We show for $2\le p<\infty$ and subspaces $X$ of quotients of $L_p$ with a $1$-unconditional finite-dimensional Schauder decomposition that $K(X,{\ell }_p)$ is an $M$-ideal in $L(X,{\ell }_p)$.

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces $X$ and $Y$ the subspace of all compact operators $𝒦(X,Y)$ is an $M(r_1{r}_2,s_1{s}_2)$-ideal in the space of all continuous linear operators $ℒ(X,Y)$ whenever $𝒦(X,X)$ and $𝒦(Y,Y)$ are $M(r_1,s_1)$- and $M(r_2,s_2)$-ideals in $ℒ(X,X)$ and $ℒ(Y,Y)$, respectively, with $r_1+s_1/2>1$ and $r_2+s_2/2>1$. We also prove that the $M(r,s)$-ideal $𝒦(X,Y)$ in $ℒ(X,Y)$ is separably determined. Among others, our results complete and improve some well-known results...

Let 𝓛 be a subspace lattice on a Banach space X and let δ: Alg𝓛 → B(X) be a linear mapping. If ⋁ {L ∈ 𝓛 : L₋ ⊉ L}= X or ⋁ {L₋ : L ∈ 𝓛, L₋ ⊉ L} = (0), we show that the following three conditions are equivalent: (1) δ(AB) = δ(A)B + Aδ(B) whenever AB = 0; (2) δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A) whenever AB + BA = 0; (3) δ is a generalized derivation and δ(I) ∈ (Alg𝓛)'. If ⋁ {L ∈ 𝓛 : L₋ ⊉ L} = X or ⋁ {L₋ : L ∈ 𝓛, L₋ ⊉ L} = (0) and δ satisfies δ(AB + BA) = δ(A)B + Aδ(B) + δ(B)A + Bδ(A)...

Nigel J. Kalton was one of the most eminent guests participating in the Józef Marcinkiewicz Centenary Conference. His contribution to the scientific aspect of the meeting was very essential. Nigel was going to prepare a paper based on his plenary lecture. The editors are completely sure that the paper would be a real ornament of the Proceedings. Unfortunately, Nigel's sudden death totally destroyed editors' hopes and plans. Every mathematician knows how unique were Nigel's mathematical achievements....

If Σ = (X,σ) is a topological dynamical system, where X is a compact Hausdorff space and σ is a homeomorphism of X, then a crossed product Banach *-algebra ℓ¹(Σ) is naturally associated with these data. If X consists of one point, then ℓ¹(Σ) is the group algebra of the integers. The commutant C(X)₁' of C(X) in ℓ¹(Σ) is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant C(X)'⁎ of C(X) in C*(Σ), the enveloping...

We study the problem of whether ${}_w\left(ⁿE\right)$, the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space (ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if ${}_w\left(ⁿE\right)$ is an M-ideal in (ⁿE), then ${}_w\left(ⁿE\right)$ coincides with ${}_{w0}\left(ⁿE\right)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if ${}_w\left(ⁿE\right){=}_{w0}\left(ⁿE\right)$ and (E) is an M-ideal in...

We establish Hölder-type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals.

We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure...