# connected stable rank

There is a beautiful formula by Leonid Vaserstein relating the Bass and topological stable rank of a commutative unital Banach algebra A to that of the matrix algebra M_n(A). Is there something similar true for the connected stable rank (defined to be the least integer n such that for all m bigger than n the set U_m(A) of unimodular/invertible m-tuples is connected).Thanks.

In [1] Proposition 2.10, and also [2] Theorem 4.7, it is shown that the connected stable rank satisfies the following estimate for the matrix algebra over a (not necessarily commutative) C*-algebra $$A$$: $$\mathrm{csr}(M_n(A)) \leq \left\lceil \frac{\mathrm{csr}(A)-1}{n}\right\rceil+1.$$ The reverse inequality seems to be unclear.