Notes on Product Systems

We summarize the basic properties of continuous tensor product systems of Hilbert spaces and their role in non-commutative dynamics. These are lecture notes, not intended for publication. 1. Concrete Product Systems In these notes we assume the reader is familiar with the definition and basic properties of E0-semigroups, and has some familiarity with their role in noncommutative dynamics. Our purpose here is to discuss product systems and their significance in the classification problem. Let α = {αt : t ≥ 0} be an E0-semigroup acting on B(H), where as always, H denotes a separable Hilbert space. The product system of α gives rise to a classifying structure for cocycle conjugacy, and is defined as follows. For every t > 0 let E(t) be the following linear space of operators E(t) = {T ∈ B(H) : αt(X)T = TX, X ∈ B(H)}. The first thing to notice is that there is a natural inner product 〈·, ·〉 on E(t) that makes it into a Hilbert space. Indeed, if S, T ∈ E(t), then one finds that for every X ∈ B(H), one has T ∗SX = T αt(X)S = (αt(X)T )∗X = (TX∗)∗S = XT ∗S, so that T ∗S must be a scalar multiple of the identity operator. Thus we can define an inner product 〈S, T 〉 by T ∗S = 〈S, T 〉1. This inner product makes the operator space E(t) into a Hilbert space with the property that the operator norm agrees with the Hilbert space norm, as one sees from T ∗T = 〈T, T 〉1, using the fact that the operator norm satisfies ‖T ∗T‖ = ‖T‖2. The second property of these inner products is also a straightforward consequence of the definition of the various spaces E(t), namely the following. For every s, t > 0, E(s)E(t) ⊆ E(s + t); moreover, for all S1, S2 ∈ E(s) and T1, T2 ∈ E(t), one has 〈S1T1, S2T2〉 = 〈S1, S2〉 · 〈T1, T2〉. Date: 7 February, 2006.