Toeplitz operators on flows

Let R act continuously on a compact Hausdorff space X giving rise to a flow on X, let θ{symbol} ε{lunate} C(X), and let T_θ{symbol}x denote the Toeplitz operator on H²(R) determined by the function θ{symbol}_x on R defined by θ{symbol}_x(t) = θ{symbol}(x + t). In this paper, we investigate the relation between the spectral properties of T_θ{symbol}x, the dynamical properties of the flow, and the value distribution theory of θ{symbol}. The analysis proceeds by imbedding T_θ{symbol}x in a type II_∞ factor and computing the real-valued index of the operator à la Connes. Our sharpest invertibility result asserts that if the flow is strictly ergodic and if the asymptotic cycle determined by the flow is injective on H¹(X, Z), then T_θ{symbol}x is invertible if and only if θ{symbol} does not vanish on X and determines the zero element in H¹(X, Z). This generalizes the classical result of Gohberg and Krein and its extension to Toeplitz operators with almost periodic symbols due to Coburn, Douglas, Schaeffer and Singer. When θ{symbol} is analytic, in the sense that θ{symbol}_x belongs to H^∞(R) for all x, we relate the II_∞ index of the Toeplitz operator determined by θ{symbol} with the density of the zeros of θ{symbol}_x in the upper half-plane. Much of our efforts to achieve this result are devoted to generalizing to arbitrary flows the value distribution theory of analytic almost periodic functions developed by Bohr, Jessen, and Tornehave and others.