In this section, we review some facts used in the main text concerning positive self-adjoint operators in Hilbert spaces, W-algebras and infinite tensor products. Throughout this section, will denote a complex Hilbert space.

Let A be a positive self-adjoint linear operator on with domain. For each, we put (see [L], 9.9) and for each, we define ; note that. From ([L], 9.9–9.11) we see, using Lebesgue's dominated convergence theorem, that a vector belongs to if and only if

if, then we put, so that the above limit gives for any vector

Recall ([L], 9.20) that the restriction of A to is the analytic generator of the so-continuous unitary representation; the operators can be also regarded as partial isometries acting on the whole of.

Let a be a subset of. A continuous function is said to be operator continuous if the mapping

is continuous with respect to the so-topology and the s-topology.

Theorem (I. Kaplansky, R. V. Kadison). Let be such that and let be a continuous function such that. Then f is operator continuous.

Proof. We first prove the theorem for. Note that the functions and are operator continuous on. Indeed, if are normal operators such that and, then

A similar argument holds for the s-topology.

Denote by the set of all operator continuous functions on and by the subset of bounded functions in. Then is a uniformly closed self-adjoint vector subspace of the *-algebra of all continuous complex functions on and. Hence is a uniformly closed *-subalgebra containing the functions and. By the Stone–Weierstrass theorem, it follows that contains the *-subalgebra of continuous complex functions vanishing at infinity on. Let be a continuous function such that sup. Then belongs to and since it follows that belongs to (actually to), so that belongs to and we conclude that f, as a sum of two functions in, also belongs to.

Consider now the general case. Let be normal operators with Sp(xi) and Sp(x) contained in and. Since, there exists a compact neighborhood of such that. Then, hence is closed in and, by the Tietze– Urysohn extension theorem, f can be extended to a continuous function g on. Let be a continuous function such that supp and.