Illinois Journal of Mathematics

Metric characterizations II

David P. Blecher and Matthew Neal

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The present paper is a sequel to our paper “Metric characterization of isometries and of unital operator spaces and systems.” We characterize certain common objects in the theory of operator spaces (unitaries, unital operator spaces, operator systems, operator algebras, and so on), in terms which are purely linear-metric, by which we mean that they only use the vector space structure of the space and its matrix norms. In the last part, we give some characterizations of operator algebras (which are not linear-metric in our strict sense described in the paper).

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Illinois J. Math., Volume 57, Number 1 (2013), 25-41.

First available in Project Euclid: 23 June 2014

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Zentralblatt MATH identifier

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25] 47L25: Operator spaces (= matricially normed spaces) [See also 46L07]
Secondary: 47L07: Convex sets and cones of operators [See also 46A55] 47L30: Abstract operator algebras on Hilbert spaces


Blecher, David P.; Neal, Matthew. Metric characterizations II. Illinois J. Math. 57 (2013), no. 1, 25--41. doi:10.1215/ijm/1403534484.

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  • C. A. Akemann, The general Stone–Weierstrass problem, J. Funct. Anal. 4 (1969), 277–294.
  • C. A. Akemann, Left ideal structure of $C^*$-algebras, J. Funct. Anal. 6 (1970), 305–317.
  • W. B. Arveson, Subalgebras of $C^{*}$-algebras, Acta Math. 123 (1969), 141–224.
  • D. P. Blecher, The Shilov boundary of an operator space and the characterization theorems, J. Funct. Anal. 182 (2001), 280–343.
  • D. P. Blecher, One-sided ideals and approximate identities in operator algebras, J. Aust. Math. Soc. 76 (2004), 425–447.
  • D. P. Blecher, Multipliers, $C^*$-modules, and algebraic structure in spaces of Hilbert space operators, Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone, Contemp. Math., vol. 365, Amer. Math. Soc., Providence, RI, 2004, pp. 85–128.
  • D. P. Blecher, D. M. Hay and M. Neal, Hereditary subalgebras of operator algebras, J. Operator Theory 59 (2008), 333–357.
  • D. P. Blecher, K. Kirkpatrick, M. Neal and W. Werner, Ordered involutive operator spaces, Positivity 11 (2007), 497–510.
  • D. P. Blecher and C. Le Merdy, Operator algebras and their modules–-An operator space approach, Oxford Univ. Press, Oxford, 2004.
  • D. P. Blecher and B. Magajna, Duality and operator algebras: Automatic weak* continuity and applications, J. Funct. Anal. 224 (2005), 386–407.
  • D. P. Blecher and B. Magajna, Duality and operator algebras II: Operator algebras as Banach algebras, J. Funct. Anal. 226 (2005), 485–493.
  • D. P. Blecher and B. Magajna, Dual operator systems, Bull. Lond. Math. Soc. 43 (2011), 311–320.
  • D. P. Blecher and M. Neal, Open partial isometries and positivity in operator spaces, Studia Math. 182 (2007), 227–262.
  • D. P. Blecher and M. Neal, Metric characterizations of isometries and of unital operator spaces and systems, Proc. Amer. Math. Soc. 139 (2011), 985–998.
  • D. P. Blecher and M. Neal, Open projections in operator algebras II: Compact projections, Studia Math. 209 (2012) 203–224.
  • D. P. Blecher and C. J. Read, Operator algebras with contractive approximate identities, J. Funct. Anal. 261 (2011), 188–217.
  • D. P. Blecher, Z.-J. Ruan and A. M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188–201.
  • C. M. Edwards and G. T. Rüttimann, Exposed faces of the unit ball in a $\mathrm{JBW}^*$-triple, Math. Scand. 82 (1998), 287–304.
  • M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama Univ. 22 (1999), 77–93.
  • X. J. Huang and C. K. Ng, An abstract characterization of unital operator spaces, J. Operator Theory 67 (2012), 289–298.
  • M. Kaneda, Extreme points of the unit ball of an operator space, available at \arxivurlarXiv:math/0408235.
  • M. Kaneda and V. Paulsen, Quasimultipliers of operator spaces, J. Funct. Anal. 217 (2004), 347–365.
  • E. Kirchberg, On restricted peturbations in inverse images and a description of normalizer algebras in $C^*$-algebras, J. Funct. Anal. 129 (1995), 1–34.
  • M. Neal and B. Russo, Operator space characterizations of $C^*$-algebras and ternary rings, Pacific J. Math. 209 (2003), 339–364.
  • M. Neal and B. Russo, A holomorphic characterization of operator algebras, to appear in Math. Scand.; available at \arxivurlarXiv:1207.7353.
  • Z.-J. Ruan, Subspaces of $C^{*}$-algebras, J. Funct. Anal. 76 (1988), 217–230.