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Geometric characterization of tripotents in real and complex JB$$^{*}$$-triples. (English) Zbl 1058.46033
The authors show that the geometric characterization of tripotent elements (in other words, partial isometries) in $$C^*$$-algebras obtained by C. A. Akemann and N. Weaver [Proc. Am. Math. Soc. 130, No. 10, 3033–3037 (2002; Zbl 1035.46038)] is also valid for $$JB^*$$-triples (every $$C^*$$-algebra is a $$JB^*$$-triple for the triple product defined by $$\{a,b,c\}= 1/2(ab^*c+ cb^*a)$$, and the same norm). As a consequence, they obtain an alternative proof of W. Kaup’s Banach-Stone theorem for $$JB^*$$-triples [Math. Z. 183, 503–529 (1983; Zbl 0519.32024)]. The basic geometric tool used in the proofs involves results on the $$M$$-structure in $$JB^*$$-triples and $$JBW^*$$-triples and on the dual $$L$$-structure in their dual and preduals.

##### MSC:
 46K70 Nonassociative topological algebras with an involution 17C65 Jordan structures on Banach spaces and algebras
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##### References:
 [1] Akemann, C; Weawer, N, Geometric characterizations of some classes of operators in C^∗-algebras and von Neumann algebras, Proc. amer. math. soc., 130, 3033-3037, (2002) · Zbl 1035.46038 [2] Arazy, J; Kaup, W, On the holomorphic rigidity of linear operators on complex Banach spaces, Quart. J. math. Oxford ser. (2), 50, 249-277, (1999) · Zbl 0964.46026 [3] Dang, T; Friedman, Y; Russo, B, Affine geometric proofs of the Banach stone theorems of kadison and Kaup, (), Rocky mountain J. math., 20, 409-428, (1990) · Zbl 0738.47029 [4] Dineen, S, The second dual of a JB∗-triple system, (), 67-69 [5] Edwards, C.M; Rüttimann, G.F, Orthogonal faces of the unit ball in a Banach space, Atti sem. mat. fis. univ. modena, 49, 473-493, (2001) · Zbl 1009.46007 [6] C.M. Edwards, R. Hügli, M-orthogonality and holomorphic rigidity in complex Banach spaces, Acta. Sci. Math., in press [7] Friedman, Y; Russo, B, Structure of the predual of a JBW^∗-triple, J. reine angew. math., 356, 67-89, (1985) · Zbl 0547.46049 [8] Friedman, Y; Russo, B, Affine structure of facially symmetric spaces, Math. proc. Cambridge philos. soc., 106, 107-124, (1989) · Zbl 0693.46010 [9] Friedman, Y; Russo, B, Classification of atomic facially symmetric spaces, Canad. J. math., 45, 33-87, (1993) · Zbl 0803.46015 [10] Hanche-Olsen, H; Størmer, E, Jordan operator algebras, Monographs and studies in mathematics, vol. 21, (1984), Pitman London · Zbl 0561.46031 [11] Horn, G, Characterization of the predual and ideal structure of a JBW^∗-triple, Math. scand., 61, 117-133, (1987) · Zbl 0659.46062 [12] R. Hügli, Contractive projections on a JBW∗-triple and its predual, Inauguraldissertation, Universität Bern, 2001 [13] Isidro, J.M; Kaup, W; Rodrı́guez, A, On real forms of JB^∗-triples, Manuscripta math., 86, 311-335, (1995) · Zbl 0834.17047 [14] Kaup, W, Algebraic characterization of symmetric complex Banach manifolds, Math. ann., 228, 39-64, (1977) · Zbl 0335.58005 [15] Kaup, W, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 183, 503-529, (1983) · Zbl 0519.32024 [16] Kaup, W; Upmeier, H, Jordan algebras and symmetric Siegel domains in Banach spaces, Math. Z., 157, 179-200, (1977) · Zbl 0357.32018 [17] Martı́nez, J; Peralta, A.M, Separate weak∗-continuity of the triple product in dual real JB∗-triples, Math. Z., 234, 635-646, (2000) · Zbl 0977.17032 [18] Pedersen, G.K, $$C\^{}\{∗\}$$-algebras and their automorphism groups, London mathematical society monographs, vol. 14, (1979), Academic Press London [19] Peralta, A.M; Stacho, L.L, Atomic decomposition of real JBW^∗-triples, Quart. J. math. Oxford, 52, 79-87, (2001) · Zbl 0982.46058 [20] A. Rodrı́guez-Palacios, Banach space characterizations of unitaries, Preprint, 2003
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