Metric characterizations of isometries and of unital operator spaces and systems

Blecher, David; Neal, Matthew

doi:10.1090/S0002-9939-2010-10670-6

Metric characterizations of isometries and of unital operator spaces and systems
HTML articles powered by AMS MathViewer

by David P. Blecher and Matthew Neal PDF

Proc. Amer. Math. Soc. 139 (2011), 985-998 Request permission

Abstract:

We give some new characterizations of unitaries, isometries, unital operator spaces, unital function spaces, operator systems, $C^*$-algebras, and related objects. These characterizations only employ the vector space and operator space structure (not mentioning products, involutions, or any kind of function on the space).

References

Charles Akemann and Nik Weaver, Geometric characterizations of some classes of operators in $C^*$-algebras and von Neumann algebras, Proc. Amer. Math. Soc. 130 (2002), no. 10, 3033–3037. MR 1908927, DOI 10.1090/S0002-9939-02-06643-1
William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
Pradipta Bandyopadhyay, Krzysztof Jarosz, and T. S. S. R. K. Rao, Unitaries in Banach spaces, Illinois J. Math. 48 (2004), no. 1, 339–351. MR 2048228
David P. Blecher, Edward G. Effros, and Vrej Zarikian, One-sided $M$-ideals and multipliers in operator spaces. I, Pacific J. Math. 206 (2002), no. 2, 287–319. MR 1926779, DOI 10.2140/pjm.2002.206.287
David P. Blecher, Kay Kirkpatrick, Matthew Neal, and Wend Werner, Ordered involutive operator spaces, Positivity 11 (2007), no. 3, 497–510. MR 2336213, DOI 10.1007/s11117-007-2086-6
David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR 2111973, DOI 10.1093/acprof:oso/9780198526599.001.0001
D. P. Blecher and B. Magajna, Dual operator systems, preprint, arXiv:0807.4250
David P. Blecher and Matthew Neal, Open partial isometries and positivity in operator spaces, Studia Math. 182 (2007), no. 3, 227–262. MR 2360629, DOI 10.4064/sm182-3-4
David P. Blecher, Zhong-Jin Ruan, and Allan M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), no. 1, 188–201. MR 1040962, DOI 10.1016/0022-1236(90)90010-I
David P. Blecher and Wend Werner, Ordered $C^*$-modules, Proc. London Math. Soc. (3) 92 (2006), no. 3, 682–712. MR 2223541, DOI 10.1017/S0024611505015595
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
Masamichi Hamana, Triple envelopes and Šilov boundaries of operator spaces, Math. J. Toyama Univ. 22 (1999), 77–93. MR 1744498
X. J. Huang and C. K. Ng, An abstract characterization of unital operator spaces, J. Operator Theory, to appear, arXiv:0805.2447.
Matthew Neal and Bernard Russo, Operator space characterizations of $C^*$-algebras and ternary rings, Pacific J. Math. 209 (2003), no. 2, 339–364. MR 1978376, DOI 10.2140/pjm.2003.209.339
Theodore W. Palmer, Banach algebras and the general theory of $*$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001. $*$-algebras. MR 1819503, DOI 10.1017/CBO9780511574757.003
Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
Vern I. Paulsen and Mark Tomforde, Vector spaces with an order unit, Indiana Univ. Math. J. 58 (2009), no. 3, 1319–1359. MR 2542089, DOI 10.1512/iumj.2009.58.3518
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
Martin E. Walter, Algebraic structures determined by 3 by 3 matrix geometry, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2129–2131. MR 1963763, DOI 10.1090/S0002-9939-02-06849-1