Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

Tilting modules under special base changes

Given a non-unit, non-zero-divisor, central element x of a ring Λ, it is well known that many properties or invariants of Λ determine, and are determined by, those of $\mathrm{\Lambda }/x\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}_x$. In the present paper, we investigate how the property of “being tilting” behaves in this situation. It turns out that any tilting module over Λ gives rise to tilting modules over ${\mathrm{\Lambda }}_x$ and $\mathrm{\Lambda }/x\mathrm{\Lambda }$ after localization and passing to quotient respectively. On the other hand, it is proved that under some mild conditions, a module over Λ is tilting if its corresponding localization and quotient are tilting over ${\mathrm{\Lambda }}_x$ and $\mathrm{\Lambda }/x\mathrm{\Lambda }$ respectively.