Abstract
In a recent paper [9], Norberto Salinas proved that the closed subset P of all hermitian idempotents of a unital C*-algebra A is a real analytic manifold, and studied some of its geometrical properties. This space, called the Grassmann manifold of A, has a simple alternative description. There is a bijective correspondence from P to the set of all unitary representations of the cyclic group Z/2 in A. More precisely, each hermitian idempotent e in A corresponds to a representation a of Z/2 = {0,1} in A, uniquely defined by α(1) = 1 – 2e.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Beyl, F.R.; Tappe, J.: Group extensions, representations, and the Schur multiplicator, Lecture Notes in Mathematics, vol. 958, Springer Verlag, Berlin, Heidelberg, New-York, 1982.
Bourbaki, N.: Variété différentielles et analytiques, Fascicule de resultats, Hermann, Paris, 1967.
Cowen, M. J.; Douglas, R.G.: Complex geometry and operator theory, Acta Math. 141(1979), 187–261.
Martin, M.: Almost product structures and derivations, Bull. Math. Soc. Sci. Math. R.S. Roumanie 23(1979), 171–176.
Martin, M.: Geometric structures on vector bundles (in Romanian), Stud. Cerc. Mat. 36(1984): I, (2), 113–142; II, (3), 171–192.
Martin, M.: Projective group representations and geometric structures, Preprint INCREST 53(1986).
Mumford, D.: Algebraic geometry. I: Complex projective varieties, Springer Verlag, Berlin, Heidelberg, New-York, 1976.
Recht, L.; Potra, H.: On the minimality of geodesies in Grassmann manifolds, Proc. Amer. Math. Soc.
Salinas, N.: The Grassmann manifold of a C*-algebra and hermitian holomorphic bundles, in Special classes of linear operators and other topics, Birkhäuser-Verlag, 1988, pp. 267–289.
Varadarajan, V.S.: Geometry of quantum theory, Springer Verlag, Berlin, Heidelberg, New-York, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Martin, M. (1990). Projective Representations of Compact Groups in C*-Algebras. In: Helson, H., Sz.-Nagy, B., Vasilescu, FH., Arsene, G. (eds) Linear Operators in Function Spaces. Operator Theory: Advances and Applications, vol 43. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7250-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7250-8_17
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-7252-2
Online ISBN: 978-3-0348-7250-8
eBook Packages: Springer Book Archive