The stable rank of some free product \(C^*\)-algebras
A unital \(C^*\)-algebra \(A\) is said to have stable rank 1 if the group of invertible elements in \(A\) is a norm dense subset of \(A\). The authors prove a general result giving sufficient conditions for a \(C^*\)-algebra \((U,\tau)\) to have stable rank 1, where \((U,\tau)\) is the reduced free product of a family \((A_i,\tau_i)\), \(i\in I\), of unital \(C^*\)-algebras \(A_i\) with normalized faithful traces \(\tau_i\). In particular, one obtains that the reduced group \(C^*\)-algebra \(C^*_{\text{red}}(G)\) has stable rank 1 if \(G\) is a discrete group arising as a free product \(G_1* G_2\) where \(| G_1|\geq 2\) and \(| G_2|\geq 3\).
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