By J. L. Morris

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**Extra resources for Conference on the Numerical Solution of Differential Equations**

**Example text**

6. In general, commutators [X, 7 ] appear in various branches of functional analysis. We will mention some of them in the following. In the lecture notes Integral Operators, Commutators, Traces, Index and Homology by Helton and 22 Operator algebras in dynamical systems Howe [75], the commutators [X, Y] are assumed to belong to the trace class. Commutators also appear in the invariant subspace problem of bounded operators on a Hilbert space. Let A be a bounded operator on an infinite-dimensional Hilbert space Jf7 and suppose that Rank [A, K] ^ 1 for some non-zero compact operator K on Jf; then A has a proper invariant closed subspace ([121], [51]).

Moreover WteJf and wfawf* = WtaW*. 5 can be extended to a general W*-algebra M. The theorem is due to H. Araki (unpublished). 7 Corollary (Araki [2]) Let t\-^octbea a-weakly continuous representation ofUn by *-automorphisms on a W*-algebra M containing the identity operator in a Hilbert space Jf. Suppose that there exists a strongly continuous unitary representation th-^uteB(^) satisfying the spectrum condition such that utaut* = at(a) (teUn,aeJi) and {ut\teW} has one eigenvalue £0 with £ 0 , a cyclic vector for Ji.

5 Application to bounded derivations In this section, we shall discuss the automatic innerness of derivations on W*-algebras and simple C*-algebras with an identity. There are few known examples of infinite-dimensional, non-commutative algebras except for operator algebras which have only inner derivations. The algebra B(E) of all bounded linear operators on a Banach space E has only inner-bounded derivations. However, there are many examples of C*-algebras which have outer-bounded derivations.