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 determined by the different involutions induced by positive invertible elements a ∈ A. The maps sending p to the unique with the same range as p and 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 . In this case q and r can be joined by a unique short geodesic along the space of idempotents Q of A.
@article{bwmeta1.element.bwnjournal-article-smv137i1p61bwm,
author = {E. Andruchow and Gustavo Corach and D. Stojanoff},
title = {Geometry of oblique projections},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {61-79},
zbl = {0951.46028},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv137i1p61bwm}
}
Andruchow, E.; Corach, Gustavo; Stojanoff, D. Geometry of oblique projections. Studia Mathematica, Tome 133 (1999) pp. 61-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv137i1p61bwm/
[00000] [1] Afriat S. N., Orthogonal and oblique projections and the characteristics of pairs of vector spaces, Proc. Cambridge Philos. Soc. 53 (1957), 800-816.
[00001] [2] Andruchow E., Corach G. and Stojanoff D., Projective spaces for C*-algebras, Integral Equations Operator Theory, to appear. | Zbl 0962.46040
[00002] [3] Brown L. G., The rectifiable metric on the set of closed subspaces of Hilbert space, Trans. Amer. Math. Soc. 337 (1993), 279-289. | Zbl 0822.46022
[00003] [4] Buckholtz D., Inverting the difference of Hilbert space projections, Amer. Math. Monthly 104 (1997), 60-61. | Zbl 0901.46019
[00004] [5] Coifman R. R., and Murray M. A. M., Uniform analyticity of orthogonal projections, Trans. Amer. Math. Soc. 312 (1989), 779-817. | Zbl 0675.42001
[00005] [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. | Zbl 0820.47019
[00006] [7] Corach G., Porta H. and Recht L., Differential geometry of systems of projections in Banach algebras, Pacific J. Math. 140 (1990), 209-228. | Zbl 0734.46031
[00007] [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. | Zbl 0799.46067
[00008] [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. | Zbl 0726.46028
[00009] [10] Dieudonné J., Quasi-hermitian operators, in: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Pergamon Press, Oxford, 1961, 115-122.
[00010] [11] Gerisch W., Idempotents, their Hermitian components, and subspaces in position p of a Hilbert space, Math. Nachr. 115 (1984), 283-303. | Zbl 0575.47002
[00011] [12] Householder A. S., and Carpenter J. A., The singular values of involutory and of idempotent matrices, Numer. Math. 5 (1963), 234-237. | Zbl 0118.01705
[00012] [13] Kerzman N., and Stein E. M., The Szegő kernel in terms of Cauchy-Fantappiè kernels, Duke Math. J. 45 (1978), 197-224. | Zbl 0387.32009
[00013] [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. | Zbl 0419.30012
[00014] [15] Kovarik Z. V., Similarity and interpolation between projectors, Acta Sci. Math. (Szeged) 39 (1977), 341-351. | Zbl 0392.47008
[00015] [16] Lax P. D., Symmetrizable linear transformations, Comm. Pure Appl. Math. 7 (1954), 633-647. | Zbl 0057.34402
[00016] [17] Mizel V. J., and Rao M. M., Nonsymmetric projections in Hilbert space, Pacific J. Math. 12 (1962), 343-357. | Zbl 0111.30703
[00017] [18] Odzijewicz A., On reproducing kernels and quantization of states, Comm. Math. Phys. 114 (1988), 577-597. | Zbl 0645.53044
[00018] [29] Odzijewicz A., On reproducing kernels and quantization of states, Coherent states and geometric quantization, ibid. 150 (1992), 385-413. | Zbl 0768.58022
[00019] [20] Pasternak-Winiarski Z., On the dependence of the orthogonal projector on deformations of the scalar product, Studia Math. 128 (1998), 1-17. | Zbl 0910.46013
[00020] [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. | Zbl 0739.46010
[00021] [22] Pasternak-Winiarski Z., Bergman spaces and kernels for holomorphic vector bundles, Demonstratio Math. 30 (1997), 199-214. | Zbl 0877.32003
[00022] [23] Phillips N. C., The rectifiable metric on the space of projections in a C*-algebra, Internat. J. Math. 3 (1992), 679-698. | Zbl 0778.46039
[00023] [24] Porta H., and Recht L., Spaces of projections in Banach algebras, Acta Cient. Venezolana 39 (1987), 408-426.
[00024] [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. | Zbl 0656.46042
[00025] [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. | Zbl 0831.46062
[00026] [27] Pták V., Extremal operators and oblique projections, Časopis Pěst. Mat. 110 (1985), 343-350. | Zbl 0611.47022
[00027] [28] Strătilă Ş., Modular Theory in Operator Algebras, Editura Academiei, Bucharest, 1981. | Zbl 0504.46043
[00028] [29] Zemánek J., Idempotents in Banach algebras, Bull. London Math. Soc. 11 (1979), 177-183. | Zbl 0429.46029