Summary: B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and “slightly colored noises”, using techniques from probability theory. Here we take a purely functional analytic approach and try to obtain a better understanding of Tsireleson’s construction and his examples.
We prove an extension of Shale’s theorem connecting the symplectic group and Weyl representation. We show that the “Shale map” respects compositions (this settles an old conjecture of K. R. Parthasarathy). Using this, we associate a product system to a sum system. This construction includes the exponential product system of Arveson as a trivial case, and the type III examples of Tsirelson.
By associating a von Neumann algebra to every “elementary set” in \([0, 1]\), in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.