Lie and Jordan Ideals in B(c 0 ) and B(l ρ )

Abstract

It is shown that ideals with respect to the canonical Lie (commutator) product in these algebras are exactly the linear manifolds that contain the images of their elements under the action of inner automorphisms induced by invertible spectral operators of scalar type. Jordan ideals in these algebras are identical with two-sided associative ideals and are also applied to a characterization of Lie ideals. It is well known that any associative algebra A becomes a Lie algebra and a Jordan algebra under the products defined by [F, G] = FG GF and F o G = '(FG + GF) for F, G E A, respectively. (The ground field is C.) A linear manifold L in A is called a Lie (Jordan) ideal if the corresponding (Lie or Jordan) product belongs to L for every F E A and G E L. It is then a two-sided ideal with respect to the corresponding product. Similarly, by an associative ideal we shall understand a two-sided ideal under the associative multiplication. Remarkable characterizations of the (not necessarily closed) Lie and Jordan ideals in the case when H is a complex infinite-dimensional separable Hilbert space and A is B(H), the (associative) algebra of all bounded linear operators in H, were described by Fong, Miers, and Sourour [3; Theorems 1 and 3] and obtained partly by Topping (see [3, 11]). Fong and Murphy [4] showed that [3, Theorem 1] is valid in the nonseparable case, too. The purpose of this paper is to prove that [3, Theorems 1 and 3] can be extended in a suitable form for the case when the underlying complex Banach space is either co or Ip (1 < p < xo) . The proofs are based on some fundamental results of Pelczynski [9] on complemented subspaces and isomorphisms in the Banach spaces above. Owing to them, we can make use of several ideas, applied earlier in the case of a separable Hilbert space, which we shall not reproduce here in detail. For a reference on spectral operators see, e.g., the book of Dunford and Schwartz [1]. For a bounded linear operator in a Banach space X, ker and im will denote its kernel and range subspace, respectively. An idempotent will be called any operator P E B(X) satisfying p2 = P. The notations (X E X D ... )x and Received by the editors June 17, 1991. 1991 Mathematics Subject Classification. Primary 47D30, 47B40, 17B60.