The author considers such a problem: Given a closed two-sided ideal J in a unital \(C^*\)-algebra A, integer n, and an invertible element \(x\in M_ n(A/J)\), under what conditions can x be lifted to a “good” element in \(M_ n(A)?\)
The main result is stated in terms of stable ranks. It implies, for instance, that for any exact sequence of \(C^*\)-algebras \[ 0\quad \to \quad B\otimes K\quad \to \quad A\quad \to \quad D\quad \to \quad 0, \] given \(u\in M_ n(D)\) unitary, one has:
(a) u can be lifted to a unitary element in \(M_ n(A)\), iff \(\partial [u]=0;\)
(b) u can be lifted to an isometry in \(M_ n(A)\), iff \(\partial [u]\leq 0;\)
(c) u can be lifted to a coisometry in \(M_ n(A)\), iff \(\partial [u]\geq 0\).