Chennai Mathematical Institute


2.00 pm,
E_0-semigroups and Product Systems

S.P. Murgan
Chennai Mathematical Institute.


An E_0-semigroup over [0,\infty) is a semigroup $\{\alpha_t\}_{t\geq 0}$ of unital, normal *-endomorphisms of the algebra B(H) of bounded operators on a Hilbert space H such that for $A \in B(H)$ and $\xi, \eta \in H$, the map $[0,\infty) \ni t \to <\alpha_t(A)\xi, \eta> \in \mathbb{C}$ is continuous. Arveson associates with every E_0-semigroup over R_+ a product system and motivated by this observation, he introduced the notion of abstract product systems. Arveson established a one-to-one correspondence between isomorphism classes of product systems of Hilbert spaces and cocycle conjugacy classes of E_0-semigroups on B(H).

In this thesis, we have generalized this theory to E_0-semigroups over closed convex cone P in R^d, where $d\geq 2.$ In the multi-parameter definition of E_0-semigroups, we replace R_+ by a closed convex cone in R^d.

Let $\alpha =\{\alpha_x\}_{x\in P}$ be an E_0-semigroup over P. Denote the interior of P by $\Omega$. For $x \in \Omega$, the intertwining space for $\alpha_{x}$ is the set

E(x):={T \in B(H): \alpha_x(A)T=TA for all A \in B(H)}.

Then $\{E(x)}_{x\in\Omega}$ forms a measurable field of Hilbert spaces which satisfies the following conditions:

1) The set E(x) is a separable Hilbert space.

2) for $x,y\in\Omega$, the map $E(x) \otimes E(y) \ni S \otimes T \to ST \in E(x+y)$ is a unitary operator.

3) There exists a sequence of measurable maps V_n: \Omega --> B(H) such that for every $x \in \Omega,$ the set {V_n(x)} is an orthonormal basis for E(x).

Any measurable field of Hilbert spaces which possess the above three conditions is called a product system over $\Omega.$ In this thesis, the following theorem is proved:


Product systems over $\Omega$ are in bijective correspondence with E_0-semigroups (up to cocycle conjugacy) over P.