Cohomology of a class of Kadison-Singer algebras

Abstract

Let L be the complete lattice generated by a nest N on an infinite-dimensional separable Hilbert space H and a rank one projection P ξ given by a vector ξ in H. Assume that ξ is a separating vector for N , the core of the nest algebra Alg(N ). We show that L is a Kadison-Singer lattice, and hence the corresponding algebra Alg(L) is a Kadison-Singer algebra. We also describe the center of Alg(L) and its commutator modulo itself, and show that every bounded derivation from Alg(L) into itself is inner, and all n-th bounded cohomology groups H n (Alg(L), B(H)) of Alg(L) with coefficients in B(H) are trivial for all n 1.

Derivations of a class of Kadison–Singer algebras

“…As a continuation of our study on the Hochshcild cohomology group and automorphisms for KS-algebras in [7,2], the purpose of this paper is to investigate the automatic continuity and (quasi-)spatiality of derivations of KS-algebras generated by double triangle lattices of projections in finite von Neumann algebras, defined in [8]. We first use the techniques and theory of unbounded operators affiliated with finite von Neumann algebras to characterize the structure of the class of KS-algebras, and relate them with transitive algebras (on different Hilbert spaces).…”

“…The celebrated result on the automatic continuity of derivations on C * -algebras was given by J. Ringrose, who proved that every derivation from a C * -algebra into its Banach bimodule is automatically continuous. Since then, many important results on the theory of derivations for operator algebras, especially for (non-selfadjoint) reflexive operator algebras, were obtained by many authors [1,3,6,7,15,17].…”

Minimal generating reflexive lattices of projections in finite von Neumann algebras

“…These results also indicate that many type I I 1 factors can be minimally generated by reflexive lattices of projections which are topologically homeomorphic to S 2 (plus zero and I ). Some examples of KS-lattices with different topological structures were also given in [8,19].…”

Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products

“…Let {e n : n ∈ N} be an orthonormal basis of H, P n = span{e i : i = 1, ..., n}, ξ = ∞ n=1 1 n e n and P ξ be the orthogonal projection from H onto the one-dimensional subspace of H generated by ξ. It follows from [20,Theorem 2.11] and [7,Lemma 3.2] that L = {0, I, P n , P ξ , P ξ ∨ P n : n = 1, 2, · · · } is a reflexive P-subspace lattice.…”