A projection-less approach to Rickart Jordan structures. (English) Zbl 1511.17069
The authors introduce weakly and weakly order Rickart \(JB^*\)-triples, and show that a \(C^*\)-algebra \(A\) is a weakly (order) Rickart \(JB^*\)-triple precisely when it is a weakly Rickart \(C^*\)-algebra. They also prove that the Pierce-2 subspace associated with any tripotent in a weakly order Rickart \(JB^*\)-triple is a Rickart \(JB^*\)-algebra in the sense of the reviewer and F. Arzikulov [“Jordan counterparts of Rickart and Baer \({}^*\)-algebras”, Uzbek Math. J. 1, 13–33 (2016); São Paulo J. Math. Sci. 13, No. 1, 27–38 (2019; Zbl 1441.17025)]. By extending a classical property of Rickart \(C^*\)-algebras, they prove that every weakly order Rickart \(JB^*\)-triple is generated by its idempotents.
Reviewer: Sh. A. Ayupov (Tashkent)
MSC:
| 17C65 | Jordan structures on Banach spaces and algebras |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 46L57 | Derivations, dissipations and positive semigroups in \(C^*\)-algebras |
| 46L05 | General theory of \(C^*\)-algebras |
| 46L60 | Applications of selfadjoint operator algebras to physics |
Keywords:
Rickart C\(^\ast\)-algebra, JB\(^\ast\)-algebra and JB\(^\ast\)-triple; Baer C\(^\ast\)-algebra and JB\(^\ast\)-algebra; weakly order Rickart JB\(^\ast\)-triple; von Neumann regularity; inner idealCitations:
Zbl 1441.17025References:
| [1] | Alfsen, E. M.; Shultz, F. W., Geometry of State Spaces of Operator Algebras, Mathematics: Theory & Applications (2003), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 1042.46001 |
| [2] | Ara, P., Left and right projections are equivalent in Rickart C^⁎-algebras, J. Algebra, 120, 433-448 (1989) · Zbl 0694.46038 |
| [3] | Ara, P.; Goldstein, D., A solution of the matrix problem for Rickart C^⁎-algebras, Math. Nachr., 164, 259-270 (1993) · Zbl 0813.46046 |
| [4] | Arzikulov, F. N., On abstract JW-algebras, Sib. Math. J., 39, 18-24 (1998) · Zbl 0917.46062 |
| [5] | Arzikulov, F. N., AJW-algebras of type I and their classification, Sib. Adv. Math., 8, 30-48 (1998) · Zbl 0924.46057 |
| [6] | Arzikulov, F. N., On an analogue of the Peirce decomposition, Sib. Math. J., 40, 413-418 (1999) · Zbl 0938.46054 |
| [7] | Ayupov, Sh. A.; Arzikulov, F. N., Jordan counterparts of Rickart and Baer^⁎-algebras, Uzbek. Mat. Zh., 1, 13-33 (2016) |
| [8] | Ayupov, Sh. A.; Arzikulov, F. N., Jordan counterparts of Rickart and Baer^⁎-algebras II, São Paulo J. Math. Sci., 13, 27-38 (2019) · Zbl 1441.17025 |
| [9] | Barton, T.; Friedman, Y., Grothendieck’s inequality for JB^⁎-triples and applications, J. Lond. Math. Soc. (2), 36, 513-523 (1987) · Zbl 0652.46039 |
| [10] | Barton, T.; Friedman, Y., Bounded derivations of JB^⁎-triples, Q. J. Math. Oxf. Ser. (2), 41, 255-268 (1990) · Zbl 0728.46046 |
| [11] | Barton, T.; Timoney, R. M., Weak^⁎-continuity of Jordan triple products and its applications, Math. Scand., 59, 177-191 (1986) · Zbl 0621.46044 |
| [12] | Berberian, S. K., Baer^⁎-Rings, Die Grundlehren der mathematischen Wissenschaften, vol. 195 (1972), Springer-Verlag: Springer-Verlag New York-Berlin · Zbl 0242.16008 |
