Non-commutative metric topology on matrix state space

Wu, Wei

doi:10.1090/S0002-9939-05-08036-6

Non-commutative metric topology on matrix state space
HTML articles powered by AMS MathViewer

by Wei Wu PDF

Proc. Amer. Math. Soc. 134 (2006), 443-453 Request permission

Abstract:

We present an operator space version of Rieffel’s theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As an application we obtain a necessary and sufficient condition for the matrix metric from an unbounded Fredholm module to give the BW-topology on the matrix state space of the $C^*$-algebra. Motivated by recent results we formulate a non-commutative Lipschitz seminorm on a matrix order unit space and characterize those matrix Lipschitz seminorms whose matrix metric topology coincides with the BW-topology on the matrix state space.

References

William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
David P. Blecher, The standard dual of an operator space, Pacific J. Math. 153 (1992), no. 1, 15–30. MR 1145913, DOI 10.2140/pjm.1992.153.15
Man Duen Choi and Edward G. Effros, Injectivity and operator spaces, J. Functional Analysis 24 (1977), no. 2, 156–209. MR 0430809, DOI 10.1016/0022-1236(77)90052-0
A. Connes, Compact metric spaces, Fredholm modules, and hyperfiniteness, Ergodic Theory Dynam. Systems 9 (1989), no. 2, 207–220. MR 1007407, DOI 10.1017/S0143385700004934
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Edward G. Effros, Aspects of noncommutative order, $\textrm {C}^{\ast }$-algebras and applications to physics (Proc. Second Japan-USA Sem., Los Angeles, Calif., 1977) Lecture Notes in Math., vol. 650, Springer, Berlin, 1978, pp. 1–40. MR 504750
Edward G. Effros, Some quantizations and reflections inspired by the Gel′fand-Naĭmark theorem, $C^\ast$-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 98–113. MR 1292011, DOI 10.1090/conm/167/1292011
Edward G. Effros and Zhong-Jin Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), no. 3, 329–337. MR 1127754, DOI 10.4153/CMB-1991-053-x
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Edward G. Effros and Corran Webster, Operator analogues of locally convex spaces, Operator algebras and applications (Samos, 1996) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 495, Kluwer Acad. Publ., Dordrecht, 1997, pp. 163–207. MR 1462680
Uffe Haagerup, Injectivity and decomposition of completely bounded maps, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 170–222. MR 799569, DOI 10.1007/BFb0074885
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR 719020
Richard V. Kadison, A representation theory for commutative topological algebra, Mem. Amer. Math. Soc. 7 (1951), 39. MR 44040
Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
Branka Pavlović, Defining metric spaces via operators from unital $C^*$-algebras, Pacific J. Math. 186 (1998), no. 2, 285–313. MR 1663810, DOI 10.2140/pjm.1998.186.285
Marc A. Rieffel, Metrics on states from actions of compact groups, Doc. Math. 3 (1998), 215–229. MR 1647515
Marc A. Rieffel, Metrics on state spaces, Doc. Math. 4 (1999), 559–600. MR 1727499
Zhong-Jin Ruan, Subspaces of $C^*$-algebras, J. Funct. Anal. 76 (1988), no. 1, 217–230. MR 923053, DOI 10.1016/0022-1236(88)90057-2
R. R. Smith and J. D. Ward, The geometric structure of generalized state spaces, J. Functional Analysis 40 (1981), no. 2, 170–184. MR 609440, DOI 10.1016/0022-1236(81)90066-5
Corran Webster and Soren Winkler, The Krein-Milman theorem in operator convexity, Trans. Amer. Math. Soc. 351 (1999), no. 1, 307–322. MR 1615970, DOI 10.1090/S0002-9947-99-02364-8
Gerd Wittstock, Ein operatorwertiger Hahn-Banach Satz, J. Functional Analysis 40 (1981), no. 2, 127–150 (German, with English summary). MR 609438, DOI 10.1016/0022-1236(81)90064-1