SPECTRAL SYNTHESIS AND MASA-BIMODULES

Abstract

The interaction between harmonic analysis and operator theory has been fruitful. In [1] Arveson related results in the theory of operator algebras to spectral synthesis. He defined synthesis for subspace lattices and proved that certain classes of lattices are synthetic. The main result in this paper is a generalization of a result of Arveson for the case of subspace maps. In order to describe in more detail the content of the present work, we need to introduce some definitions and facts from [2] and [9].

Let ${\cal H}_1$ and ${\cal H}_2$ be separable complex Hilbert spaces, and ${\cal P}_i$ be the lattice of all (orthogonal) projections on ${\cal H}_i, i = 1, 2$ . Following Erdos [1], we let ${\cal M}({\cal P}_1, {\cal P}_2)$ denote the set of all maps $\varphi : {\cal P}_1 \longrightarrow {\cal P}_2$ which are 0-preserving and $\vee$ -continuous (that is, they preserve arbitrary suprema). We will call such maps subspace maps. It was shown in [2] that each $\varphi \in {\cal M}({\cal P}_1, {\cal P}_2)$ uniquely defines semi-lattices ${\cal S}_{1\varphi} \subseteq {\cal P}_1$ and ${\cal S}_{2\varphi} = \varphi({\cal P}_1) \subseteq {\cal P}_{2}$ such that $\varphi$ is a bijection between ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ and is uniquely determined by its restriction to ${\cal S}_{1\varphi}$ . Moreover, ${\cal S}_{1\varphi}$ is meet-complete and contains the identity projection while ${\cal S}_{2\varphi}$ is join-complete and contains the zero projection. If ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ are commutative, we say that $\varphi$ is a commutative subspace map.