Lie ideals in triangular operator algebras

Abstract

We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if 2 is a closed Lie ideal of the triangular operator algebra A, then there exist a closed associative ideal IC and a closed subalgebra Or of the diagonal A ni A* so that K C 2 C IC + Oc. INTRODUCTION Let A be an associative complex algebra. Under the Lie multiplication [x, y] xy yx, A becomes a Lie algebra. A Lie ideal in A is a linear manifold Z in A for which [a, k] c ? for every a E A and k c 2. In many instances, there is a close connection between the Lie ideal structure and the associative ideal structure of A. This connection has been investigated for prime rings in [6], in [3] for B($5) the set of bounded operators on a Hilbert space Si, and in [10] for certain von Neumann algebras. (See also [9, 4, 11].) In this paper we pursue this line of investigation for two classes of triangular operator algebras, namely nest algebras anld triangular UHF algebras. The authors would like to thank Frank A. Zorzitto for many helpful conversations. 1. WEAKLY CLOSED LIE IDEALS IN NEST ALGEBRAS Recall that a nest JK on a Hilbert space Si is a chain of closed subspaces of Si which is closed under the operations of arbitrary intersections and closed linear spans, and which includes {O} and S5. The nest algebra T(KJ) is the algebra of all operators on Se leaving every member of JK invariant. This is always closed in the weak operator topology. The diagonal D(KJ) of a nest algebra T(JK) is the von Neumann subalgebra 7(K) n T(K)*. If E, F E KV with E < F, then F E is called an interval of the nest. The nonzero minimal intervals are called atoms. A nest is atomic if the atoms of the nest span 55. We refer the reader to [1] for more information on nest algebras. Our main result, Theorem 12, shows that for every weakly closed Lie ideal Z in 7(JK), there exist a corresponding weakly closed associative ideal IC and a von Neumann subalgebra SK of O(A/) such that