| [13] | Braun, R.; Kaup, W.; Upmeier, H., A holomorphic characterisation of Jordan-C^⁎-algebras, Math. Z., 161, 277-290 (1978) · Zbl 0385.32002 |
| [14] | Brown, L. G.; Pedersen, G. K., On the geometry of the unit ball of a C^⁎-algebra, J. Reine Angew. Math., 469, 113-147 (1995) · Zbl 0834.46041 |
| [15] | Bunce, L. J., On compact action in JB-algebras, Proc. Edinb. Math. Soc. (2), 26, 353-360 (1983) · Zbl 0531.46038 |
| [16] | Bunce, L. J.; Chu, C.-H.; Zalar, B., Structure spaces and decomposition in JB^⁎-triples, Math. Scand., 86, 17-35 (2000) · Zbl 1073.46513 |
| [17] | Burgos, M.; Fernández-Polo, F. J.; Garcés, J. J.; Martínez Moreno, J.; Peralta, A. M., Orthogonality preservers in C^⁎-algebras, JB^⁎-algebras and JB^⁎-triples, J. Math. Anal. Appl., 348, 220-233 (2008) · Zbl 1156.46045 |
| [18] | Burgos, M.; Fernández-Polo, F. J.; Garcés, J. J.; Peralta, A. M., Orthogonality preservers revisited, Asian-Eur. J. Math., 2, 387-405 (2009) · Zbl 1207.46061 |
| [19] | Burgos, M.; Garcés, J. J.; Peralta, A. M., Automatic continuity of biorthogonality preservers between weakly compact JB^⁎-triples and atomic JBW^⁎-triples, Stud. Math., 204, 97-121 (2011) · Zbl 1234.46055 |
| [20] | Burgos, M.; Kaidi, A.; Morales, A.; Peralta, A. M.; Ramírez, M. I., von Neumann regularity and quadratic conorms in JB^⁎-triples and C^⁎-algebras, Acta Math. Sin. Engl. Ser., 24, 2, 185-200 (2008) · Zbl 1157.46042 |
| [21] | Burgos, M. J.; Márquez-García, A. C.; Morales-Campoy, A.; Peralta, A. M., Linear maps between C^⁎-algebras preserving extreme points and strongly linear preservers, Banach J. Math. Anal., 10, 3, 547-565 (2016) · Zbl 1385.47013 |
| [22] | Cabrera García, M.; Rodríguez Palacios, Á., Non-Associative Normed Algebras, vol. 1, Encyclopedia of Mathematics and Its Applications, vol. 154 (2014), Cambridge University Press: Cambridge University Press Cambridge, The Vidav-Palmer and Gelfand-Naimark theorems · Zbl 1322.46003 |
| [23] | Cabrera García, M.; Rodríguez Palacios, Á., Non-Associative Normed Algebras, vol. 2, Encyclopedia of Mathematics and Its Applications, vol. 167 (2018), Cambridge University Press: Cambridge University Press Cambridge, Representation theory and the Zel’manov approach · Zbl 1390.17001 |
| [24] | Dineen, S., The second dual of a JB^⁎-triple system, (Múgica, J., Complex Analysis, Functional Analysis and Approximation Theory. Complex Analysis, Functional Analysis and Approximation Theory, North-Holland Math. Stud., vol. 125 (1986), North-Holland: North-Holland Amsterdam-New York), 67-69 · Zbl 0653.46053 |
| [25] | Dixmier, J., Sur certains espaces consideres par M. H. Stone, Summa Bras. Math., 2, 151-182 (1951) · Zbl 0045.38002 |
| [26] | Edwards, C. M., Ideal theory in JB-algebras, J. Lond. Math. Soc. (2), 16, 507-513 (1977) · Zbl 0368.46044 |
| [27] | Edwards, C. M.; Rüttimann, G. T., On the facial structure of the unit balls in a JBW^⁎-triple and its predual, J. Lond. Math. Soc. (2), 38, 317-322 (1988) · Zbl 0621.46043 |
| [28] | Edwards, C. M.; Rüttimann, G. T., Inner ideals in C^⁎-algebras, Math. Ann., 290, 621-628 (1991) · Zbl 0743.46049 |
