Fredholm theories in von Neumann algebras. I

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.

On the dimension theory of von Neumann algebras

“…This all works for L p modules [16], too, and is closely related to the K 0 functor [14], [35]. (Most of the ideas of the preceding two paragraphs were discussed by Breuer [1], [2], without making reference to standard forms. He focused on the monoid generated by equivalence classes of finite projections, because the associated Grothendieck group, called the index group of M, is the natural carrier for the Fredholm theory of M. Olsen [27] Proof.…”

“…The description of the quotient (P(M)/∼) is known as the dimension theory for M. This is essentially the first invariant in the subject, going back to Murray and von Neumann's initial observations [24,Part II]. Among other uses, dimension theory leads directly to the type decomposition, classifies representations (see Section 7), and supports the generalized Fredholm theory [1], [2], [27] required for noncommutative geometry. In this paper we prove basic results about three aspects of dimension theory: topology, parameterization, and order.…”

Теория Индекса И Некоммутативная Геометрия На Многообразиях Со Слоением

“…Теория операторов Брейера-Фредгольма была разработана в работах [55], [56] в случае, когда N -фактор, и распространена на случай, когда N не является фактором, в работе [57].…”

Elementary operators on prime C*-algebras II

“…. For W*-algebras there is another concept of compact elements due to Breuer [6]. An element a of a W*-algebra A is called compact relative to A if it belongs to the closed ideal m(A) generated by all finite projections in A.…”