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Toeplitz operators on flows
Journal article   Open access   Peer reviewed

Toeplitz operators on flows

Raúl E Curto, Paul S Muhly and Jingbo Xia
Journal of functional analysis, Vol.93(2), pp.391-450
1990
DOI: 10.1016/0022-1236(90)90133-6
url
https://doi.org/10.1016/0022-1236(90)90133-6View
Published (Version of record) Open Access

Abstract

Let R act continuously on a compact Hausdorff space X giving rise to a flow on X , let ϑ ϵ C ( X ), and let T ϑ x denote the Toeplitz operator on H 2 (R) determined by the function ϑ x on R defined by ϑ x ( t ) = ϑ ( x + t ). In this paper, we investigate the relation between the spectral properties of T ϑ x , the dynamical properties of the flow, and the value distribution theory of ϑ. The analysis proceeds by imbedding T ϑ 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 H 1 ( X , Z), then T ϑ x is invertible if and only if ϑ does not vanish on X and determines the zero element in H 1 ( 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 is analytic, in the sense that ϑ x belongs to H ∞ (R) for all x , we relate the II ∞ index of the Toeplitz operator determined by ϑ with the density of the zeros of ϑ 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.

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