Spherical isometries of finite dimensional \(C^{\ast}\)-algebras. (English) Zbl 1371.46008
Summary: In this paper, it is shown that every surjective isometry between the unit spheres of two finite dimensional \(C^\ast\)-algebras extends to a real-linear Jordan isomorphism followed by multiplication operator by a fixed unitary element. This gives an affirmative answer to D. Tingley’s problem [Geom. Dedicata 22, 371–378 (1987; Zbl 0615.51005)] between two finite-dimensional \(C^\ast\)-algebras. Moreover, we show that if two finite dimensional \(C^\ast\)-algebras have isometric unit spheres, then they are isomorphic.
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B25 | Classical Banach spaces in the general theory |
| 46L05 | General theory of \(C^*\)-algebras |
Citations:
Zbl 0615.51005References:
| [1] | Akemann, C. A.; Pedersen, G. K., Facial structure in operator algebra theory, Proc. Lond. Math. Soc. (3), 64, 418-448 (1992) · Zbl 0759.46050 |
| [2] | Cheng, L.; Dong, Y., On a generalized Mazur-Ulam question: extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl., 377, 464-470 (2011) · Zbl 1220.46006 |
| [3] | Connes, A., A factor not anti-isomorphic to itself, Ann. Math. (2), 101, 536-554 (1975) · Zbl 0315.46058 |
| [4] | Ding, G. G., On isometric extension problem between two unit spheres, Sci. China Ser. A, 52, 2069-2083 (2009) · Zbl 1190.46013 |
| [5] | Ding, G. G., The isometric extension problem between unit spheres of two separable Banach spaces, Acta Math. Sin. (Engl. Ser.), 31, 1872-1878 (2015) · Zbl 1351.46011 |
| [6] | Ding, G. G.; Li, J. Z., Sharp corner points and isometric extension problem in Banach spaces, J. Math. Anal. Appl., 405, 297-309 (2013) · Zbl 1306.46011 |
| [7] | 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, 317-332 (1988) · Zbl 0621.46043 |
| [8] | Hatori, O.; Molnár, L., Isometries of the unitary group, Proc. Amer. Math. Soc., 140, 2127-2140 (2012) · Zbl 1244.47032 |
| [9] | Hatori, O.; Molnár, L., Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in \(C^\ast \)-algebras, J. Math. Anal. Appl., 409, 158-167 (2014) · Zbl 1306.46053 |
| [10] | Kadets, V.; Martín, M., Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces, J. Math. Anal. Appl., 396, 441-447 (2012) · Zbl 1258.46004 |
| [11] | Kadison, R. V., A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2), 56, 494-503 (1952) · Zbl 0047.35703 |
| [12] | Kadison, R. V.; Pedersen, G. K., Means and convex combinations of unitary operators, Math. Scand., 57, 249-266 (1985) · Zbl 0573.46034 |
| [13] | Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras. Vol. II, Advanced Theory, Pure and Applied Mathematics, vol. 100 (1986), Academic Press, Inc.: Academic Press, Inc. Orlando, FL · Zbl 0601.46054 |
| [14] | Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras. Vol. IV, Special Topics. Advanced Theory - An Exercise Approach (1992), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA · Zbl 0869.46029 |
| [15] | Mankiewicz, P., On extension of isometries in normed linear spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., 20, 367-371 (1972) · Zbl 0234.46019 |
| [16] | Mascioni, V.; Molnár, L., Linear maps on factors which preserve the extreme points of the unit ball, Canad. Math. Bull., 41, 434-441 (1998) · Zbl 0919.47024 |
| [17] | Megginson, R. E., An Introduction to Banach Space Theory (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0910.46008 |
| [18] | Sourour, A. R., Invertiblity preserving linear maps on \(L(X)\), Trans. Amer. Math. Soc., 348, 13-30 (1996) · Zbl 0843.47023 |
| [19] | Tanaka, R., A further property of spherical isometries, Bull. Aust. Math. Soc., 90, 304-310 (2014) · Zbl 1312.46021 |
| [20] | Tanaka, R., The solution of Tingley’s problem for the operator norm unit sphere of complex \(n \times n\) matrices, Linear Algebra Appl., 494, 274-285 (2016) · Zbl 1382.15034 |
| [21] | Tingley, D., Isometries of the unit sphere, Geom. Dedicata, 22, 371-378 (1987) · Zbl 0615.51005 |
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