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Journal of the London Mathematical Society 2002 65(3):733-744; doi:10.1112/S0024610702003150
© 2002 by London Mathematical Society
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© The London Mathematical Society

Spectral Synthesis and Masa-Bimodules

I. G. Todorov

Department of Mathematics, University of Aegean 83200 Karlovassi, Samos, Greece

Received 17 January 2001. Revision received 9 October 2001.

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 {varphi} : 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 {varphi} isin M(P1, P2) uniquely defines semi-lattices S1{varphi} {subseteq} P1 and S2{varphi} = {varphi}(P1) {subseteq} P2 such that {varphi} is a bijection between S1{varphi} and S2{varphi} and is uniquely determined by its restriction to S1{varphi}. Moreover, S1{varphi} is meet-complete and contains the identity projection while S2{varphi} is join-complete and contains the zero projection. If S1{varphi} and S2{varphi} are commutative, we say that {varphi} is a commutative subspace map.


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