Conditional expectations and the corona problem of ergodic Hardy spaces (Q1066422)

Let X be a compact Hausdorff space. Suppose that the one-parameter group \(\{\alpha_ t: t\in {\mathbb{R}}\}\) of homeomorphisms acts on X as a minimal flow. Let m be an ergodic probability measure of the flow. Let A(X) be the collection of continuous functions \(\phi\) such that for \(x\in X\), \(\phi (\alpha_ t(x))\), as a function on \({\mathbb{R}}\), extends to a bounded analytic function on the upper half plane. This A(X) is an analogue of the disc algebra. The weak-* closure of A(X) will be denoted by \(H^{\infty}(X)\), an analogue of \(H^{\infty}\) of the unit disc. Using the corona theorem for the upper half plane, we obtain the following corona theorem for \(H^{\infty}(X).\) Theorem. Given \(\phi_ 1,...,\phi_ N\in H^{\infty}(X)\). Then there are \(\psi_ 1,...,\psi_ N\in H^{\infty}(X)\) satisfying \(\phi_ 1\psi_ 1+...+\phi_ N\psi_ N=1\) if and only if there is a \(\delta >0\) such that for any \(r>0\) and a.e. \(x\in X\), \(\sum^{N}_{j=1}| (\phi_ j*P_{ir})(x)| \geq \delta\) where \(P_ z(t)=Imz/\pi | t-z|\) and * denotes the convolution via the flow.