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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.

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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.

[1] Nistor. Stable range for tensor products of extensions of K by C(X), J. Operator Th. 16 (1986)

[2] Rieffel. The homotopy groups of the unitary groups of non-commutative tori, J. Oper. Theory 17 (1987)

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