Operator ranges and completely bounded homomorphisms

Mathes, D. Benjamin

doi:10.1090/S0002-9939-1989-0979229-5

Operator ranges and completely bounded homomorphisms
HTML articles powered by AMS MathViewer

by D. Benjamin Mathes PDF

Proc. Amer. Math. Soc. 107 (1989), 155-164 Request permission

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.

References

Edward A. Azoff, Invariant linear manifolds and the self-adjointness of operator algebras, Amer. J. Math. 99 (1977), no. 1, 121–137. MR 435886, DOI 10.2307/2374011
A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
Kenneth R. Davidson, Invariant operator ranges for reflexive algebras, J. Operator Theory 7 (1982), no. 1, 101–107. MR 650195
Jacques Dixmier, Étude sur les variétés et les opérateurs de Julia, avec quelques applications, Bull. Soc. Math. France 77 (1949), 11–101 (French). MR 32937
P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
Ciprian Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 20 (1971), no. 10, 897–900. MR 399893, DOI 10.1512/iumj.1971.20.20074
Uffe Haagerup, Solution of the similarity problem for cyclic representations of $C^{\ast }$-algebras, Ann. of Math. (2) 118 (1983), no. 2, 215–240. MR 717823, DOI 10.2307/2007028

On the domains of closed linear transformations in Hilbert space

52

D. Benjamin Mathes, On invariant operator ranges of abelian strictly cyclic algebras, J. Operator Theory 23 (1990), no. 2, 293–300. MR 1066808
E. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, On invariant operator ranges, Trans. Amer. Math. Soc. 251 (1979), 389–398. MR 531986, DOI 10.1090/S0002-9947-1979-0531986-6
Eric Nordgren and Charles Roy, On invariant operator ranges of the analytic Toeplitz operators, Indiana Univ. Math. J. 37 (1988), no. 2, 369–377. MR 963507, DOI 10.1512/iumj.1988.37.37018

Operator algebras and invariant operator ranges

Sing Cheong Ong, Invariant operator ranges of nest algebras, J. Operator Theory 3 (1980), no. 2, 195–201. MR 578939
Vern I. Paulsen, Completely bounded maps on $C^{\ast }$-algebras and invariant operator ranges, Proc. Amer. Math. Soc. 86 (1982), no. 1, 91–96. MR 663874, DOI 10.1090/S0002-9939-1982-0663874-4
Vern I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), no. 1, 1–17. MR 733029, DOI 10.1016/0022-1236(84)90014-4
Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472

Operator ranges of shifts and

-algebras

Dan Virgil Voiculescu, Sur les sous-espaces parafermés invariants d’une algèbre de von Neumann, Bull. Sci. Math. (2) 96 (1972), 161–168 (French). MR 322526