Tag Archives: Hilbert space

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 xth 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.

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