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.
KeywordsHilbert Space Banach Space Operator Algebra Uniform Algebra Disc Algebra
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Notes and Remarks on Chapter 4
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