We establish interpolation formulæ for operator spaces that are components of a given quasi-normed operator ideal. Sometimes we assume that one of the couples involved is quasi-linearizable, some other times we assume injectivity or surjectivity in the ideal. We also show the necessity of these suppositions.

A certain class of Arens-Michael algebras having no non-zero injective topological ⨶-modules is introduced. This class is rather wide and contains, in particular, algebras of holomorphic functions on polydomains in ${ℂ}^n$, algebras of smooth functions on domains in ${ℝ}^n$, algebras of formal power series, and, more generally, any nuclear Fréchet-Arens-Michael algebra which has a free bimodule Koszul resolution.

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

Let ${[X_0,X_1]}_t$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_0$ and $X_1$ will act boundedly on each ${[X_0,X_1]}_t$. Let $T_t$ denote such an operator when considered on ${[X_0,X_1]}_t$, and $\sigma \left(T_t\right)$ denote its spectrum. We are motivated by the question of whether or not the map $t\to \sigma \left(T_t\right)$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t\to {\left(\sigma \left(T_t\right)\right)}^{\wedge }$ (polynomially convex hull) and $t\to {\partial }_e\left(\sigma \left(T_t\right)\right)$ (boundary of the polynomially convex...