Abstract
Let \(C_1(H)\) denote the space of all trace class operators on an arbitrary complex Hilbert space H. We prove that \(C_1(H)\) satisfies the \(\lambda \)-property, and we determine the form of the \(\lambda \)-function of Aron and Lohman on the closed unit ball of \(C_1(H)\) by showing that
for every a in \({C_1(H)}\) with \(\Vert a\Vert _1 \le 1\). This is a non-commutative extension of the formula established by Aron and Lohman for \(\ell _1\).
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References
Aron, R.M., Lohman, R.H.: A geometric function determined by extreme points of the unit ball of a normed space. Pac. J. Math. 127, 209–231 (1987)
Bhatia, R.: Perturbation Bounds for Matrix Eigenvalues. Society for Industrial and Applied Mathematics, Philadelphia (2007)
Bhatia, R., Semrl, P.: Distance between hermitian operators in Schatten classes. Proc. Edinb. Math. Soc. 3(9), 377–380 (1996)
Blecher, D.P., Le Merdy, C.: Operator Algebras and Their Modules—An Operator Space Approach. London Mathematical Society Monographs. New Series, vol. 30. Oxford University Press, Oxford (2004)
Brown, L.G., Pedersen, G.K.: On the geometry of the unit ball of a \(\text{ C }^*\)-algebra. J. Reine Angew. Math. 469, 113–147 (1995)
Brown, L.G., Pedersen, G.K.: Approximation and convex decomposition by extremals in \(\text{ C }^*\)-algebras. Math. Scand. 81, 69–85 (1997)
Burgos, M., Kaidi, A., Morales Campoy, A., Peralta, A.M., Ramírez, M.: Von Neumann regularity and quadratic conorms in \(\text{ JB }^*\)-triples and \(\text{ C }^*\)-algebras. Acta Math. Sin. 24, 185–200 (2008)
Dunford, N., Schwartz, J.T.: Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. Interscience Publishers John Wiley & Sons, New York (1963)
Effros, E.G., Ruan, Z.J.: Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. Oxford University Press, New York (2000)
Fernández-Polo, F.J., Jordá, E., Peralta, A.M.: Tingley’s problem for \(p\)-Schatten von Neumann classes. J. Spectr. Theory (2018) (accepted) arxiv:1803.00763v2
Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969)
Jamjoom, F.B., Siddiqui, A.A., Tahlawi, H.M.: On the geometry of the unit ball of a \(\text{ JB }^*\)-triple. Abstr. Appl. Anal. 2013, 1–8 (2013). https://doi.org/10.1155/2013/891249 (Article ID 891249)
Jamjoom, F.B., Siddiqui, A.A., Tahlawi, H.M.: The \(\lambda \)-function in \(\text{ JB }^{*}\)-triples. J. Math. Anal. Appl. 414, 734–741 (2014)
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. (Oxf.) 66, 583–603 (2015)
Jamjoom, F.B., Peralta, A.M., Siddiqui, A.A., Tahlawi, H.M.: Extremally rich \(\text{ JB }^*\)-triples. Ann. Funct. Anal. 7, 578–592 (2016)
Kaup, W.: A Riemann Mapping Theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983)
Markus, A.S.: The eigen and singular values of the sum and product of linear operators. Russ. Math. Surv. 19, 92–120 (1964)
McCarthy, C.A.: \(C_p\). Isr. J. Math. 5, 249–271 (1967)
Mirsky, L.: Symmetric gauge functions and unitarily invariant norms. Q. J. Math. Oxf. Ser. 2(11), 50–59 (1960)
Pedersen, G.K.: The \(\lambda \)-function in operator algebras. J. Oper. Theory 26, 345–381 (1991)
Pisier, G.: Introduction to Operator Space Theory. London Mathematical Society Lecture Note Series, vol. 294. Cambridge University Press, Cambridge (2003)
Sakai, S.: \(\text{ C }^*\)-Algebras and \(W^*\)-Algebras. Springer, Berlin (1971)
Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (2003)
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The author is partially supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund project no. MTM2014-58984-P and Junta de Andalucía grant FQM375.
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Peralta, A.M. The \(\lambda \)-Function in the Space of Trace Class Operators. Mediterr. J. Math. 15, 217 (2018). https://doi.org/10.1007/s00009-018-1260-3
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DOI: https://doi.org/10.1007/s00009-018-1260-3