Conditional expectations and operator decompositions (Q1892345)

For a unital \(C^*\)-algebra \({\mathcal A}\) and a conditional expectation \(\Phi\) from \({\mathcal A}\) onto a subalgebra \({\mathcal B}\) consider the decomposition \({\mathcal A}= {\mathcal B}\oplus H\), the corresponding selfadjoint parts \({\mathcal A}^s= {\mathcal B}^s \oplus H^s\), and the corresponding images under the exponential function: \(G^+= \exp ({\mathcal A}^s)\) the set of positive invertibles of \({\mathcal A}\), \({\mathcal B}^+\) the set of positive invertibles of \({\mathcal B}\), and \(C= \{\exp h\mid h\in H\}\). Here is the main result: The maps \[ {\mathcal B}^+ \times C\ni (b,c) \mapsto (bcc^* b^*)^{1/2}\in G^+, \qquad {\mathcal B}^+ \times C \ni (bc)\mapsto bcb\in G^+ \] are analytic diffeomorphisms, and so is \[ {\mathcal B}^+\times C\times U\ni (b, c, u)\mapsto bcu\in G, \] where \(G\) and \(U\) denote the set of invertibles and of unitaries of \({\mathcal A}\), respectively.