Thanks to a result of Herman and Vaserstein in [[3](https://doi.org/10.1007/BF01388839)], Rieffel's notion of stable rank [[4](https://doi.org/10.1112/plms/s3-46.2.301)] coincides with the Bass stable rank [[1](http://www.numdam.org/item/PMIHES_1964__22__5_0/?source=BSMF_1962__90__323_0)] 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](https://doi.org/10.1007/3-540-28517-2)])** 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 $A$).


1. We have $\mathrm{sr}(pAp)\ge\mathrm{sr}(A)$.
2. 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]](http://www.numdam.org/item/PMIHES_1964__22__5_0/?source=BSMF_1962__90__323_0) H. Bass, K-theory and stable algebra, Publications Mathématiques de l’IHÉS 22 (1964), pp. 5–60.


[[2]](https://doi.org/10.1007/3-540-28517-2) B. Blackadar, *Operator Algebras*, Encyclopaedia of Mathematical Sciences 122 (2006), Springer, pp. 444-452.


[[3]](https://doi.org/10.1007/BF01388839) R. Herman and L. N. Vaserstein, *The stable range of $C^\ast$-algebras*, Inventiones Mathematicae 77 (1984) pp. 553-555.


[[4]](https://doi.org/10.1112/plms/s3-46.2.301) 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.



Stable rank one and corners of $C^\ast$-algebras
For separable $C^\ast$-algebras this phenomenon cannot happen, since any corner of a separable stable rank one $C^\ast$-algebra has stable rank one. More generally, this holds for $C^\ast$-algebras $A$ such that all its two-sided closed ideals are $\sigma$-unital.


In fact, suppose $A$ is separable and has stable rank one. Then all of its two-sided closed ideals have stable rank one [4, Theorem 4.4]. By Brown's stable isomorphism theorem (L. G. Brown, Stable isomorphism of hereditary subalgebras of C∗-algebras, Pacific J. Math 71 (1977), 335-348.), any corner $B$ of $A$ is stably isomorphic to the two-sided closed ideal in $A$ generated by $B$. As stable rank one is preserved by stable isomorphism [4, Theorem 3.6], any corner of $A$ has stable rank one.


