Abstract
The author generalizes the Arzelà-Ascoli theorem to the setting of matrix order unit spaces, extending the work of Antonescu-Christensen on unital C*-algebras. This gives an affirmative answer to a question of Antonescu and Christensen.
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References
Connes, A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergodic Theory Dynam. Systems, 9(2) (1989), 207–220.
Rieffel, M. A.: Metrics on states from actions of compact groups. Doc. Math., 3, 215–229 (1998)
Rieffel, M. A.: Metrics on state spaces. Doc. Math., 4, 559–600 (1999)
Rieffel, M. A.: Group C*-algebras as compact quantum metric spaces. Doc. Math., 7, 605–651 (2002)
Rieffel, M. A.: Gromov-Hausdorff distance for quantum metric spaces. Mem. Amer. Math. Soc., 168, 1–65 (2004)
Rieffel, M. A.: Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance. Mem. Amer. Math. Soc., 168, 67–91 (2004)
Antonescu, C., Christensen, E.: Metrics on group C*-algebras and a non-commutative Arzelà-Ascoli theorem. J. Funct. Anal., 214, 247–259 (2004)
Effros, E. G.: Advances in quantized functional analysis, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 906–916
Wu, W.: Non-commutative metric topology on matrix state spaces. Proc. Amer. Math. Soc., 134, 443–453 (2006)
Wu, W.: Non-commutative metrics on matrix state spaces. J. Ramanujan Math. Soc., 20, 215–254 (2005)
Wu, W.: Quantized Gromov-Hausdorff distance. J. Funct. Anal., 238, 58–98 (2006)
Choi, M. D., Effros, E. G.: Injectivity and operator spaces. J. Funct. Anal., 24, 156–209 (1977)
Webster, C., Winkler, S.: The Krein-Milman theorem in operator convexity. Trans. Amer. Math. Soc., 351, 307–322 (1999)
Effros, E. G., Webster, C.: Operator analogues of locally convex spaces, Operator algebras and applications (Samos, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 495, Kluwer Acad. Publ., Dordrecht, 1997, 163–207
Arveson, W. B.: Subalgebras of C*-algebras. Acta Math., 123, 141–224 (1969)
Paulsen, V. I.: Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986
Kerr, D.: Matricial quantum Gromov-Hausdorff distance. J. Funct. Anal., 205, 132–167 (2003)
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Project sponsored by the Shanghai Leading Academic Discipline Project (Project No. B407), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the National Natural Science Foundation of China (Grant No. 10671068)
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Wu, W. An operator Arzelà-Ascoli theorem. Acta. Math. Sin.-English Ser. 24, 1139–1154 (2008). https://doi.org/10.1007/s10114-007-6354-y
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DOI: https://doi.org/10.1007/s10114-007-6354-y