Weak Limits of Projections and Compactness of Subspace Lattices

A strongly closed lattice of projections on a Hubert space is compact if the associated algebra of operators has a weakly dense subset of compact operators. If the lattice is commutative, there are necessary and sufficient conditions for compactness, one in terms of the structure of the lattice, and the other in terms of a measure on the lattice. There are many examples of compact lattices, and two main types of examples of noncompact lattices. Compactness is also related to the study of weak limits of certain projections. In [A3], Arveson generalized N. Andersen's important theorem on continuous nests [An, Theorem 3.5.5] to certain commutative subspace lattices associated with "2-continuous" measures. An important property of these lattices is that they are compact in the strong operator topology. Since compactness is also a property of nests, and plays an important role in the similarity theory for nests [Da2, La, An], we investigate in this paper the problem of determining which subspace lattices are compact. A greater understanding of this problem will hopefully help in the development of similarity theory for other lattices. It should be noted that it is well-known that there are also noncompact lattices, namely orthocomplemented commutative subspace lattices. In §2, we give a main result (Theorem 2.2) which provides a connection between compactness of a lattice 7£ and the presence of compact operators in the associated operator algebra alg 7£. This result provides many examples of compact lattices. It also shows that there is a relationship between complete distributivity, compactness, and other nice properties, especially if the lattice is commutative. §3 deals with properties of noncompact lattices. Included are two conditions equivalent to noncompactness for commutative lattices: one is essentially measuretheoretic, and the other is a condition on the structure of the lattice. We also give two quite different classes of examples of noncompact lattices. In the process, we obtain a representation theorem for certain infinite tensor products of commutative subspace lattices, which extends a result in [GHL]. We then show that "most" infinite tensor products are noncompact. One simple condition determining compactness is whether or not the lattice has any nonprojection weak limits (Proposition 2.1). This property is related to a paper Received by the editors June 27, 1986. Some of the results in this paper were presented at the NSF-CBMS Conference on Quasitriangularity and Analyticity in Operator Algebras, Texas Tech University, August 1, 1983. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25. ©1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page 515 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use