Strongly exposed points in quotients of Douglas algebras (Q1064519)

An essentially sup-norm closed subalgebra between \(H^{\infty}\) and \(L^{\infty}\) on the unit circle is called a Douglas algebra. For a Banach space Y, ball(Y) denotes the closed unit ball of Y. A point y in ball(Y) is called strongly exposed if there is a bounded linear functional L of Y satisfying the following conditions; i) \(\| L\|_{Y^*}=L(y)=1\), and ii) if a sequence \(\{y_ n\}\) in ball(Y) satisfies \(L(y_ n)\to 1\) (n\(\to \infty)\), then \(\| y-y_ n\| \to 0\) (n\(\to \infty).\) Theorem. Let \(B_ 1\) and \(B_ 2\) be Douglas algebras with \(H^{\infty}\subset B_ 1\subsetneqq B_ 2\subset L^{\infty}\). Then there are no strongly exposed points in \(ball(B_ 2/B_ 1)\).