Chennai Mathematical Institute


2.00 pm,
E_0-semigroups over closed convex cones

Anbu Arjunan
Chennai Mathematical Institute.


The thesis focuses on the study of multi-parameter E_0-semigroups. The one parameter theory of E_0-semigroups were initiated by R.T. Powers and developed extensively by Arveson. Let P be a closed convex cone in R^{d}. Let H be a Hilbert space and let B(H) be the algebra of all bounded operators of H. An E_0-semigroup over P on B(H) is a $\sigma$-weakly continuous semigroup of unital normal *-endomorphisms of B(H).

There are significant differences between the one parameter theory and the multiparameter theory. The first difference is that In the one-parameter case there is only one automorphism E_0-semigroup. In contrast to this, in the multiparameter case, automorphism groups are classified by strictly upper triangular matrices.

The class of CCR flows considered in this thesis is given below.

Let A be a closed subset of $\mathbb{R}^d$ which is invariant by under P and we call such subsets as P-modules. Let K be a Hilbert space of dimension k with $k\in\mathbb{N}$. We define an isometric representation V of P associated to a P-module A of multiplicity k. Similar to the one-parameter case we construct the CCR flow $\alpha$ associated to the isometric representation V, called the CCR flow associated to a P-module A of multiplicity k.

In the one-parameter case it is known that there are only countable many CCR flows. This is based on the existence of enough units. But in the multiparameter CCR flows, there is only one unit upto a scalar multiple. We use another invariant called the gauge group to study the CCR flows described above.

We prove the following theorem in this thesis.


Let A_1, A_2 be P-modules and let K_1, K_2 be two Hilbert spaces of dimensions k_1, k_2. Let $\alpha^{(A_i,K_i)}$ be the CCR flow associated to the module A_i with multiplicity k_i. Then the following statements are equivalent.

1) The CCR flows $\alpha^{(A_1,K_1)}$ and $\alpha^{(A_2,K_2)}$ are cocycle conjugate.

2) A_1 and A_2 are translates of each other and k_1=k_2.

We demonstrate the idea of the proof of the above theorem for a specific collection of modules when P=R_{+}^{2}.