ec 2 00 5 Lattices of Projections and Operator Ranges in C * - algebras
نویسنده
چکیده
We say that a C*-algebra A has the ‘sublattice property’ if the set P(A) of all projections in A is a sublattice of the projection lattice of its enveloping von Neumann algebra A∗∗. It is shown that this property is equivalent to the requirement that all pairs of projections have strictly positive ‘angle’. We discuss this property in relation to some results obtained by C. Akemann and by A. Lazar. The AW* algebras with the sublattice property are characterized. We say that a C*-algebra A satisfies property L if for any representation π, the collection Or(π(A)) of ranges of operators in π(A) is a lattice with respect to intersections and (non-closed) linear spans. Property L is strictly stronger than the sublattice property. We exhibit classes of algebras possessing property L. It is shown that the set of all central covers c(π) of representations π for which Or(π(A)) is a lattice is an ideal in the lattice of central projections of A∗∗.
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