, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany (email: neeb@mathematik.tu-darmstadt.de)
*
✉For correspondence; e-mail: Daniel.Beltita@imar.ro
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Abstract

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A hermitian algebra is a unital associative ℂ-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ℝ. In the case of an algebra 𝒜 endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra 𝒜 is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups Un(𝒜) on each flag manifold is transitive.

Keywords

algebra with continuous inversionflag manifoldlocally convex Lie group

MSC classification

Secondary: 46K05: General theory of topological algebras with involution 58B10: Differentiability questions 46H30: Functional calculus in topological algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 96 - 113
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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