Geometry of oblique projections

E. Andruchow; Gustavo Corach; D. Stojanoff

Studia Mathematica (1999)

  • Volume: 137, Issue: 1, page 61-79
  • ISSN: 0039-3223

Abstract

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Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections P a determined by the different involutions a induced by positive invertible elements a ∈ A. The maps φ : P → P a sending p to the unique q ∈ P a with the same range as p and Ω a : P a → P a sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that there exists a positive element a ∈ A satisfying q , r ∈ P a . In this case q and r can be joined by a unique short geodesic along the space of idempotents Q of A.

How to cite

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Andruchow, E., Corach, Gustavo, and Stojanoff, D.. "Geometry of oblique projections." Studia Mathematica 137.1 (1999): 61-79. <http://eudml.org/doc/216675>.

@article{Andruchow1999,
abstract = {Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections $P_a$ determined by the different involutions $#_a$ induced by positive invertible elements a ∈ A. The maps $φ:P → P_a$ sending p to the unique $q ∈ P_a$ with the same range as p and $Ω_a : P_a → P_a$ sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that there exists a positive element a ∈ A satisfying $q,r ∈ P_a$. In this case q and r can be joined by a unique short geodesic along the space of idempotents Q of A.},
author = {Andruchow, E., Corach, Gustavo, Stojanoff, D.},
journal = {Studia Mathematica},
keywords = {-algebra; oblique projections; idempotents; geodesic},
language = {eng},
number = {1},
pages = {61-79},
title = {Geometry of oblique projections},
url = {http://eudml.org/doc/216675},
volume = {137},
year = {1999},
}

TY - JOUR
AU - Andruchow, E.
AU - Corach, Gustavo
AU - Stojanoff, D.
TI - Geometry of oblique projections
JO - Studia Mathematica
PY - 1999
VL - 137
IS - 1
SP - 61
EP - 79
AB - Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections $P_a$ determined by the different involutions $#_a$ induced by positive invertible elements a ∈ A. The maps $φ:P → P_a$ sending p to the unique $q ∈ P_a$ with the same range as p and $Ω_a : P_a → P_a$ sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that there exists a positive element a ∈ A satisfying $q,r ∈ P_a$. In this case q and r can be joined by a unique short geodesic along the space of idempotents Q of A.
LA - eng
KW - -algebra; oblique projections; idempotents; geodesic
UR - http://eudml.org/doc/216675
ER -

References

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  1. [1] Afriat S. N., Orthogonal and oblique projections and the characteristics of pairs of vector spaces, Proc. Cambridge Philos. Soc. 53 (1957), 800-816. 
  2. [2] Andruchow E., Corach G. and Stojanoff D., Projective spaces for C*-algebras, Integral Equations Operator Theory, to appear. Zbl0962.46040
  3. [3] Brown L. G., The rectifiable metric on the set of closed subspaces of Hilbert space, Trans. Amer. Math. Soc. 337 (1993), 279-289. Zbl0822.46022
  4. [4] Buckholtz D., Inverting the difference of Hilbert space projections, Amer. Math. Monthly 104 (1997), 60-61. Zbl0901.46019
  5. [5] Coifman R. R., and Murray M. A. M., Uniform analyticity of orthogonal projections, Trans. Amer. Math. Soc. 312 (1989), 779-817. Zbl0675.42001
  6. [6] Corach G., Operator inequalities, geodesics and interpolation, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 101-115. Zbl0820.47019
  7. [7] Corach G., Porta H. and Recht L., Differential geometry of systems of projections in Banach algebras, Pacific J. Math. 140 (1990), 209-228. Zbl0734.46031
  8. [8] Corach G., Porta H. and Recht L., Differential geometry of systems of projections in Banach algebras, The geometry of spaces of projections in C*-algebras, Adv. Math. 101 (1993), 59-77. Zbl0799.46067
  9. [9] Corach G., Porta H. and Recht L., The geometry of spaces of selfadjoint invertible elements of a C*-algebra, Integral Equations Operator Theory 16 (1993), 771-794. Zbl0726.46028
  10. [10] Dieudonné J., Quasi-hermitian operators, in: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Pergamon Press, Oxford, 1961, 115-122. 
  11. [11] Gerisch W., Idempotents, their Hermitian components, and subspaces in position p of a Hilbert space, Math. Nachr. 115 (1984), 283-303. Zbl0575.47002
  12. [12] Householder A. S., and Carpenter J. A., The singular values of involutory and of idempotent matrices, Numer. Math. 5 (1963), 234-237. Zbl0118.01705
  13. [13] Kerzman N., and Stein E. M., The Szegő kernel in terms of Cauchy-Fantappiè kernels, Duke Math. J. 45 (1978), 197-224. Zbl0387.32009
  14. [14] Kerzman N., and Stein E. M., The Szegő kernel in terms of Cauchy-Fantappiè kernels, The Cauchy kernel, the Szegő kernel, and the Riemann mapping function, Math. Ann. 236 (1978), 85-93. Zbl0419.30012
  15. [15] Kovarik Z. V., Similarity and interpolation between projectors, Acta Sci. Math. (Szeged) 39 (1977), 341-351. Zbl0392.47008
  16. [16] Lax P. D., Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633-647. Zbl0057.34402
  17. [17] Mizel V. J., and Rao M. M., Nonsymmetric projections in Hilbert space, Pacific J. Math. 12 (1962), 343-357. Zbl0111.30703
  18. [18] Odzijewicz A., On reproducing kernels and quantization of states, Comm. Math. Phys. 114 (1988), 577-597. Zbl0645.53044
  19. [29] Odzijewicz A., On reproducing kernels and quantization of states, Coherent states and geometric quantization, ibid. 150 (1992), 385-413. Zbl0768.58022
  20. [20] Pasternak-Winiarski Z., On the dependence of the orthogonal projector on deformations of the scalar product, Studia Math. 128 (1998), 1-17. Zbl0910.46013
  21. [21] Pasternak-Winiarski Z., On the dependence of the orthogonal projector on deformations of the scalar product, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), 110-134. Zbl0739.46010
  22. [22] Pasternak-Winiarski Z., Bergman spaces and kernels for holomorphic vector bundles, Demonstratio Math. 30 (1997), 199-214. Zbl0877.32003
  23. [23] Phillips N. C., The rectifiable metric on the space of projections in a C*-algebra, Internat. J. Math. 3 (1992), 679-698. Zbl0778.46039
  24. [24] Porta H., and Recht L., Spaces of projections in Banach algebras, Acta Cient. Venezolana 39 (1987), 408-426. 
  25. [25] Porta H., and Recht L., Spaces of projections in Banach algebras, Minimality of geodesics in Grassmann manifolds, Proc. Amer. Math. Soc. 100 (1987), 464-466. Zbl0656.46042
  26. [26] Porta H., and Recht L., Spaces of projections in Banach algebras, Variational and convexity properties of families of involutions, Integral Equations Operator Theory 21 (1995), 243-253. Zbl0831.46062
  27. [27] Pták V., Extremal operators and oblique projections, Časopis Pěst. Mat. 110 (1985), 343-350. Zbl0611.47022
  28. [28] Strătilă Ş., Modular Theory in Operator Algebras, Editura Academiei, Bucharest, 1981. Zbl0504.46043
  29. [29] Zemánek J., Idempotents in Banach algebras, Bull. London Math. Soc. 11 (1979), 177-183. Zbl0429.46029

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