Sphere and projective space of a C*-algebra with a faithful state

• Autorzy: Antunez Andrea C.

Identyfikatory

DOI

10.1515/dema-2019-0036

• Języki publikacji: EN

Abstrakty

EN

Let A be a unital C*-algebra with a faithful state φ. We study the geometry of the unit sphere Sφ = {x∈A : φ(x*x) = 1} and the projective space Pφ = Sφ/T. These spaces are shown to be smooth manifolds and homogeneous spaces of the group Uφ(A) of isomorphisms acting in A which preserve the inner product induced by φ, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in Pφ, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

Słowa kluczowe

EN

homogeneous space; minimal curves; C*-algebra; projective space

PL

przestrzeń jednorodna; C*-algebra; przestrzeń rzutowa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2019
• Tom: Vol. 52, nr 1
• Strony: 410--427
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Antunez Andrea C.

• Instituto de Ciencias, Universidad Nacional de General Sarmiento

Bibliografia

• [1] Krein M. G., Compact linear operators on functional spaces with two norms, Translated from the Ukranian (Dedicated to the memory of Mark Grigorievich Krein 1907-1989), Integral Equations Operator Theory, 1998, 30(2), 140-162
• [2] Lax P. D., Symmetrizable linear transformations, Comm. Pure Appl. Math., 1954, 7, 633-647
• [3] Corach G., Porta H., Recht L., The geometry of spaces of projections in C*-algebras, Adv. Math., 1993, 101(1), 59-77
• [4] Porta H., Recht L., Minimality of geodesics in Grassmann manifolds, Proc. Amer. Math. Soc., 1987, 100(3), 464-466
• [5] Gohberg I. C., Zambickii M. K., On the theory of linear operators in spaces with two norms (Russian) Ukrain. Mat. Ž., 1966, 18(1), 11-23
• [6] Porta H., Recht L., Spaces of projections in a Banach algebra, Acta Cient. Venezolana, 1987, 38(4), 408-426
• [7] Beltiţă D., Smooth homogeneous structures in operator theory, Chapman and Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics, 2006, 137
• [8] Raeburn I., The relationship between a commutative Banach algebra and its maximal ideal space, J. Funct. Anal., 1977, 25(4),366-390
• [9] Andruchow E., Larotonda G., Hopf-Rinow theorem in the Sato Grassmannian, J. Funct. Anal., 2008, 255(7), 1692-1712
• [10] Andruchow E., Operators which are the difference of two projections, J. Math. Anal. Appl., 2014, 420(2), 1634-1653

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).