FACTORIZATION PROBLEMS AND THE K(,O) GROUPS OF NEST ALGEBRAS (OPERATORS)

Let M be a von Neumann algebra. A nest (beta) in M is a commuting family of projections containing O and I which is linearly ordered as a lattice and which is strongly closed. Let Alg((beta)) be the set of all elements of M which leave every projection of (beta) invariant and let D((beta)) be the set of all elements of M which commute with every number of (beta). A nest is injective if there exists a faithful normal projection (PHI) of norm one from M onto D((beta)).;We consider the problem of factoring an invertible S (ELEM) M as S = WA where W is a unitary operator in M and both A and A('-1) belong to Alg (beta). Through a variation of the LU decomposition for matrices we prove the following:;Theorem A. Let (beta) be an injective nest in M and let S (ELEM) M be invertible. Let S(beta) be the nest in M given by SP : P (ELEM) (beta) ; here SP denotes the projection whose range equals the range of SP. Then S factors as above if and only if S(beta) is injective.;Theorem B. Let (beta) be an injective nest in M and let I = Alg (beta) (INTERSECT) ker (PHI). The following are equivalent: (i) Every invertible S factors as above. (ii) Every left invertible element of I with a unit adjoined is right invertible. (iii) I contains no non-zero idempotent. (iv) Given an idempotent e (ELEM) Alg (beta) there exists an invertible A with A, A('-1) (ELEM) Alg (beta) such that AeA('-1) is a projection.;Theorem C. Let (M,tr) be a semifinite von Neumann algebra. Let T (ELEM) M be positive, invertible such that S = I - T satisfies tr(S*S) ) K(,o)(QD((beta))) is an isomorphism.;Since it is straightforward to compute K(,o)(QD((beta))), theorem D allows us to compute K(,o)(Alg (beta)) for any nest (beta) in L(H). (Abstract shortened with permission of author.)...