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Operator ranges and completely bounded homomorphisms


Author: D. Benjamin Mathes
Journal: Proc. Amer. Math. Soc. 107 (1989), 155-164
MSC: Primary 47D25; Secondary 47A15, 47C05
DOI: https://doi.org/10.1090/S0002-9939-1989-0979229-5
MathSciNet review: 979229
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Abstract: In this paper it is shown that the set of invariant operator ranges that induce completely bounded homomorphisms is a sublattice of $ \operatorname{Lat}_{1/2}\mathfrak{A}$, when $ \mathfrak{A}$ is any norm closed algebra of operators on a Hilbert space. A characterization of this sublattice is given, and several concrete examples are discussed.


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  • [1] E. Azoff, Invariant linear manifolds and the selfadjointness of operator algebras, Amer. J. Math. 99 (1977), 121-138. MR 0435886 (55:8837)
  • [2] A. Connes, Classification of injective factors, Ann. of Math. 104 (1976), 73-115. MR 0454659 (56:12908)
  • [3] K. R. Davidson, Invariant operator ranges for reflexive algebras, J. Operator Theory 7 (1982), 101-108. MR 650195 (83e:47004)
  • [4] J. Dixmier, Étude sur les variétés et les opérateurs de Julia, Bull. Soc. Math. France 77 (1949), 11-101. MR 0032937 (11:369f)
  • [5] P. A. Fillmore and J. P. Williams, On operator ranges, Adv. in Math. 7 (1971), 254-281. MR 0293441 (45:2518)
  • [6] C. Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 20 (1972), 897-900. MR 0399893 (53:3734)
  • [7] U. Haagerup, Solution of the similarity problem for cyclic representations of $ {C^*}$-algebras, Ann. of Math. 118 (1983), 215-240. MR 717823 (85d:46080)
  • [8] G. W. Mackey, On the domains of closed linear transformations in Hilbert space, Bull. Amer. Math. Soc. 52 (1946), 1009.
  • [9] B. Mathes, On invariant operator ranges of abelian strictly cyclic algebras, J. Operator Theory, (to appear). MR 1066808 (91h:47045)
  • [10] E Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, On invariant operator ranges, Trans. Amer. Math. Soc. 251 (1979), 389-398. MR 531986 (81c:47010)
  • [11] E. Nordgren and C. Roy, On invariant ranges of the analytic Toeplitz operators, Indiana Univ. Math. J. (to appear). MR 963507 (89k:47043)
  • [12] S.-C. Ong, Operator algebras and invariant operator ranges, Ph. D. dissertation, Dalhousie University, Halifax, Nova Scotia, 1979.
  • [13] -, Invariant operator ranges of nest algebras, J. Operator Theory 3 (1980), 195-201. MR 578939 (81f:47008)
  • [14] V. I. Paulsen, Completely bounded maps on $ {C^*}$-algebras and invariant operator ranges, Proc. Amer. Math. Soc. 86 (1982), 91-96. MR 663874 (83i:46064)
  • [15] -, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), 1-17. MR 733029 (86c:47021)
  • [16] -, Completely bounded maps and dilations, Pitman Research Notes in Math. Series, no. 146, Wiley, New York, 1986. MR 868472 (88h:46111)
  • [17] C. Roy, Operator ranges of shifts and $ {C^*}$-algebras, Ph. D. dissertation, University of New Hampshire, Durham, N. H., 1986.
  • [18] D. Voiculescu, Sur les sous-espaces parafermés invariants d'une algèbre de von Neumann, Bull. Sci. Math. (2) 96 (1972), 161-168. MR 0322526 (48:888)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0979229-5
Keywords: Invariant operator range, completely bounded homomorphism
Article copyright: © Copyright 1989 American Mathematical Society

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