\(K\)-continuity is equivalent to \(K\)-exactness (Q2795652)

A \(C^*\)-algebra \(A\) is exact when the functor \(B \mapsto A \otimes B\) preserves short exact sequences, where \(\otimes\) is the spatial tensor product. It is continuous when \(\lim (A \otimes B_n) = A \otimes \lim B_n\) for any inductive sequence \(B_0 \to B_1 \to \cdots\) of \(C^*\)-algebras. By a theorem of \textit{E. Kirchberg} [J. Funct. Anal. 129, No.~1, 35--63 (1995; Zbl 0912.46059)], continuity is equivalent to exactness.NEWLINENEWLINEThis paper considers a \(K\)-theoretic analogue. A \(C^*\)-algebra \(A\) is called \(K\)-exact if the functor \(B \mapsto K_0(A \otimes B)\) preserves short exact sequences. It is called \(K\)-continuous if \(\lim K_0(A \otimes B_n) = K_0(A \otimes \lim B_n)\) for any inductive sequence \(B_n\) of \(C^*\)-algebras. The main result is that \(K\)-continuity is equivalent to \(K\)-exactness.