LIE IDEALS IN TRIDIAGONAL ALGEBRA ALG𝓛∞

Abstract

Abstract. We give examples of Lie ideals ina tridiagonal algebra AlgL ∞ and study some properties of Lie ideals in AlgL ∞ . We also investigaterelationships between Lie ideals in AlgL ∞ . Let k be a fixed naturalnumber. Let A be a linear manifold in AlgL ∞ such that T (2 k−1,2 ) = 0for all T ∈A. Then Ais a Lie ideal if and only if T (2k−1,2k−1) = T (2k,2k) for all T ∈A. 1. IntroductionLet H be an infinite-dimensional separable Hilbert space with a fixed or-thonormal base {e 1 ,e 2 ,...} and let B(H) be the algebra of all bounded oper-ators on H. If x 1 ,x 2 ,...,x k are vectors in H, we denote by [x 1 ,x 2 ,...,x k ]the closed subspace spanned by the vectors x 1 ,x 2 ,...,x k . A subspace lat-tice L is a strongly closed lattice of orthogonal projections acting on H. Wedenote by L ∞ the subspace lattice generated by the subspaces [e 1 ], [e 3 ],...,[e 2n−1 ],...,[e 1 ,e 2 ,e 3 ], [e 3 ,e 4 ,e 5 ],..., [e 2n−3 ,e 2n−2 ,e 2n−1 ],.... By AlgL ∞ , wemean the algebra of bounded operators which leave invariant all of the sub-spaces in L