Thanks to a result of Herman and Vaserstein in [3], Rieffel's notion of stable rank [4] coincides with the Bass stable rank [1] for every $C^\ast$-algebra $A$: we denote it by $\mathrm{sr}(A)$ and we simply call it the stable rank of $A$.
The following fact yields a convenient characterization of the stable rank one case.
Lemma (folklore I suppose) A $C^\ast$-algebra has stable rank one if and only if the invertible elements are dense in $\widetilde{A}$, where $\widetilde{A}$ denotes $A$ if it is unital, and the unitization $A\oplus\mathbb{C}1$ of $A$ otherwise. For short, denoting the group of all invertible elements in $\widetilde{A}$ by $G\left(\widetilde{A}\right)$, we have: $$\mathrm{sr}(A)=1\quad \Leftrightarrow\quad\overline{G\left(\widetilde{A}\right)}=\widetilde{A}.$$
Now here are the facts I'm interested in. They deal with the stable rank of the corners $pAp$ of a given $C^\ast$-algebra $A$.
Proposition (Corollary V.3.1.18 in [2]) Let $A$ be a unital $C^\ast$-algebra and $p$ a full projection in $A$ (i.e. the closed ideal of $A$ generated by $p$ is all of $A$).
- We have $\mathrm{sr}(pAp)\ge\mathrm{sr}(A)$.
- If $\mathrm{sr}(A)<\infty$, then $\mathrm{sr}(pAp)<\infty$ and $$\mathrm{sr}(A)=1\quad \Leftrightarrow\quad\mathrm{sr}(pAp)=1.$$
What if we remove the assumption that $p$ is full?
- Question A: What are the known examples of $C^\ast$-algebras $A$ and projections $p\in A\setminus\{0,1\}$ such that $$\mathrm{sr}(A)=1<\mathrm{sr}(pAp)?$$
- Question B: What are the known examples of unital $C^\ast$-algebras $A$ and projections $p\in A\setminus\{0,1\}$ such that $$\mathrm{sr}(A)=1<\min\{\mathrm{sr}(pAp),\mathrm{sr}((1-p)A(1-p))\}?$$
[1] H. Bass, K-theory and stable algebra, Publications Mathématiques de l’IHÉS 22 (1964), pp. 5–60.
[2] B. Blackadar, Operator Algebras, Encyclopaedia of Mathematical Sciences 122 (2006), Springer, pp. 444-452.
[3] R. Herman and L. N. Vaserstein, The stable range of $C^\ast$-algebras, Inventiones Mathematicae 77 (1984) pp. 553-555.
[4] M. A. Rieffel, Dimension and stable rank in the K-theory of $C^\ast$-algebras, Proceedings of the London Mathematical Society 46 (1983), pp. 301–333.