A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property...

Under the assumption that m is a non-doubling measure on Rd, the authors obtain the (Lp,Lq)-boundedness and the weak type endpoint estimate for the multilinear commutators generated by fractional integrals with RBMO (m) functions of Tolsa or with Osc exp Lr(m) functions for r greater than or equal to 1, where Osc exp Lr(m) is a space of Orlicz type satisfying that Osc exp Lr(m)=RBMO(m) if r=1 and Osc exp Lr(m) is a subset of RBMO(m) if r>1.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

Let $𝒳$ be a Banach space of dimension $n>1$ and $𝔄\subset ℬ\left(𝒳\right)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d:𝔄\to 𝔄$ (not necessarily linear) satisfies $$d\left(\right[[U,V],W\left]\right)=\left[\right[d\left(U\right),V],W]+\left[\right[U,d\left(V\right)],W]+\left[\right[U,V],d(W\left)\right]$$ for all $U,V,W\in 𝔄$, then $d=\psi +\tau$, where $\psi$ is an additive derivation of $𝔄$ and $\tau :𝔄\to 𝔽I$ vanishes at second commutator $\left[\right[U,V],W]$ for all $U,V,W\in 𝔄$. Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S\in 𝔄$ and a linear mapping $\tau$ from $𝔄$ into $𝔽I$ satisfying $\tau \left(\right[[U,V],W\left]\right)=0$ for all $U,V,W\in 𝔄$, such that $d\left(U\right)=SU-US+\tau \left(U\right)$ for all $U\in 𝔄$.