The developments of the present paper center around the observation that the ring B of bounded everywhere defined operators in Hilbert space contains non-trivial two-sided ideals.' This fact, which has escaped all but oblique notice in the development of the theory of operators, is of course fundamental from the point of view of algebra and at the same time differentiates 93 sharply from the ring of all linear operators over a unitary space with finite dimension number. As examples of two-sided ideals in 93 we may mention here the class of all operators A such that J(A), the range of A, has a finite dimension number, the class of all operators of Hilbert-Schmidt type,2 and the class fT of all totally continuous operators. Except for the ideal (0), every two-sided ideal in 93 contains the first ideal mentioned, and except for the ideal J3 itself, every twosided ideal in 93 is contained in the ideal f. Moreover, on the basis of the special spectral properties of the self-adjoint members of sJ, it is possible to characterize every two-sided ideal in ?B very simply in terms of the spectra of its nonnegative self-adjoint elements; for both the formulation and the proof of this result, which together with the facts mentioned above is discussed in ?1, the author is indebted to J. v. Neumann. The restriction of our attention to those ideals in if which are two-sided is basic for the points which we wish to develop; the two-sidedness compensates for the absence of commutativity in 93 in such a way as to permit the construction of quotient rings by the standard methods of abstract algebra.3 These rings, which are of course homomorphs of B with respect to addition and multiplication, are also homomorphs of. @ with respect to the operation *, and exhibit all of the formal properties of matrix algebras. This is established in ?2, and there also various properties of the associated congruences in gB are discussed. The remainder of the paper deals solely with the quotient ring @/{, where fJ is the ideal of totally continuous operators, and the associated congruence in i3. For essentially topological reasons, this is the only one of the quotient rings in

http://www.fa.uni-tuebingen.de/lehre/ss-2008/banachalgebren/scripts/calkin_math_ann_1942.pdf

Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space

The bounded resolutions of the identity in a von Neumann algebra can be ordered by I Es(u) } - ET(U), uER. The selfadjoint operators in the algebra are partially ordered by this relation and are shown to form a conditionally complete lattice. The lattice operations are (essentially) defined by Evsa(u) = AESj(U) for all u contained in R. This order is called spectral order and agrees with the usual order on commutative subalgebras. For positive operators, S is greater than or equal to T in spectral order if and only if Sn is greater than or equal to Tn in the usual order for all n >1. Kadison's well-known counterexample is shown to fail. The operator lattice defined by spectral order differs from a vector lattice in the fact that S>T does not imply that S+C>-T+C.

The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice

Spectral order of operators and range projections

That is, we establish directly the well-known spectral theorem for unbounded self-adjoint operators using only simple geometric intrinsic properties of Hilbert space. Of course the fundamental facts about the spectral representation of bounded as well as unbounded operators have been known in substance since the appearance in 1906 [3] of Hilbert's memoir on integral equations and in 1929 [8] of von Neumann's fundamental paper on unbounded operators. Since that time many papers have been published in this subject using a variety of methods. Some of these methods apply only to bounded operators, while others are suited to the general (unbounded) case. However, all but a few of these methods use techniques and principles which lie outside of Hilbert space theory proper, such as Helly's selection principle, Riesz's second representation theorem and so on. For further examples of technique and for a comprehensive list of references on the spectral theorem we refer the reader to [1], p. 927 and [5]. It was not until 1935 that Lengyel and Stone [5] gave a new proof of the spectral theorem which was strictly elementary in the sense that it depended only upon intrinsic properties of Hilbert space. Their paper dealt with the case of bounded operators and the authors remarked that they could not handle the general case in the same way. In their introduction they wrote: "Indeed our method, which requires the study of powers of an operator, is not suited to the case of unbounded operators" ([5], p. 853). Just this sentence stimulated the author to try to handle the general case in the same way and indeed it is possible. The fundamental idea of our proof of the spectral theorem as well as that of Lengyel and Stone consists in considering invariant subspaces F(A,2)= {xenlxeD(A"), ]lA"xl[ <2" . Itxll for n = 1,2, 3 .... }.

M. H. Stone

Papers

Elementary Proof of the Spectral Theorem