Abstract
The main goal of this article is to set off and to study some genuine diffential geometric objects that naturally occur in the framework of C*-algebras. The intent was to develop a unified treatment of a few specific situations that were considered by the authors in previous articles (cf. [M3–4], [MS1–2], [S]), as well as by many others (cf. [ARS], [AS], [A], [CPR1–4],[LM],[Ma],[MR],[PR1–2],[W1–3]). The advantage of our present approach seems to be that that instead of certain, more or less, ad-hoc methods, we tried to find an appropriate setting and suitable simple tools which facilitate the introduction of techniques from differential geometry into operator algebras. It should be mentioned that our contribution in this paper is strongly motivated by the program initiated by M. J. Cowen and R. G. Douglas (cf. [CD1–2]). More evidence for these aspects can be found in [S], and also in [M2–3] and [MS1–2].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andruchow, E.; Recht, L.; Stojanoff, D., The space of spectral measures is a homogeneous reductive space, Integral Equations Operator Theory, 16(1993), 1–14.
Andruchow, E.; Stojanoff, D., Geometry of unitary orbits, J. Operator Theory, 26(1991), 25-41.
AUPETIT, B., Projections in real Banach algebras, Bull. London Math. Soc., 13 (1981), 412–414.
Bishop, R. L.; Crittenden, R. J., Geometry of Manifolds, Academic Press, New York, 1964.
Bourbaki, N., Variete Differentielles et Analytiques, Fascicule de results, Hermann, Paris, 1967.
Corach, G.; Porta, H.; Recht, L., Differential geometry of systems of projections in Banach algebras, Pacific J. Math., 143(1990), 209–228.
Corach, G.; Porta, H.; Recht, L., Differential geometry of spaces of relatively regular operators, Integral Equations Operator Theory, 13(1990), 771–794.
Corach, G.; Porta, H.; Recht, L., The geometry of the space of self-adjoint invertible elements in a C*-algebra, Integral Equations Operator Theory, 16(1993), 333–359.
Corach, G.; Porta, H.; Recht, L., The geometry of spaces of projections in C*-algebras, Advances in Math., to appear.
Cowen, M. J.; Douglas, R.G., Complex geometry and operator theory, Acta Math., 141(1978), 187–261.
Cowen, M. J.; Douglas, R.G., Operators possessing an open set of eigenvalues, in Colloquia Math., 35, North Holland, 1980, pp. 323–341.
Fell, J. M. G.; Doran, R. S., Representations of Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, vol. 2: Banach *-Algebraic Bundles, Induced Representations, and the Generalized Mackey Analysis, Academic Press, Inc., 1988.
Greub, W.; Halperin, S.; Vanstone, R., Connections, Curvature And Cohomology, vol. 2, Academic Press, New York, 1973.
Helgason, S., Differential Geometry and Symmetric Spaces, Academic Press, 1962.
Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry, vol. I, Interscience, New York, 1963.
Lubotzky, A.; Magid, A. R., Varietes of representations and finitely generated groups, Mem. Amer. Math. Soc., 336(1985).
Magid, A. R., Deformations of representations of discrete groups, in Classical Groups and Related Topics, Contemporary Mathematics, Vol. 82, 1989, pp. 79–87.
Martin, M., Almost product structures and derivations, Bull. Math. Soc. Sci. Math. Roumanie, 23(1979), 171–176.
Martin, M., Hermitian geometry and involutive algebras, Math. Z., 188(1985), 359–382.
Martin, M., An operator theoretic approach to analytic functions into the Grassmann manifold, Math. Balkanica, 1(1987), 45–58.
Martin, M., Projective representations of compact groups in C*-algebras, in Operator Theory: Advances and Applications, vol. 43, Birkhäuser Verlag, Basel, 1990, pp. 237–253.
Martin, M.; Salinas, N., The canonical complex structure of flag manifolds in a C*-algebra, to appear.
Martin, M.; Salinas, N., Flag manifolds and the Cowen-Douglas theory, in preparation.
Mata, L.; Recht, L., Infinite dimensional homogeneous reductive spaces, Acta Cientifica Venezolana, 43 (1992). 76–90.
Mumford, D., Algebraic Geometry I: Complexe Projective Varieties, Springer Verlag, Berlin — Heidelberg — New York, 1976.
Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. J. Math., 76 (1954), 33–65.
Porta, H.; Recht, L., Spaces of projections in Banach algebras, Acta Cientifica Venezolana, 39(1987), 408–426.
Porta, H.; Recht, L., Minimality of geodesies in Grassmann manifolds, Proc. Amer. Math. Soc., 100(1987), 464–466.
Potapov, V. P., The multiplicative structure of J-contractive matrix functions, Amer. Math. Soc. Transl., Ser. II, 15(1960), 131–244.
Salinas, N., The Grassmann manifold of a C*-algebra and hermitian holomorphic bundles, in Operator Theory: Advances and Applications, vol. 28, Birckhäuser Verlag, Basel, 1988, pp. 267–289.
Upmeier, H., Symmetric Banach manifolds and Jordan C*-algebras, North-Holland Math. Studies, 104, North-Holland, Amsterdam, 1985.
Wilkins, D. R., The Grassmann manifold of a C*-algebra, Proc. of the Royal Irish Acad., 90A (1990), 99–116.
Wilkins, D. R., On infinite-dimensional symmetric spaces, to appear.
Wilkins, D. R., On the classification of Grassmann manifolds in C*-algebras, to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Martin, M., Salinas, N. (1995). Differential Geometry of Generalized Grassmann Manifolds in C*-Algebras 241. In: Gohberg, I., Langer, H. (eds) Operator Theory and Boundary Eigenvalue Problems. Operator Theory: Advances and Applications, vol 80. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9106-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9106-6_13
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9909-3
Online ISBN: 978-3-0348-9106-6
eBook Packages: Springer Book Archive