We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this.

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

We construct a weak Hilbert space that is a twisted Hilbert space.

Let ${\it\varphi}$ be a homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$. We define a multiplication on the Cartesian product space ${\mathcal{A}}\times {\mathcal{B}}$ and obtain a new Banach algebra ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$. We show that biprojectivity as well as biflatness of ${\mathcal{A}}\times _{{\it\varphi}}{\mathcal{B}}$ are stable with respect to ${\it\varphi}$.

We define the filtrated $\text{K}$-theory of a ${{\text{C}}^{*}}$-algebra over a finite topological space $X$ and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over $X$ in terms of filtrated $\text{K}$-theory.

For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification.

We also exhibit an example where filtrated $\text{K}$-theory is not yet a complete invariant. We describe two ${{\text{C}}^{*}}$-algebras over a space $X$ with four points that have isomorphic filtrated $\text{K}$-theory without being $\text{KK}\left( X \right)$-equivalent. For this space $X$, we enrich filtrated $\text{K}$-theory by another $\text{K}$-theory functor to a complete invariant up to $\text{KK}\left( X \right)$-equivalence that satisfies a Universal Coefficient Theorem.