Abstract
Let M be a unital \(\hbox {JB}^*\)-algebra whose closed unit ball is denoted by \({\mathcal {B}}_M\). Let \(\partial _e({\mathcal {B}}_M)\) denote the set of all extreme points of \({\mathcal {B}}_M\). We prove that an element \(u\in \partial _e({\mathcal {B}}_M)\) is a unitary if and only if the set
contains an isolated point. This is a new geometric characterisation of unitaries in M in terms of the set of extreme points of \({\mathcal {B}}_M\).
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References
Akemann, C.A., Weaver, N.: Geometric characterizations of some classes of operators in \(\text{ C }^*\)-algebras and von Neumann algebras. Proc. Am. Math. Soc. 130(10), 3033–3037 (2002)
J. Becerra Guerrero, M. Cueto-Avellaneda, F.J. Fernández-Polo, A.M. Peralta, On the extension of isometries between the unit spheres of a JBW\(^*\) -triple anda Banach space. J. Inst. Math. Jussieu. https://doi.org/10.1017/S1474748019000173
Braun, R., Kaup, W., Upmeier, H.: A holomorphic characterisation of Jordan-\(\text{ C }^*\)-algebras. Math. Z. 161, 277–290 (1978)
Cabello-Sánchez, J.: A reflection on Tingley’s problem and some applications. J. Math. Anal. Appl. 476(2), 319–336 (2019)
M. Cabrera García, A. Rodríguez Palacios, Non-associative normed algebras. In: Vol. 1, vol. 154 of Encyclopedia of Mathematics and its Applications. The Vidav–Palmer and Gelfand–Naimark theorems. Cambridge University Press, Cambridge (2014)
Fernández-Polo, F.J., Martínez, J., Peralta, A.M.: Geometric characterization of tripotents in real and complex \(\text{ JB }^*\)-triples. J. Math. Anal. Appl. 295, 435–443 (2004)
Fernández-Polo, F.J., Peralta, A.M.: Partial isometries: a survey. Adv. Oper. Theory 3(1), 75–116 (2018)
Friedman, Y., Russo, B.: Structure of the predual of a JBW \(^*\)-triple. J. Reine Angew. Math. 356, 67–89 (1985)
Hamhalter, J., Kalenda, O.F.K., Peralta, A.M., Pfitzner, H.: Measures of weak non-compactness in preduals of von Neumann algebras and JBW\(^*\)-triples. J. Funct. Anal. 278(1), 108300 (2020)
Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitman, London (1984)
Isidro, J.M., Kaup, W., Rodríguez-Palacios, A.: On real forms of \(\text{ JB }^*\)-triples. Manuscripta Math. 8(6), 311–335 (1995)
Isidro, J.M., Rodríguez-Palacios, A.: Isometries of JB-algebras. Manuscripta Math. 86, 337–348 (1995)
Jamjoom, F.B., Siddiqui, A.A., Tahlawi, H.M., Peralta, A.M.: Approximation and convex decomposition by extremals and the \(\lambda \)-function in \(\text{ JBW }^*\)-triples. Q. J. Math. (Oxford) 66, 583–603 (2015)
Kadison, R.V.: Isometries of operator algebras. Ann. Math. 54, 325–338 (1951)
Kato, Y.: Some theorems on projections of von Neumann algebras. Math. Japon. 21(4), 367–370 (1976)
Kaup, W.: A Riemann mapping theorem for bounded symmentric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983)
Kaup, W., Upmeier, H.: Jordan algebras and symmetric Siegel domains in Banach spaces. Math. Z. 157, 179–200 (1977)
Leung, C.-W., Ng, C.-K., Wong, N.-C.: Geometric unitaries in JB-algebras. J. Math. Anal. Appl. 360, 491–494 (2009)
Li, B.: Real operator algebras. World Scientific Publishing Co. Inc, River Edge (2003)
Mori, M.: Tingley’s problem through the facial structure of operator algebras. J. Math. Anal. Appl. 466(2), 1281–1298 (2018)
Mori, M., Ozawa, N.: Mankiewicz’s theorem and the Mazur-Ulam property for \(\text{ C }^*\)-algebras. Studia Math. 250(3), 265–281 (2020)
Peralta, A.M.: A survey on Tingley’s problem for operator algebras. Acta Sci. Math. (Szeged) 84, 81–123 (2018)
A. Rodríguez Palacios, Banach space characterizations of unitaries: a survey, J. Math. Anal. Appl. 369(1), 168–178 (2010)
Siddiqui, A.A.: Average of two extreme points in \(\text{ JBW }^*\)-triples. Proc. Jpn. Acad. 83, 176–178 (2007)
Wright, J.D.M.: Jordan \(\text{ C }^*\)-algebras. Michigan Math. J. 24, 291–302 (1977)
Wright, J.D.M., Youngson, M.A.: On isometries of Jordan algebras. J. London Math. Soc. 17, 339–44 (1978)
Acknowledgements
Authors partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund Project no. PGC2018-093332-B-I00, Programa Operativo FEDER 2014-2020 and Consejería de Economía y Conocimiento de la Junta de Andalucía Grant number A-FQM-242-UGR18, and Junta de Andalucía grant FQM375. First author also supported by Universidad de Granada, Junta de Andalucía and Fondo Social Europeo de la Unión Europea (Iniciativa de Empleo Juvenil), grant number 6087.
The authors thank the anonymous peer reviewers for their valuable comments.
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Cueto-Avellaneda, M., Peralta, A.M. Metric Characterisation of Unitaries in \(\hbox {JB}^*\)-Algebras. Mediterr. J. Math. 17, 124 (2020). https://doi.org/10.1007/s00009-020-01556-w
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DOI: https://doi.org/10.1007/s00009-020-01556-w
Keywords
- Unitaries
- Geometric unitaries
- Vertex
- Extreme points
- \(\hbox {C}^*\)-algebra
- \(\hbox {JB}^*\)-algebra
- \(\hbox {JB}^*\)-triple