# Direct Integral Decomposition

Let $\{H_x\}_{x\in X}$ be a collection of Hilbert spaces such that $(X,\Sigma,\mu)$ is a measure space.  Now define

$\displaystyle H=\left\{s\in\bigoplus_{x\in X} H_x:\int_X |s(x)|^2\,d\mu<\infty\right\}$

where $s(x)=s_x$ (the $x$th component of $s$).  Then

$\displaystyle\langle s,t\rangle=\int_X\langle s_x,t_x\rangle_x\,d\mu$

defines a pre inner product on $H$.  Now let $H_0=span\{s-t:s=t\,a.e.\}$.  Then $H/H_0$ is an inner product space, and its completion is a Hilbert space called the direct integral of $\{H_x\}_{x\in X}$.  We denote the direct integral by

$\displaystyle H_X^\oplus=\int_X^\oplus H_x\,d\mu.$

Now suppose $\{T_x\}_{x\in X}$ is a collection of linear operators where $T_x\in L(H_x)$ such that $\{\|T_x\|\}$ is uniformly bounded.  Then there is an operator $T\in L(H_X^\oplus)$ where $T(s)_x=T_x(s_x)$ and where we can define

$\displaystyle \|T\|={\mbox{ess}\sup}_{x\in X}\{\|T_x\|\}.$

This gives a representation $\rho:L^\infty(X,\mu)\to L(H_X^\oplus)$ defined by

$\left(\rho(f)(s)\right)_x=f(x)s_x,$

which is essentially just a component-wise left action of $f$ on $s$$\rho(L^\infty(X,\mu))$ is called the algebra of diagonalizable operators of $H_X^\oplus$, which we will denote $D(H_X^\oplus)$.

Theorem 1.  Let $\rho:A\to L(H)$ be a representation of von Neumann algebra $A$ on a separable Hilbert space $H$ such that $B$ is a von Neumann subalgebra of $A'$.  Then there exists a measure space $(X,\Sigma,\mu)$, a collection of Hilbert spaces $\{H_x\}_{x\in X}$, and a  unitary map $U:H\to H_X^\oplus$ such that $U\left(\rho(B)(s)\right)=D(H_X^\oplus)(s')$ for all $s\in H$ and corresponding $s'\in H_X^\oplus$ and

$\displaystyle UTU^*=\int_X^\oplus T_x\,d\mu\in L(H_X^\oplus)$

for all $T\in\rho(B')$.

Thus if we let $B=A'$ from above, then $L(H_x)$ is a factor and we also write

$\displaystyle\rho(A')=\int_X^\oplus\rho(A')_x\,d\mu=\int_X^\oplus\rho(A_x)'\,d\mu.$

This is called the central decomposition of $A$.  This also gives a representation of $A$ on $H_X^\oplus$ defined by $a\mapsto U\rho(a)U^*$.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Springer-Verlag.  2006.