An elementary proof of this result is given using only those properties that result from the fact that ReM is a Dedekind complete Riesz space with many normal integrals. This thesis will be primarily concerned with those properties of an Abelian W•-algebra M which follow from the fact that ReM is a Dedekind complete Riesz space. An ordered vector space Lis called a Riesz space if, for every pair f, g E L, sup(f, g) exists in L.
The linear subspace K of a Riesz space L is called a Riesz subspace of L whenever, for each pair f, gin K, the elements. The Riesz space L is called Dedekind complete if every non-empty subset of L which is bounded from above has a supremum. The real linear functional cp on the Riesz space Lis is said to be positive when cp(f).
A regularly bounded linear functional cp on the Riesz space Lis is said to be a normal integral if u I 0 implies inf ~(u ) I = 0. normal integrals on L will be denoted by L. The space denoted by L will be denoted by L* Banach dual for L. then l.B is an ideal in L. SJJI), equipped with the usual operator norm, is a B*-algebra. For any operator SE.t(N') we denote by N(S) the null space Sanda with R(S) the closure in J{ of the range S.
INTRODUCTORY REMARKS ON VON NEUMANN ALGEBRAS Let Jibe a complex Hilbert space;!("' the algebra of all bounded
The proof is elementary because it does not depend on the spectral theorem for bounded operators, which is then derived due to the Riesz space structure of Re M. It is possible to obtain a relatively simple proof of this result in the special case. of a w·-algebra. We will denote the set of all (real) linear functionals on ReM with (ReM)I, the Banach dual of ReM with (ReM)*, the band of normal integrals on ReM with (ReM)~, and the order dual of ReM by (ReM)~. The notation and terminology of the Riesz space will be as in f!O.
Notation: For each subset D of the Riesz space ReM we will denote by (D) (respectively (D}) the order ideal (respectively band) generated by D. If N is a subset of M we will denote this by N w the closure of N in the weak operator topology.6: (i) If N is an algebraic ideal (respectively weakly closed algebraic ideal) in M, then ReN is an order ideal (respectively band) in ReM. ii) If K is an order ideal (respectively band) in ReM, then K + i K is an algebraic ideal (respectively weakly closed algebraic ideal) in M.
If N is weakly closed, then O ~ T. ii) If K is an ideal of order in ReM, then K+ in K is definitely a linear subspace of M. Assume that TE K+ in K and that U is a unit element of M. From the uniqueness of the square root follows. In particular, N is an algebraic ideal of M. iii), (iv) follow easily and the proof will be omitted.
Finally, if M is abelian, let I-P denote the identity component on the band M = {TEReM: cp(! Tl)= O}. Proof: Let fEi }iE.11 be any family of mutually disjoint projections of M with Ei s: Ecp for eai:h iE .P. 6 that if cp is a positive linear normal functional on an Abelian W*-algebra M, then there exists an xE7' such that cp - For any C in MEcp, let RC denote. right) multiplication operator on MEcp defined by RC (A) = AC for A E MEcp. In later sections we will show that if M is Abelian and the normal positive linear functional cp satisfies cp-< w, then cp = WT T. More precisely, in [17] a B*-algebra Mis is called a W*-algebra if there exists a Banach space F such that M = Fifr. If Fis is canonically embedded as a norm - closed subspace of M*, then it can be shown that F is generated by the set of normal positive linear functions on M. It can further be shown that if M is a W' *-algebra in the above sense then M can be faithfully represented as a weak closure :f:-subalgebra of !('4C') for a Hilbert space 'ii and that in such a representation the topology cr(M, F ) is equivalent to the weak operator topology on bounded spheres. 0 we will denote the set of all linear densely defined closed transformations T that suffice TU 2 UT for every unitary operator U in M. Tl':_e proof is elementary, since it uses only those properties that are consequences of the structure of the Riesz space ReM. It will be shown in later sections that if algebraic operations are properly defined, then ReM. 0 is itself a Dedekind complete Riesz space, which is also the universal termination of the Dedekind complete Riesz space ReM. The next lemma is a substitution of the spectral theorem for the unbound self-adjoint transformations of M. 3: Let H be a positive self-adjoint transformation of M 0. There exists a sequence {F } of projections in M with the following. ii) The restriction of H to F 'JI is bounded and belongs to M. Proof of the consequence: Immediate. We only state the result, the details of the proof are exactly as in [12], p. 6: Any closed linear tightly defined transformation T in M. 0 can be represented in one and only one way in the form T = VH where H is a positive self-adjoint transformation in M. 0 and V is a partial isometry in M. 8: Let!Jt be an essentially dense linear manifold in"·. We now have available the following results from Von Neumann and Murray regarding the algebraic properties of M. 0, without resorting to the general form of the spectral theorem. real extensions do not exist in M. 0, then A+B, AB have unique extensions on elements of M. Denote these extensions by [A+B], [AB] respectively. In this section it will be shown that the natural ordering in ReM can be extended to a partial ordering of ReM. i is therefore a Riesz isomorphism, and in the following we do not distinguish between ReM and i(ReM). 0• It is clear from the construction of ReM 0 that ReM is order dense in ReM. 3 gives the existence of a sequence of projections P E M such that P j I such that SP EM for each n. It is clear that any upper bound of the system [A'TB'T 1] is an upper bound of the system [A'TB'T. R(B), it suffices to note that the closure of the graph in V x '}{ limits B to '1. The restriction of 'f to (ReM)+ defines a positive normal linear functional on ReM. subsets of the index set ,JI. Proof: An outline of the proof is given, the details follow exactly as in Theorem 10.10. The restriction of 'f to (ReM)+Ei defines a positive faithful normal linear functional on ME., whence. Finally if (3'} denotes the family of all finite subsets of the index set .;, then for each TE(R.eMJ+,. 0 be a normal faithful semi-finite trace of{ReM 0) • Let E be any projection of M such that t. The restriction of t to ReM defines a positive normal linear functional on M if and only if t 0(T) < +oo. Set T =~EJ x Ti and note that TE(ReM . 0f.° Let f:JJ denote the family of finite subgroups of the set of index J) and let S be any element of (ReMJ: Note that lT. In this section we will show that the family of semi-finite traces embedded in X can be endowed with a Riesz space structure. This leads 'immediately to a representation of the elements of ReM 0 as normal integrals defined on a dense ideal of the order of ReM. With ~ we will denote the family of all kinds of dense ideals. Then ~ is the basis of the filter. The set of classes of equivalent elements will be denoted by r(ReM . 0) and its elements by (cp). It is clear that qi is a normal semifinite trace on (ReMJ~) and that the restriction of qi on ( ReMf is a normal semifinite trace on ReM in the usual sense {[lJ p. 79). Let 'f o be a faithful normal semifinite trace on (ReMd~ and let 0 ~ cpEct>, cp be an extension of cp on (ReMJ~) By Theorem 10. Since the restrictions of tT+, cps on ReM are normal semifinite traces on ReM, with Corollary 10 .That I'(ReM), I'(ReM . 0) are isomorphic Riesz spaces immediately follows from the fact that ReM is a series ideal in ReM.
