Linear maps between \(C^*\)-algebras whose adjoints preserve extreme points of the dual ball (Q1272798)

Let \(\mathcal{A}\) and \(\mathcal{B}\) be C\({^{*}}\)-algebras; \(\operatorname {ext} X_{1}\) denotes the closed unit ball of a Banach space \(X\). The authors' main concern is, to characterise a linear mapping \(\psi: {\mathcal{A}}\to {\mathcal{B}}\) with the property that \(\rho \circ \psi \in\text{ext} {\mathcal{A}}^{*}_{1}\) whenever \(\rho \in \operatorname {ext}{\mathcal{B}}^{*}_{1}\). Their main result generalises Section 7 of \textit{E. Størmer} [Acta Math. 110, 233-278 (1963; Zbl 0173.42105)]. Assuming that the double commutant \({\mathcal{B}}''\subset {\mathcal{R}}_{E}\), for some projection \(E\) of a von Neumann algebra \({\mathcal{R}}\), \(\psi\) is characterised under the condition to have mutually orthogonal central projections (in order to get a density property slightly weaker than \((\psi({\mathcal{A}})'' = {\mathcal{R}}''\)), in terms of a Jordan *-homomorphism (or anti-*homomorphism) \(\varphi:A \to{\mathcal{R}}\) and \(E\). Up to *-isomorphism \(\psi\) is characterised `globally' by the fact that it admits a decomposition into a degenerate part \(A \to (I-E)\psi(A)(I - E)\) and a non-degenerate part \( A \to E\psi(A)E\). The non-degenerate part is *-isomorphic to a mapping \(A \to F \varphi(A)W F\), \(F\) the initial projection of a partial isometry \(W \in {\mathcal{R}}\) (or \(F\) the final projection of \(W\)). The degenerate part seems to be a nuisance, an irreducible representation \(\pi_0\), of the restriction of \({\mathcal{B}}\) to \(I-E\) being of the form \(V(\alpha(A)w)\) where \(\alpha_0\) is an irreducible representation of \({\mathcal{A}}\) in \(k\), \(w \in k\) and \( V_0\) an embedding of \(k\) into \(h\). The authors give a `local' characterisation of \(\psi\) such that, for an irreducible representation \(\pi\) of \({\mathcal{B}}\) in \(h\), basically \(\pi\circ\psi\) is (modulo similar formulations) either of the form \(V{^{*}}\alpha(A)U\) where \(V\) and \(U\) are isometries, \(\alpha\) a Jordan *-homomorphism of \({\mathcal{A}}\) into \({\mathcal{L}}(k)\), or of the form \(JV(\alpha(A)\omega)\), where \(J\) denotes the embedding of Hilbert-Schmidt operators of \(k\) into \({\mathcal{L}}(k)\). Here use is made of the adjoint \(\psi ^{*}\) acting on finite-rank Hilbert-Schmidt operators. From their results the authors can conclude that if \(\omega\) pure implies \(\omega\circ\psi\) pure, then also \(\rho \in\text{ext}{\mathcal{B}}{^{*}}_1\) implies \(\rho\circ\psi \in\text{ext} {\mathcal A}^{*}_{1}\). The final section comprises applications to extremal points of uniform algebras over a compact \(X\), in particular the disc algebra and \(H_{\infty}\).