On absorbing extensions (Q2701636)

Let \(A\) and \(B\) be \(C^*\)-algebras. An extension \(\tau\) of \(A\) by \(B\) is absorbing [unital-absorbing with \(A\) unital] if \(\tau\) is strongly equivalent to \(\tau \oplus \theta ,\) for any trivial extension [strongly unital trivial extension] \(\theta .\) Note that if \(\theta\) is an absorbing trivial extension, then \(\tau \oplus \theta\) is absorbing extension for any \(\tau .\) Absorbing trivial extensions play an important role in the operator K-Theory. The existence of an absorbing trivial extension has only been established by G.G.Kasparov in the case where at least one of the \(C^*\)-algebras involved is nuclear. NEWLINENEWLINENEWLINEThe purpose of the paper is to show that such extensions always exist when both \(C^*\)-algebras are separable. If \(A\) is unital there is a unital trivial extension which is unitally absorbing.