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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Commutativity in operator algebras


Author: David P. Blecher
Journal: Proc. Amer. Math. Soc. 109 (1990), 709-715
MSC: Primary 46L05; Secondary 47D25
MathSciNet review: 1009985
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Abstract: By an 'operator algebra' we shall mean a subalgebra $ \mathcal{A}$ of the algebra $ \mathcal{B}(\mathcal{H})$ of bounded operators on a Hilbert space $ \mathcal{H}$, together with the matrix normed structure $ \mathcal{A}$ inherits from $ \mathcal{B}(\mathcal{H})$. A unital operator algebra is an operator algebra with an identity of norm 1. Note that we do not require the algebra to be self-adjoint or uniformly closed. Such algebras were characterized abstractly by Ruan, Sinclair, and the author up to complete isometric isomorphism. In this paper we study commutativity for operator algebras, and we give a characterization of commutative unital operator algebras and a characterization of unital uniform algebras.


References [Enhancements On Off] (What's this?)

  • [1] D. P. Blecher, Z-J. Ruan and A. M. Sinclair, A characterization of operator algebras, J. Funct. Anal. (to appear).
  • [2] D. P. Blecher and V. I. Paulsen, Tensor products of operator spaces, preprint. MR 1121615 (93d:46095)
  • [3] T. K. Carne, Not all $ H'$-algebras are operator algebras, Math. Proc. Cambridge Philos. Soc. 86 (1979), 243-249. MR 538746 (81m:46063)
  • [4] H. B. Cohen, Injective envelopes of Banach spaces, Bull. Amer. Math. Soc. 70 (1964), 723-726. MR 0184060 (32:1536)
  • [5] E. G. Effros, On multilinear completely bounded maps, Contemp.Math. 62 (1987), 450-479. MR 878396 (88h:46112)
  • [6] E. G. Effros and Z-J. Ruan, On matricially normed spaces, Pacific J. Math. 132 (1988), 243-264. MR 934168 (90a:46150)
  • [7] -, On non-self-adjoint operator algebras, preprint.
  • [8] A. Grothendieck, Resume de la theorie metrique des produits tensoriels topologiques, Bull. Soc. Mat. Sao Paulo 8 (1956), 1-79. MR 0094682 (20:1194)
  • [9] M. Hamana, Injective envelopes of operator systems, Publ. RIMS Kyoto Univ. 15 (1979), 773-785. MR 566081 (81h:46071)
  • [10] J. R. Isbell, Three remarks on injective envelopes of Banach spaces, J. Math. Anal. 27 (1969), 516-518. MR 0251512 (40:4739)
  • [11] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325-338. MR 0043392 (13:256a)
  • [12] M. Nagasawa, Isomorphisms between commutative Banach algebras with applications to rings of analytic functions, Kōdai Math. Sem Rep. 11 (1959), 182-188. MR 0121645 (22:12379)
  • [13] T. Okayusu, Some cross norms which are not uniformly cross, Proc. Japan Acad. 46 (1970), 54-57. MR 0264414 (41:9009)
  • [14] S. K. Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970), 481-490. MR 0268710 (42:3607)
  • [15] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., Longman, London, 1986. MR 868472 (88h:46111)
  • [16] V. I. Paulsen and R. R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73 (1987), 258-276. MR 899651 (89m:46099)
  • [17] G. Pisier, Factorization of linear operators and geometry of Banach spaces, C.B.M.S. Series no. 60, Amer. Math. Soc., Providence, RI, 1986. MR 829919 (88a:47020)
  • [18] Z-J. Ruan, Subspaces of $ {C^*}$-algebras, J. Funct. Anal. 76 (1988), 217-230. MR 923053 (89h:46082)
  • [19] -, Injectivity and operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89-104. MR 929239 (91d:46078)
  • [20] M. Takesaki, Theory of operator algebras I, Springer-Verlag, Berlin, 1979. MR 548728 (81e:46038)
  • [21] J. Tomiyama, On the transpose map of matrix algebras, Proc. Amer. Math. Soc. 88 (1983), 635-638. MR 702290 (85b:46064)
  • [22] A. M. Tonge, Banach algebras and absolutely summing operators, Math. Proc. Cambridge Philos. Soc. 80 (1976), 465-473. MR 0438152 (55:11071)
  • [23] W. B. Arveson, Subalgebras of $ {C^*}$-algebras I, Acta Math. 123 (1969), 142-224; II, 128 (1972), 271-308. MR 0394232 (52:15035)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1009985-X
PII: S 0002-9939(1990)1009985-X
Article copyright: © Copyright 1990 American Mathematical Society