Completely bounded homomorphisms and derivations
In this chapter, we study completely bounded homomorphisms u: A → B(H) when A ⊂ B(ℋ) is a subalgebra. We first consider the case when H and ℋ are Banach spaces but mostly concentrate on the Hilbert space case. In the latter case, we prove the fundamental result that a unital homomorphism is completely bounded if it is similar to a completely contractive one. Let δ: A → B(H) be a derivation on a C*-algebra. We show that δ is completely bounded if it is inner. When A is the disc algebra, we prove that an operator T on H is similar to a contraction iff it is completely polynomially bounded, or in other words if the associated homomorphism f → f(T) is completely bounded. We discuss a variant for operators on a Banach space and give several related facts. Finally, we give examples showing that a bounded (and actually contractive) unital homomorphism on a uniform algebra is not necessarily completely bounded.
Unable to display preview. Download preview PDF.
Notes and Remarks on Chapter 4
- [Pa1]Paulsen V. Completely bounded maps and dilations. Pitman Research Notes in Math. 146, Longman, Wiley, New York, 1986.Google Scholar
- [J]Johnson B. E.: Cohomology in Banach algebras. Memoirs Amer. Math. Soc. 127 (1972).Google Scholar
- [SS]Sinclair A. and Smith R.: Hochschild cohomology of von Neumann algebras. LMS Lecture notes series. Cambridge Univ. Press, 1994.Google Scholar
- [Di2]Dixmier J.: les algèbres d’opérateurs dans l’espace Hilbertien (Algèbres de von Neumann). Gauthier-Villars, Paris, 1969. (Translated into: von Neumann algebras, North-Holland, 1981.)Google Scholar
- [Ar2]Arveson W.: Ten lectures on operator algebras. CBMS 55. Amer. Math. Soc. Providence (RI ) 1984.Google Scholar
- [SN]Sz: Nagy B.: Completely continuous operators with uniformly bounded iterates. Publ. Math. Inst. Hungarian Acad. Sci. 4 (1959) 89–92.Google Scholar
- [DP]Douglas V. and Paulsen V.: Hilbert modules over function algebras. Pitman Longman 1989.Google Scholar
- [CCFW]Carlson J., Clark D., Foias C. and Williams J. Projective Hilbert A(D)modules. Preprint.Google Scholar