Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by(1)We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the formFor example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

ABSTRACT. It is shown that every strongly closed algebra of operators acting on a separable Hubert space can be expressed as a direct integral of irreducible algebras. In particular, every reductive algebra is the direct integral of transitive algebras. This decomposition is used to study the relationship between the transitive and reductive algebra problems. The final section of the paper shows how to view direct integrals of algebras as measurable algebra-valued functions.

Let ~ be a complex Hilbert space, and let ~d(~) and ~(oug) denote, respectively, the algebra of compact (linear) operators and that of all (bounded linear) operators on ~. If Te ~(Jt~), we denote by {s.(T)} the sequence of characteristic numbers of T, i.e., the eigenvalues of the positive compact operator (T'T) 1/2 arranged in decreasing order and repeated according to multiplicity. The yon Neumann-Schatten class cgp is the algebra of all T with ~ (s.(T)) p < oo. By an ideal in oU(dcg) or in N(Jg) we shall mean a two-sided ideal. A Lie ideal 5f in ~(~r is a linear subspace of N(~) with the property that A e ~(~f), B e ~o imply AB-BA ~ ~. An ideal J in JU(dr ~) is not necessarily an ideal in ~(oug). At least in one interesting special case, J is an ideal in ~(H) if and only if it is a Lie ideal. In this case, i.e., the case where J is generated by a single positive operator P, we also give other conditions equivalent to ~r being an ideal in N(oug), including one in terms of the sequence {s.(P)}. Our Theorem 1 will demonstrate, for example, that if s.(P) = 2-", then J is an ideal in ~(~f), but it is not if s.(P)=n-2. If {a.} and {b.} are sequences of numbers we find it convenient to use the usual notation a. = o(b.) to mean lim(a./b.) = 0.

1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius1of a Hilbert space operator T. Among the many interesting developments, we may mention:(a) C. Berger's proof of the “power inequality”2(b) R. Bouldin's result that3for any isometry V commuting with T;(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

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