| [29] | Edwards, C. M.; Rüttimann, G. T., A characterization of inner ideals in JB^⁎-triples, Proc. Am. Math. Soc., 116, 1049-1057 (1992) · Zbl 0783.46033 |
| [30] | Edwards, C. M.; Rüttimann, G. T., Peirce inner ideals in Jordan^⁎-triples, J. Algebra, 180, 41-66 (1996) · Zbl 0847.17028 |
| [31] | Fernández López, A.; García Rus, E.; Sánchez Campos, E.; Siles Molina, M., Strong regularity and generalized inverses in Jordan systems, Commun. Algebra, 20, 7, 1917-1936 (1992) · Zbl 0759.17019 |
| [32] | Fernández López, A.; García Rus, E.; Sánchez Campos, E.; Siles Molina, M., Strong regularity and Hermitian Hilbert triples, Q. J. Math. Oxf. Ser. (2), 45, 177, 43-55 (1994) · Zbl 0807.46066 |
| [33] | Friedman, Y.; Russo, B., Structure of the predual of a JBW^⁎-triple, J. Reine Angew. Math., 356, 67-89 (1985) · Zbl 0547.46049 |
| [34] | Garcés, J. J.; Peralta, A. M., One-parameter groups of orthogonality preservers on JB^⁎-algebras, Adv. Oper. Theory, 6 (2021), Paper No. 43, 25 pp · Zbl 1529.47069 |
| [35] | Goldstein, D., Polar decomposition in Rickart C^⁎-algebras, Publ. Mat., 39, 5-21 (1995) · Zbl 0836.46047 |
| [36] | Grove, K.; Pedersen, G. K., Sub-Stonean spaces and corona sets, J. Funct. Anal., 56, 124-143 (1984) · Zbl 0539.54029 |
| [37] | Friedman, Y.; Russo, B., The Gelfand-Naimark theorem for JB^⁎-triples, Duke Math. J., 53, 139-148 (1986) · Zbl 0637.46049 |
| [38] | Hamhalter, J.; Kalenda, O.; Peralta, A. M.; Pfitzner, H., Grothendieck’s inequalities for JB^⁎-triples: proof of the Barton-Friedman conjecture, Trans. Am. Math. Soc., 374, 1327-1350 (2021) · Zbl 1482.46089 |
| [39] | Hanche-Olsen, H.; Størmer, E., Jordan Operator Algebras (1984), Pitman: Pitman London · Zbl 0561.46031 |
| [40] | Handelman, D., Finite Rickart C^⁎-Algebras and Their Properties. Studies in Analysis, Adv. in Math. Suppl. Stud., vol. 4, 171-196 (1979), Academic Press: Academic Press New York-London · Zbl 0511.46054 |
| [41] | Harte, R.; Mbekhta, M., On generalized inverses in C^⁎-algebras, Stud. Math., 103, 1, 71-77 (1992) · Zbl 0810.46062 |
| [42] | Harte, R.; Mbekhta, M., Generalized inverses in C^⁎*-algebras. II, Stud. Math., 106, 2, 129-138 (1993) · Zbl 0810.46063 |
| [43] | Jamjoom, F. B.; Peralta, A. M.; Siddiqui, A. A.; Tahlawi, H. M., Approximation and convex decomposition by extremals and the λ-function in JBW^⁎-triples, Q. J. Math., 66, 2, 583-603 (2015) · Zbl 1341.46010 |
| [44] | Jamjoom, F. B.; Peralta, A. M.; Siddiqui, A. A.; Tahlawi, H. M., Extremally rich JB^⁎-triples, Ann. Funct. Anal., 7, 4, 578-592 (2016) · Zbl 1385.46008 |
| [45] | Jordan, P.; von Neumann, J.; Wigner, E. P., On an algebraic generalization of the quantum mechanical formalism, Ann. Math. (2), 35, 29-64 (1934) · JFM 60.0902.02 |
| [46] | Kaplansky, I., Projections in Banach algebras, Ann. Math. (2), 53, 235-249 (1951) · Zbl 0042.12402 |
| [47] | Kaplansky, I., Algebras of type I, Ann. Math. (2), 56, 460-472 (1952) · Zbl 0047.35701 |
| [48] | Kaplansky, I., Modules over operator algebras, Am. J. Math., 75, 839-858 (1953) · Zbl 0051.09101 |
