We identify a class of smooth Banach *-algebras that are differential subalgebras of commutative C*-algebras whose openness of multiplication is completely determined by the topological stable rank of the target C*-algebra. We then show that group algebras of Abelian groups of unbounded exponent fail to have uniformly open convolution. Finally, we completely characterize in the complex case (uniform) openness of multiplication in algebras of continuous functions in terms of the covering dimension.

The paper investigates the algebraic properties of weakly inverse-closed complex Banach function algebras generated by functions of bounded variation on a finite interval. It is proved that such algebras have Bass stable rank 1 and are projective-free if they do not contain nontrivial idempotents. These properties are derived from a new result on the vanishing of the second Čech cohomology group of the polynomially convex hull of a continuum of a finite linear measure described by the classical H. Alexander theorem.

Let $X$ be a compact metric space. A lower bound for the radius of comparison of the ${{\text{C}}^{*}}$-algebra $\text{C}\left( X \right)$ is given in terms of ${{\dim}_{\mathbb{Q}}}\,X$, where ${{\dim}_{\mathbb{Q}}}\,X$ is the cohomological dimension with rational coefficients. If ${{\dim}_{\mathbb{Q}}}\,X\,=\,\dim\,X\,=\,d$, then the radius of comparison of the ${{\text{C}}^{*}}$-algebra $C\left( X \right)$ is $\max \left\{ 0,\,\left( d\,-\,1 \right)/\,2\,-\,1 \right\}$ if $d$ is odd, and must be either ${d}/{2}\;\,-\,1$ or ${d}/{2}\;\,-\,2$ if $d$ is even (the possibility ${d}/{2}\;\,-\,1$ does occur, but we do not know if the possibility ${d}/{2}\;\,-\,2$ can also occur).

We analyze the decomposition rank (a notion of covering dimension for nuclear C*-algebras introduced by E. Kirchberg and the author) of subhomogeneous C*-algebras. In particular, we show that a subhomogeneous C*-algebra has decomposition rank $n$ if and only if it is recursive subhomogeneous of topological dimension $n$, and that $n$ is determined by the primitive ideal space.

As an application, we use recent results of Q. Lin and N. C. Phillips to show the following. Let $A$ be the crossed product C*-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then the decomposition rank of $A$ is dominated by the covering dimension of the underlying manifold.

We shall discuss relations between rectangularity and piecewise rectangularity of product spaces. In particular, we show that for each positive integer n there exists an n-dimensional, collectionwise normal, non-piecewise rectangular product X × Y which satisfies the inequality dim (X × Y) ≤ dim X + dim Y.

The central area of investigation is in the isolation of conditions on mappings which leave invariant the classes of locally finite-dimensional metric spaces and strongly countable-dimensional metric spaces. Examples of such properties are open and closed with discrete point-inverses, open and finite-to-one, or open, closed, and countable-to-one.