Let AlgN be a nest algebra associated with the nest N on a (real or complex) Banach space X. Suppose that there exists a non-trivial idempotent P ∈ AlgN with range P (X) ∈ N and δ : AlgN → AlgN is a continuous linear mapping (generalized) left derivable at P , i.e. δ(ab) = aδ(b)+ bδ(a) (δ(ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ AlgN with ab = P . we show that δ is a (generalized) Jordan left derivation. Moreover, we characterize the strongly operator topology continuous linear maps δ on some nest algebra AlgN with property that δ(P ) = 2P δ(P ) or δ(P ) = 2P δ(P ) − P δ(I) every idempotent P in AlgN .MSC(2010): 47B47; 47L35.
Let X be a Hilbert C*-module over a C*-algebra B. In this paper we introduce two classes of operator algebras on the Hilbert C*-module X called operator algebras with property and operator algebras with property ℤ, and we study the first (continuous) cohomology group of them with coefficients in various Banach bimodules under several conditions on B and X. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.
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