| [49] | Kaup, W., A Riemann Mapping Theorem for bounded symmentric domains in complex Banach spaces, Math. Z., 183, 503-529 (1983) · Zbl 0519.32024 |
| [50] | Kaup, W., On spectral and singular values in JB^⁎-triples, Proc. R. Ir. Acad., A Math. Phys. Sci., 96, 1, 95-103 (1996) · Zbl 0904.46039 |
| [51] | Kaup, W., On Grassmanian associated with JB^⁎-triples, Math. Z., 236, 567-584 (2001) · Zbl 0988.46048 |
| [52] | Liebmann, M.; Rühaak, H.; Henschenmacher, B., Non-associative algebras and quantum physics. A historical perspective |
| [53] | Martínez, 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 |
| [54] | Murphy, G. J., C^⁎-Algebras and Operator Theory (1990), Academic Press, Inc.: Academic Press, Inc. Boston, MA · Zbl 0714.46041 |
| [55] | Pedersen, G. K., C^⁎-Algebras and Their Automorphism Groups, London Mathematical Society Monographs, vol. 14 (1979), Academic Press: Academic Press London · Zbl 0416.46043 |
| [56] | Pedersen, G. K., SAW^⁎-algebras and Corona C^⁎-algebras, contributions to non-commutative topology, J. Oper. Theory, 15, 15-32 (1986) · Zbl 0597.46060 |
| [57] | Peralta, A. M.; Rodríguez Palacios, A., Grothendieck’s inequalities for real and complex \(\operatorname{JBW}^\ast \)-triples, Proc. Lond. Math. Soc. (3), 83, 605-625 (2001) · Zbl 1037.46058 |
| [58] | Peralta, A. M.; Russo, B., Automatic continuity of triple derivations on C^⁎-algebras and JB^⁎-triples, J. Algebra, 399, 960-977 (2014) · Zbl 1325.46051 |
| [59] | Rickart, C. E., Banach algebras with an adjoint operation, Ann. Math. (2), 47, 528-550 (1946) · Zbl 0060.27103 |
| [60] | Rodríguez Palacios, Á., On the strong^⁎ topology of a JBW^⁎-triple, Q. J. Math. Oxf. Ser. (2), 42, 165, 99-103 (1991) · Zbl 0723.17026 |
| [61] | Rodríguez-Palacios, A., Jordan structures in analysis, (Jordan Algebras. Jordan Algebras, Oberwolfach, 1992 (1994), de Gruyter: de Gruyter Berlin), 97-186 · Zbl 0818.17036 |
| [62] | Saitô, K.; Wright, J. D.M., Monotone Complete C^⁎-Algebras and Generic Dynamics, Springer Monographs in Mathematics (2015), Springer: Springer London · Zbl 1382.46003 |
| [63] | Saitô, K.; Wright, J. D.M., On defining AW^⁎-algebras and Rickart C^⁎-algebras, Q. J. Math., 66, 979-989 (2015) · Zbl 1342.46052 |
| [64] | Sakai, S., A characterization of W^⁎-algebras, Pac. J. Math., 6, 763-773 (1956) · Zbl 0072.12404 |
| [65] | Sakai, S., C^⁎- and W^⁎-Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 60 (1971), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0219.46042 |
| [66] | Takesaki, M., Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences, vol. 124, Operator Algebras and Non-commutative Geometry, vol. 5 (2002), Springer-Verlag: Springer-Verlag Berlin, Reprint of the first (1979) edition · Zbl 0990.46034 |
| [67] | Topping, D., Jordan algebras of self-adjoint operators, Mem. Am. Math. Soc., 53 (1965), 48 pp · Zbl 0137.10203 |
| [68] | Wright, J. D.M., Wild AW^⁎-factors and Kaplansky-Rickart algebras, J. Lond. Math. Soc. (2), 13, 83-89 (1976) · Zbl 0319.46047 |
| [69] | Wright, J. D.M., Jordan C^⁎-algebras, Mich. Math. J., 24, 291-302 (1977) · Zbl 0384.46040 |
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