Von Neumann algebras and Boolean valued analysis
Gaisi Takeuti · Journal of the Mathematical Society of Japan · 1983
The aim of this paper is to apply Boolean valued analysis developed in [5],[6], [7], [8] to von Neumann algebras.Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{Z}$ its center, and $\mathcal{B}$ the complete Boolean algebra of all projections in Z. Then $\mathcal{B}$ -valued analysis gives a machinery to transfer theorems on factors into theo- rems on von Neumann algebras whose centers are $\mathcal{Z}$ .Our machinery is a kind of a dictionary which gives a translation of the notions on factors into the notions on von Neumann algebras with the center $\mathcal{Z}$ .E. $g.$ , the translations of type I, type $II_{1}$ , type $II_{\infty}$ and type III are exactly type I, type $II_{1}$ , type $II_{\infty}$ and type III themselves.However, the translations of trace and weight become generalized center valued trace and generalized center valued weight respectively.This machinery immediately reduces many properties of center valued traces or weights to properties of traces or weights on factors.In \S 5, we state direct translations of Murray and von Neumann's theorem on factors of type $II_{\infty}$ and Takesaki's theorem on factors of type III in order to show how our machin- ery works.Our theory is closely related to von Neumann's reduction theory.We prove that every von Neumann algebra is a factor in a Boolean valued sense while the reduction theory proves that every von Neumann algebra with a countability condition is a direct integral of factors.Thus a systematic interpretation of Boolean valued notion provides us with a machinery of automatic translations of notions on factors into notions on von Neumann algebras with the center $\mathcal{Z}$ , while the reduction theory reduces many problems on von Neumann algebras to problems on factors.It is not difficult to eliminate $\mathcal{B}$ -valued analysis in our machinery.In order to do so, we have to introduce the following S-valued Hilbert space, where $\overline{\mathcal{Z}}$ is the unbounded extension of $\mathcal{Z}$ .If we express $\mathcal{Z}$ as $L^{\infty}(\Omega, \mu)$ , then $\overline{\mathcal{Z}}$ is the set of all measurable functions.$\hat{\mathcal{H}}$ is called a $\overline{\mathcal{Z}}$ -valued Hilbert space if $\hat{\mathcal{H}}$ is a $\overline{\mathcal{Z}}-$ module with the inner product satisfying the following properties: 1) $\zeta,$ $\eta\in\hat{\mathcal{H}}\Rightarrow(\zeta|\eta)\in\overline{\mathcal{Z}}$2) Let $\zeta,$ $\eta,$ $\xi\in\hat{\mathcal{H}}$ and $f,$ $g\in\overline{\mathcal{Z}}$ .Then the following hold.2.1) $(f\zeta|\eta)=f\cdot(\zeta|\eta)$ $a$ .$e$ .G. TAKEUTI 2.2) $(\zeta|\eta)=\overline{(\eta|\zeta)}a.e$ .2.3) $(f\zeta+g\eta|\xi)=f\cdot(\zeta|\xi)+g\cdot(\eta|\xi)$ $a.e$ .3) $(\zeta|\zeta)\geqq 0$ $a.e.$ , and $\Vert\zeta\Vert=0$ $a.e.$ , iff $\zeta=0$ , where $\Vert\zeta\Vert=\sqrt{(\zeta|\zeta)}$ .4) $\hat{\mathcal{H}}$ is complete in the following sense.If $j arrow\infty\lim_{iarrow\infty}\Vert\zeta_{i}-\zeta_{j}\Vert=0a.e.$ , then there exists $\zeta\in\hat{\mathcal{H}}$ such that $\lim_{iarrow\infty}\Vert\zeta_{i}-\zeta\Vert=0a$ .$e$ .However, it is our belief that $\mathcal{B}$ -valued analysis makes our machinery much simpler.Besides, there is a possibility that some problem on von Neumann algebras can be proved independent from set theory by our method, though we have not found any good candidate of this sort of problems.Let us summarize the basic facts on Boolean valued universe of sets here- with.Let $\mathcal{B}$ be a complete Boolean algebra.Then $V^{(\mathcal{B})}$ the universe of $\mathcal{B}$ -valued sets satisfies the following properties.1. Let $u$ and $v$ be members of $V^{(\mathcal{B})}$ .Then $u\in v$ or $u=v$ has a truth value in $\mathcal{B}$ .If the truth value of $u\in v$ or $u=v$ is $b\in \mathcal{B}$ , then we write $\ovalbox{\tt\small REJECT} u\in v\ovalbox{\tt\small REJECT}=b$ or [ $u=v\ovalbox{\tt\small REJECT}=b$ respectively.We can assign truth values [ $\varphi\ovalbox{\tt\small REJECT}$ in $\mathcal{B}$ to all set theoretic statement $\varphi$ .If [ $\varphi\ovalbox{\tt\small REJECT}=1$ , then we say $\varphi$ is true' or $\varphi$ holds'.If [ $\varphi\ovalbox{\tt\small REJECT}=0$ , then we say $\varphi$ is false' or $\varphi$ does not hold'.If [ $\varphi J=b$ and $0<b<1$ , then $\varphi$ takes an intermediate truth value and we say $\varphi$ holds as much as $b'$ .2. $V^{(\mathcal{B})}$ is a model of ZFC.Let $\varphi$ be a theorem in modern mathematics i.e., a theorem in ZFC.Then [$\varphi I=1$ is also a theorem in ZFC i.e., [ $\varphi\ovalbox{\tt\small REJECT}=1$ is a different theorem in modern mathematics.Therefore, this procedure provides us with a machinery to produce a new theorem from an old theorem $\varphi$ .3. Since $V^{(\mathcal{B})}$ satisfies ZFC, we can construct real numbers in $V^{(\mathcal{B})}$ by Dedekind's cuts.