Topological stable rank of $$\mathcal {E}'(\mathbb {R})$$

The set $$\mathcal {E}'(\mathbb {R})$$ E ′ ( R ) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of $$\mathcal {E}'(\mathbb {R})$$ E ′ ( R ) is 2.

On the Bézout equation in the ring of periodic distributions

A corona type theorem is given for the ring D'A(Rd) of periodic distributions in Rd in terms of the sequence of Fourier coefficients of these distributions,which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of D'A(Rd) are both equal to 1.

Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

We determine the generalized $E$-stable ranks for the real algebra, $C(\boldsymbol{D})_{\mathrm{sym}}$, of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra $A(\boldsymbol{D})_{\mathsf R}$ of holomorphic functions. A characterization of those invertible functions in $C(E)$ is given that can be uniformly approximated on $E$ by invertibles in $A(\boldsymbol {D})_{\mathsf R}$. Finally, we compute the Bass and topological stable rank of $C(K)_{\mathrm{sym}}$ for real symmetric compact planar sets $K$.

STABLE RANK FOR INCLUSIONS OF C*-ALGEBRAS

When a unital C*-algebra A has topological stable rank one (write tsr (A) = 1), we know that tsr (pAp) = 1 for a non-zero projection p ∈ A. When, however, tsr (A) ≥ 2, it is generally false. We prove that if a unital C*-algebra A has a simple unital C*-subalgebra D of A with common unit such that D has Property (SP) and sup p ∈ P(D) tsr (pAp) < ∞, then tsr (A) ≤ 2. As an application let A be a simple unital C*-algebra with tsr (A) = 1 and Property (SP), [Formula: see text] finite groups, αk actions from Gk to Aut ((⋯((A × α1 G1) ×α2 G2)⋯) ×αk-1 Gk-1). (G0 = {1}). Then [Formula: see text]