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2007 | 5 | 4 | 639-653
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Metrics in the sphere of a C*-module

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EN
Abstrakty
EN
Given a unital C*-algebra $$\mathcal{A}$$ and a right C*-module $$\mathcal{X}$$ over $$\mathcal{A}$$ , we consider the problem of finding short smooth curves in the sphere $$\mathcal{S}_\mathcal{X} $$ = {x ∈ $$\mathcal{X}$$ : 〈x, x〉 = 1}. Curves in $$\mathcal{S}_\mathcal{X} $$ are measured considering the Finsler metric which consists of the norm of $$\mathcal{X}$$ at each tangent space of $$\mathcal{S}_\mathcal{X} $$ . The initial value problem is solved, for the case when $$\mathcal{A}$$ is a von Neumann algebra and $$\mathcal{X}$$ is selfdual: for any element x 0 ∈ $$\mathcal{S}_\mathcal{X} $$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x 0 and $$\dot \gamma $$ (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0, x 1 ∈ $$\mathcal{S}_\mathcal{X} $$ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f 0 the selfadjoint projection I − x 0 ⊗ x 0, if the algebra f 0 $$\mathcal{L}_\mathcal{A} (\mathcal{X})$$ f 0 is finite dimensional, then there exists a curve γ joining x 0 and x 1, which is minimizing along its path.
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Czasopismo
Rocznik
Tom
5
Numer
4
Strony
639-653
Opis fizyczny
Daty
wydano
2007-12-01
online
2007-12-01
Twórcy
Bibliografia
  • [1] E. Andruchow, G. Corach and M. Mbekhta: “On the geometry of generalized inverses”, Math. Nachr., Vol. 278, (2005), no. 7–8, pp. 756–770. http://dx.doi.org/10.1002/mana.200310270
  • [2] E. Andruchow, G. Corach and D. Stojanoff: “Geometry of the sphere of a Hilbert module”, Math. Proc. Cambridge Philos. Soc., Vol. 127, (1999), no. 2, pp. 295–315. http://dx.doi.org/10.1017/S0305004199003771
  • [3] E. Andruchow, G. Corach and D. Stojanoff: “Projective spaces of a C*-algebra”, Integral Equations Operator Theory, Vol. 37, (2000), no. 2, pp. 143–168. http://dx.doi.org/10.1007/BF01192421
  • [4] E. Andruchow and A. Varela: “C*-modular vector states”, Integral Equations Operator Theory, Vol. 52, (2005), pp. 149–163. http://dx.doi.org/10.1007/s00020-002-1280-y
  • [5] C.J. Atkin: “The Finsler geometry of groups of isometries of Hilbert space”, J. Austral. Math. Soc. Ser. A, Vol. 42, (1987), pp. 196–222. http://dx.doi.org/10.1017/S1446788700028202
  • [6] C. Davis, W.M. Kahan and H.F. Weinberger: “Norm preserving dilations and their applications to optimal error bounds”, SIAM J. Numer. Anal., Vol. 19, (1982), pp. 445–469. http://dx.doi.org/10.1137/0719029
  • [7] C.E. Durán, L.E. Mata-Lorenzo and L. Recht: “Metric geometry in homogeneous spaces of the unitary group of a C*-algebra. Part I. Minimal curves”, Adv. Math., Vol. 184, (2004), no. 2, pp. 342–366. http://dx.doi.org/10.1016/S0001-8708(03)00148-8
  • [8] C.E. Durán, L.E. Mata-Lorenzo and L. Recht: “Metric geometry in homogeneous spaces of the unitary group of a C*-algebra. Part II. Geodesics joining fixed endpoints”, Integral Equations Operator Theory, Vol. 53, (2005), no. 1, pp. 33–50. http://dx.doi.org/10.1007/s00020-003-1305-1
  • [9] S. Kobayashi and K. Nomizu: Foundations of differential geometry, Vol. II. Reprint of the 1969 original, Wiley Classics Library, John Wiley & Sons, New York, 1996.
  • [10] M.G. Krein: “The theory of selfadjoint extensions of semibounded Hermitian transformations and its applications”, Mat. Sb., Vol. 20, (1947), pp. 431–495, Vol. 21, (1947), pp. 365–404 (in Russian).
  • [11] E.C. Lance: “Hilbert C*-modules, A toolkit for operator algebraists”, London Math. Soc. Lecture Note Ser., Vol. 210, Cambridge University Press, Cambridge, 1995.
  • [12] P.R. Halmos and J.E. McLaughlin: “Partial isometries”, Pacific J. Math., Vol. 13, (1963), pp. 585–596.
  • [13] L.E. Mata-Lorenzo and L. Recht: “Infinite-dimensional homogeneous reductive spaces”, Acta Cient. Venezolana, Vol. 43, (1992), pp. 76–90.
  • [14] S. Parrott: “On a quotient norm and the Sz.-Nagy-Foias lifting theorem”, J. Funct. Anal., Vol. 30, (1978), no. 3, pp. 311–328. http://dx.doi.org/10.1016/0022-1236(78)90060-5
  • [15] W.L. Paschke: “Inner product modules over B*-algebras”, Trans. Amer. Math. Soc., Vol. 182, (1973), pp. 443–468. http://dx.doi.org/10.2307/1996542
  • [16] F. Riesz and B. Sz.-Nagy: Functional Analysis, Frederick Ungar Publishing Co., New York, 1955.
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Bibliografia
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