The general stable rank in nonstable \(K\)-theory (Q5928706)

For a \(C^*\)-algebra \(A\) denote by \({\mathcal U}(A)\) its unitary group (if \(A\) is non-unital, one has to use its unitalization \(A^+\) instead) and by \({\mathcal U}_0(A)\) the connected component of the unit in \({\mathcal U}(A)\) and put \(U(A)={\mathcal U}(A)/{\mathcal U}_0(A)\). Put \(p_1=\text{diag}(1,0)\in M_2(A^+)\) and let \(1_k\) denote the unit of \(M_k(A^+)\). A \(C^*\)-algebra \(A\) has 1-cancellation if for any projection \(p\in M_2(A^+)\) the Murray-von Neumann equivalence of projections \(\text{diag}(p,1_k)\) and \(\text{diag}(p_1,1_k)\) implies this equivalence for \(p\) and \(p_1\). The general stable rank of \(A\), gsr\((A)\), was defined by \textit{M. A. Rieffel} [Proc. Lond. Math. Soc., III. Ser. 46, 301-333 (1983; Zbl 0533.46046)] as the minimal integer \(n\) such that for any \(m\geq n\) the group \({\mathcal U}(M_m(A))\) acts transitively on the set \(\{(a_1,\ldots,a_m)\in (A^+)^m: \sum_{i=1}^m a_i^*a_i=1\}\). NEWLINENEWLINENEWLINEThe author shows that the natural homomorphism \(i_A:U(A)\to K_1(A)\) is injective iff \(i_{M_n(A)}\) is injective for all \(n\) iff the suspension \(C^*\)-algebra \(SA=C_0(0,1)\otimes A\) has 1-cancellation iff \(\text{gsr}(SA)=1\). As applications the author calculates \(\text{gsr}(SA)\) and \(\text{gsr}(C(S^1)\otimes A)\) for some \(C^*\)-algebras and describe some classes of \(C^*\)-algebras \(A\) such that the map \(i_A\) is an isomorphism.