The Riesz space structure of an Abelian W*-algebra
<p>Let M be an Abelian W*-algebra of operators on a Hilbert space <i>H</i>. Let M<sub>0</sub> be the set of all linear, closed, densely defined transformations in <i>H</i> which commute with every unitary operator in the commutant M’ of M. A well known resul...
| Summary: | <p>Let M be an Abelian W*-algebra of operators on a Hilbert space <i>H</i>. Let M<sub>0</sub> be the set of all linear, closed, densely defined transformations in <i>H</i> which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in <i>H</i>, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M<sub>0</sub>, and an elementary proof is given of the fact that a positive self-adjoint transformation in M<sub>0</sub> has a unique positive square root in M<sub>0</sub>. It is then shown that when the algebraic operations are suitably defined, then M<sub>0</sub> becomes a commutative algebra. If ReM<sub>0</sub> denotes the set of all self-adjoint elements of M<sub>0</sub>, then it is proved that ReM<sub>0</sub> is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM<sub>0</sub> which characterizes the normal integrals on the order dense ideals of ReM<sub>0</sub>. It is then shown that ReM<sub>0</sub> may be identified with the extended order dual of ReM, and that ReM<sub>0</sub> is perfect in the extended sense. </p>
<p>Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.</p>
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