This paper is an exposition of the so-called injective Morita contexts (in which the connecting bimodule morphisms are injective) and Morita α-contexts (in which the connecting bimodules enjoy some local projectivity in the sense of ZimmermannHuisgen). Motivated by situations in which only one trace ideal is in action, or the compatibility between the bimodule morphisms is not needed, we introd...

In [CY98] Crane and Yetter introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter in [Yet98] is a proper generalization of Gerstenhaber’s deformation theory for associative algebras [Ger63, Ger64, GS88]. In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that u...

For a Banach algebra $A$, $A''$ is $(-1)$-Weakly amenable if $A'$ is a Banach $A''$-bimodule and $H^1(A'',A')={0}$. In this paper, among other things,  we study the relationships between the $(-1)$-Weakly amenability of $A''$ and the weak amenability of $A''$ or $A$. Moreover, we show that the second dual of every $C^ast$-algebra is $(-1)$-Weakly amenable.

Every ring extension of A by R induces a pair of group homomorphisms L : R → EndZ(A)/L(A);R : R → EndZ(A)/R(A), preserving multiplication, satisfying some certain conditions. A such 4-tuple (R,A,L,R) is called a ring pre-extension. Each ring pre-extension induces a R-bimodule structure on bicenter KA of ring A, and induces an obstruction k, which is a 3-cocycle of Z-algebra R, with coefficients...

Let $H$ be a Hopf algebra and $A$ an $H$-bimodule algebra‎. ‎In this paper‎, ‎we investigate Gorenstein global dimensions for Hopf‎ ‎algebras and twisted smash product algebras $Astar H$‎. ‎Results from‎ ‎the literature are generalized‎. 

let a be a unital r-algebra and m be a unital a-bimodule. it is shown that every jordan derivation of the trivial extension of a by m, under some conditions, is the sum of a derivation and an antiderivation.

Let A be a unital Banach algebra. We give a characterization of the left Banach A-modules X for which there exists a commutative unital C-algebra C(K), a linear isometry i:X → C(K), and a contractive unital homomorphism θ:A → C(K) such that i(a· x) = θ(a)i(x) for any a ∈ A, x ∈ X . We then deduce a “commutative” version of the Christensen-Effros-Sinclair characterization of operator bimodules. ...