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 H1 and H2 be separable complex Hilbert spaces, and Pi be the lattice of all (orthogonal) projections on Hi, i = 1, 2. Following Erdos [1], we let M(P1, P2) denote the set of all maps φ : P1 → P2 which are 0‐preserving and v‐continuous (that is, they preserve arbitrary suprema). We will call such maps subspace maps. It was shown in [2] that each φ ∈ M(P1, P2) uniquely defines semi‐lattices S1φ ⊆ P1 and S2φ = φ(P1) ⊆ P2 such that φ is a bijection between S1φ and S2φ and is uniquely determined by its restriction to S1φ. Moreover, S1φ is meet‐complete and contains the identity projection while S2φ is join‐complete and contains the zero projection. If S1φ and S2φ are commutative, we say that φ is a commutative subspace map.