On extensions of stably finite \(C^*\)-algebras (Q2891147)

A \(C^*\)-algebra \(A\) is called finite if it admits an approximate unit of projections and all projections in \(A\) are finite. If \(A\otimes\mathcal K\) is finite, then \(A\) is called stably finite. The main result of the paper under review is that any \(C^*\)-algebra \(A\) with an approximate unit of projections contains a smallest ideal \(I\) such that the quotient \(A/I\) is stably finite. This complements a previous result due to \textit{J. S. Spielberg} [J. Funct. Anal. 81, No. 2, 325--344 (1988; Zbl 0678.46047)].