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Similar commutative subspace lattices (CSL's) are shown to be unitarily equivalent if certain sublattices (which may be taken to be nests!) are unitarily equivalent and a technical condition is satisfied. This result provides a connection between existing results for arbitrary similarities of countable CSL's and similarities of general CSL's by operators near the identity. One consequence is the generalizaton to CSL's of a theorem of David Pitts on the relationship between similarity and unitary equivalence of nests he calls "injective." H denotes a separable Hilbert space with real or complex scalars. B(H) is the space of all bounded, linear operators on H. The word "projection" always refers to an operator of orthogonal projection. For any A in B(H), rp(A) denotes the projection of H on the closure of the range of A. R(A) is the range of A, and I is the identity operator. If D is a linear operator on B(H) satisfying (i) for all X E B(H) such that X > 0, ?(X) ? 0, (ii) ?(X*) = ?D(X)* for all X E B(H), (iii) D is idempotent, and (iv) 'D(X)'D(Y) = ?D(X(?D(Y)) for all X and Y in B(H), then D is a conditional expectation from B(H) onto R(:D). Remarks. 1. (ii) is a consequence of (i) when the scalars are complex. 2. The identity ?(X)?(Y) = ?(DID(X)Y) is easily derived from (iv) and (ii). 3. R(D) is the set of fixed points of (D. 4. [11; Chapter II] is a good reference for the properties of conditional expectation operators. Let F c 7 c B(H). A function (D from 7 into itself is F-homogeneous if D(CX) = C'D(X) and D(XC) = 'D(X)C for all C in W and X in F. (Thus we do not require any linearity or continuity.) Of course, conditional expectation operators are examples of homogeneous maps, and the significance of Received by the editors September 6, 1988. Presented to the Great Plains Operator Theory Seminar in Indianapolis, Indiana in May, 1988. 1980 Mathemnatics Subject Classification ( 1985 Revision). Primary 47C05, 47A 1 5, 47A68. Ket' words and phrases. Commutative subspace lattice algebra, operator factorization, nest, conditional expectations on von Neumann algebras. (? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

Arveson nests and operator factorization along commutative subspace lattices