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ORDER EMBEDDING OF A MATRIX ORDERED SPACE

  • ANIL K. KARN (a1)

Abstract

We characterize certain properties in a matrix ordered space in order to embed it in a C*-algebra. Let such spaces be called C*-ordered operator spaces. We show that for every self-adjoint operator space there exists a matrix order (on it) to make it a C*-ordered operator space. However, the operator space dual of a (nontrivial) C*-ordered operator space cannot be embedded in any C*-algebra.

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References

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[1]Blecher, D. P. and Neal, M., ‘Open partial isometries and positivity in operator spaces’, Studia Math. 182 (2007), 227262.
[2]Choi, M. D. and Effros, E. G., ‘Injectivity and operator spaces’, J. Funct. Anal. 24 (1977), 156209.
[3]Effros, E. G. and Ruan, Z. J., ‘On the abstract characterization of operator spaces’, Proc. Amer. Math. Soc. 119 (1993), 579584.
[4]Jameson, G. J. O., Ordered Linear Spaces, Lecture Notes in Mathematics, 141 (Springer, New York, 1970).
[5]Kadison, R. and Ringrose, J., Fundamentals of the Theories of Operator Algebras, I (Academic Press, New York, 1983).
[6]Karn, A. K., ‘A p-theory of ordered normed spaces’, Positivity 14 (2010), 441458.
[7]Karn, A. K. and Vasudevan, R., ‘Approximate matrix order unit spaces’, Yokohama Math. J. 44 (1997), 7391.
[8]Karn, A. K. and Vasudevan, R., ‘Matrix duality for matrix ordered spaces’, Yokohama Math. J. 45 (1998), 118.
[9]Karn, A. K. and Vasudevan, R., ‘Characterizations of matricially Riesz normed spaces’, Yokohama Math. J. 47 (2000), 143153.
[10]Ruan, Z. J., ‘Subspaces of C *-algebras’, J. Funct. Anal. 76 (1988), 217230.
[11]Schreiner, W. J., ‘Matrix regular operator spaces’, J. Funct. Anal. 152 (1998), 136175.
[12]Werner, W., ‘Subspaces of L(H) that are *-invariant’, J. Funct. Anal. 193 (2002), 207223.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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