The author uses set-theoretic methods to study the module structure of Von Neumann algebras by replacing the field of complex numbers by a commutative \(AW^*\)-algebras Z. A normed Z-module X is a Z-module with norm \(\| \cdot \|\) which satisfies \(\| ax\| \leq \| a\|_{\infty}\| x\|\) for all a in Z and x in X. Denote the space of all bounded Z-linear maps from the normed Z-module X to the normed Z-module Y by \(Hom_ Z(X,Y)\). Let \(X^{\#}\) be \(Hom_ Z(X,Z)\), the Z-dual of X. A \(C^*\)-algebra which contains Z as a unital \(sub\)- C\({}^*\)-algebra of the centre Z(A) of A is said to be
Z-dual if there is a normed Z-module X such that A is isometrically Z- isomorphic to \(X^{\#},\)
Z-bi-dual if there is a normed Z-module X such that A is isometrically Z- isomorphic to \(X^{\#\#}\), and
Z-self-dual if A is isometrically Z-isomorphic to \(A^{\#}.\)
A \(C^*\)-algebra B is said to be Z-embeddable if there exists a Type I \(AW^*\)-algebra L with centre Z and a *-isomorphism \(\pi\) from B to L such that \(\pi\) (B) coincides with its double commutant \(\pi (B)''.\)
The first three main applications of the author’s theory are:
(i) A is Z-dual if and only if A is Z-embeddable;
(ii) If Z coincides with the centre of A then A is Z-bi-dual if and only if it is a Type I \(AW^*\)-algebra;
(iii) If Z coincides with the centre of A then A is Z-self-dual if and only if it is a finite Type I \(AW^*\)-algebra.
Let X be a Banach Z-module. Then a function \(\| \cdot \|_ Z: X\to Z\) is called a Z-valued norm on X if: \[ \| x+y\|_ Z\leq \| x\|_ Z+\| y\|_ Z,\quad \| ax\|_ Z=| a| \| x\|_ Z,\quad \| x\|_ Z\geq 0,\quad \| x\|_ Z=0\quad if\text{ and } only\quad if\quad x=0. \] The Z-valued norm defines an induced norm \(\| \cdot \|\) on X by \(\| x\| =\| \| x\|_ Z\|_{\infty}.\) Clearly X is a Banach Z-module with its induced norm. Such an X is said to be a Kaplansky-Banach Z-module if and only if X is a Banach space with respect to its induced norm and, for each partition of unity \((b_ j)\) in Z and bounded family \((x_ j)\) in X there exists a (unique) x in X such that, for all j, \(b_ jx=b_ jx_ j\). The last main result is the following:
(iv) Let A be a Z-embeddable \(C^*\)-algebra and let \(A_{\#}\) be a Banach Z-module such that \((A_{\#})^{\#}\) is isometrically z- isomorphic to A. Then \(A_{\#}\) is a Kaplansky-Banach Z-module and if A is the Z-dual of another Kaplansky-Banach Z-module X then X is isometrically Z-isomorphic to \(A_{\#}.\)
This is a generalization of Sakai’s characterization of \(W^*\)- algebras.