By Peter A. Fillmore

The topic of operator algebras has skilled super progress lately with major purposes to components inside algebraic arithmetic in addition to allied components resembling unmarried operator idea, non-self-adjoint operator algegras, K-theory, knot idea, ergodic idea, and mathematical physics. This publication makes fresh advancements in operator algebras available to the non-specialist.

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6) To check that 11"( a)x E E 2 , it suffices to show that 11"( a)x E gl. We have then PE1I"(a)x = nl~ (PE1I"(a)xn + ~ PE1I"(a)1I"(af)xf) = nl~~ (1I"E(a)X n + ~ 1I"E(aaf)x f ) hence since 1I"E is assumed a homomorphism 26 1. 6) PE7I"(a)x proof. = O. This shows that 7I"(a)x E EJ. and concludes the 0 Remark. -Nagy's dilation theorem (or Ando's). For any contraction T there is a unitary operator U admitting invariant subspaces E 2 , El with E2 C El such that T = PE16E,UIE16E,. This explains why the structure of the invariant subspaces of unitary operators is so important to understand the structure of general contractions.

We prove a fundamental factorization/extension theorem for completely bounded maps, and give several consequences. In this viewpoint, the underlying idea is the same in both cases (completely bounded maps or operators factoring through Hilbert space). At the end of this chapter, we give several examples of bounded linear maps which are not completely bounded, and related norm estimates. In this chapter we will prove the basic factorization theorem of completely bounded maps. As many factorization theorems in Functional Analysis it can be derived rather directly from the Hahn-Banach theorem.

3. There are non-unitarizable uniformly bounded representations on IF 00' Remark. It is easy to modify the argument to construct for each t: > 0 a representation uniformly bounded by 1 + t: and still non-unitarizable. We will use a very convenient amenability criterion due to Hulanicki (based on previous work of Kesten), as follows. 4. The following properties of a discrete group are equivalent. (i) G is amenable. (G))' tEG (iii) Same as (ii) with C = 1. Proof: Assume (i). Let cp be an invariant mean for G.

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