Lifting automorphisms (Q1284143)

Let \(E\) be an essential extension \(0\to{\mathcal K}\to E@>\pi>> A\to 0\) given by a monomorphism \(\tau: A\to\text{End}(\ell^2)/{\mathcal K}\), where \(A\) is a separable \(C^*\)-algebra, \({\mathcal K}\) is the \(C^*\)-algebra of compact operators on the Hilbert space \(\ell^2\). For any unital \(C^*\)-algebra \(C\) the connected component of the automorphism group \(\Aut(C)\) is denoted by \(\Aut_0(C)\). The automorphism \(\beta\in \Aut(C)\) is said to be inner, if there is a unitary \(u\in C\) such that \(\beta(x)= \text{ad}(u)(x)= u^*xu\) \((x\in C)\), and it is said to be approximately inner, if there are unitaries \(u_n\in C\) such that \(\lim_{n\to\infty}\text{ad}(u_n)(x)= \beta(x)\) \((x\in C)\). Let \(\alpha: E\to E\) be an automorphism. It is well known that there is a unitary \(U\in\text{End}(\ell^2)\) such that \(\alpha= \text{ad}(U)\). Set \(E_\alpha= C^*(E,U)\), the \(C^*\)-algebra generated by \(E\) and \(U\). Since \(\alpha({\mathcal K})={\mathcal K}\) the automorphism \(\alpha\) induces the automorphism \(\overline\alpha\) on \(A\). Denote by \(E\times_\alpha{\mathbb{Z}}\) the cross product of \(E\) with \({\mathbb{Z}}\) using the action \(\alpha\). The \(K\)-groups \(K_i(E_\alpha)\) and \(K_i(E\times_\alpha{\mathbb{Z}})\) are computed and relations between them are established. Let \(s:\text{ext}^1_{\mathbb{Z}}(K_1(A),{\mathbb{Z}})\to \text{ext}^1_{\mathbb{Z}}(K_1(A),{\mathbb{Z}}/\text{im}(\delta))\) be the natural surjective map in group theory, where \(\delta: K_1(A)\to{\mathbb{Z}}\) is the index map, and let \(\omega(E)= \ker(\delta)\). The following two theorems are the main results of the paper. Theorem 3.5. If \(\omega(E)= 0\), \(\alpha\in \Aut(E)\), \(\overline\alpha\in \Aut_0(E)\) then \(\alpha\in \Aut_0(E)\) iff \([\alpha]= [\text{id}_E]\) in \(\text{KK}(E,E)\). Moreover, the automorphism \(\alpha\) is approximately inner if \([\alpha]= [\text{id}_E]\) in \(\text{KK}(E,E)\). Let \(\Aut_e(E)\) be the subgroup of \(\Aut(E)\), consisting of those automorphisms \(\alpha\) with \(\overline\alpha= \text{id}_E\) and \(\Aut_{e0}(E)\) be the connected component of \(\Aut_e(E)\). The special trivial essential extension \(\widehat E\) induced by \(\tau: A\to\text{End}(\ell^2)/{\mathcal K}\) and its ``inverse'' \((\tau)^{-1}\) is constructed. The automorphism \(\alpha\) induces the automorphism \(\widehat\alpha\in \Aut_e(\widehat E)\). Theorem 4.2. Let \(\alpha\in \Aut(E)\) with \(\overline\alpha= \text{id}_E\). Then: (a) \(\alpha\in \Aut_{e0}(E)\) iff \([\widehat\alpha]= [\text{id}_{\widehat E}]\) in \(\text{KK}(\widehat E,\widehat E)\); (b) if \(\alpha\) is inner, then \([\widehat\alpha]= [\text{id}_{\widehat E}]\); (c) \(\alpha\) is approximately inner iff \([\widehat\alpha]= [\text{id}_{\widehat E}]\).