We present new break-up and coherent data for h meson photoproduction on the deuteron, using a deuterium target and tagged bremsstrahlung photons up to 1 GeV. The differential cross sections for the coherent process were measured from threshold to 800 MeV. They are much smaller than those previously reported. The break-up channel provides a direct measurement of the neutron to proton differential cross section ratios. At the S 11 ͑1535͒ resonance peak, s n ͞s p 0.68 6 0.06 leading to an isoscalar to isovector amplitude ratio of A s ͞A y 0.096 6 0.02. [S0031-9007 (97)03324-3]

In the first part of this paper (Breuer [2]) the foundations of a generalized theory of compact operators and Fredholm operators relative to a yon Neumann algebra A were laid. The index group I(A) of A and the index map Index: ..~(A)-* I(A)of the space f'(A) of Fredholm elements of A were defined. Also the generalized Fredholm alternatives stating that 1 -C is Fredholm relative to A of index zero if C is compact relative to A were proved.In the second part of this paper the generalized Fredholm theory will be further developed. Some of the classical theorems of Atkinson [1], GohbergKrein [6] and Cordes-Labrousse [3] will be generalized. These generalizations will characterize the relative Fredholm elements modulo the relative compact elements, state the additivity of the index, and relate the index group I(A) to the group n0~Z-(A) of connected components of ~-(A). In particular, it will be shown that if A is properly infinite, then the algebraic invariant I(A) of A is canonically isomorphic to the topological invariant ~o~(A) of ~(A). The von Neumann algebra analogue of a theorem on the multipticativity of the index (Palais [9]) is also included, although it is not used in the following.Of course, the proofs are modeled after the classical proofs. To avoid misunderstandings the following two points should be mentioned. First the compact elements of A (or more precisely, relative to A) are not necessarily compact operators in the usual sense. Second the range of a Fredholm element of A is not necessarily closed. It is mainly the lack of these two properties that required the modification of the classical proofs.The following notations will be used. i This simplified notation is justified by Proposition 2 of the appendix saying that the index group of A is independent of A'.

No abstract

The theory of compact and of Fredholm operators of a Hilbert space has been generalized in various directions by several authors (Cordes [2,3], Neubauer [12,13] and others). In the present paper the foundations of a generalization of this theory to von Neumann algebras are laid. This generalization depends heavily on the notion of the finiteness of a projection relative to a von Neumann algebra.Using the equivalence classes of finite projections of a v o n Neumann algebra A (of continuous linear operators of a complex Hilbert space H) one can canonically construct an ordered commutative group F and a canonical map Dim of the finite projections of A into F. The function Dim has all the formal properties of the relative dimension function which Murray and von Neumann [11] defined for factors.The definition and the existence proof of the null projection N T and of the range projection R T of T e A are straightforward. Call T finite if R r is finite and compact if it is a limit in the norm of finite elements. For T t o be Fredholm it is required that (i) N r is finite and (ii) there is a finite projection E of A such that the range of 1 -E is contained in the range of T. Condition (ii) does not imply that the range of T is closed 1, unless the finiteness of E relative to A implies the finiteness of E relative to the full operator algebra L,e(/-/). 2 It is easy to see that (ii) implies the finiteness of Nr,, so that the index This paper concludes with the study of the increasing sequence of the null projections of the elements ( 1 -T)", n = 1, 2, 3, ..., where T e A is compact. A well-known decomposition theorem of F. Riesz for compact operators (see 2 E is finite relative to ~'(H) lff the linear space E(H) is finite-dimensional in the usual sense.