×

The space of spectral measures is a homogeneous reductive space. (English) Zbl 0798.47009

Authors’ abstract: We prove that the space of spectral measures on a \(W^*\)-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.

MSC:

47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
58B10 Differentiability questions for infinite-dimensional manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Apostol, C.; Fialkow, L.A.; Herrero, D.A. and Voiculescu, D.; Approximation of Hilbert space operators, Vol 2, Pitman, Boston, 1984. · Zbl 0572.47001
[2] Arveson, W.; Analicity in Operator Algebras, American J. Math. 89 3 (1967), 578-642. · Zbl 0183.42501 · doi:10.2307/2373237
[3] Corach, G.; Porta, H. and Recht, L.; The geometry of spaces of projections in C*-algebras, Advances in Math. (to appear). · Zbl 0799.46067
[4] Corach, G.; Porta, H. and Recht, L.; Multiplicative integrals and geometry of spaces of projections, Rev. de la UMA 34 (1988) 132-149. · Zbl 0756.46041
[5] Dixmier, J.; Les Algebres d’operateurs dans l’espace Hilbertien. Gauthier-Villars, Paris, 1957. · Zbl 0088.32304
[6] Kadison, R.V. and Ringrose, J.R.; Fundamentals of the Theory of Operator Algebras, Vol 2, Academic Press, New York, 1986. · Zbl 0601.46054
[7] Kobayashi, S. and Nomizu, K.; Foundations of Differential Geometry, Vol 2, Interscience Publishers, New York, 1969. · Zbl 0175.48504
[8] Neumann J. von; On rings of operators, III, Ann. of Math. 41 (1949), 94-161. · Zbl 0023.13303
[9] Recht, L.; Stojanoff, D. and Suarez, D.; Conditional expectations inL 2[0,1] (preprint).
[10] Umegaki, H.; Conditional expectation in an operator algebra, III, Kodai Math. Sem. Rep. 11 (1959), 51-64. · Zbl 0102.10801 · doi:10.2996/kmj/1138844157
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.