Alladi Ramakrishnan Hall

Some Results on Multiparameter Decomposable Product Systems and CAR Flows

Namitha CH

A product system is a measurable field of Hilbert spaces (indexed
over a semigroup), along with a compatible product rule. Product systems were
introduced by William Arveson in his study of E0- semigroups over [0, ∞). He
showed that the product system associated with an E0- semigroup is a complete
invariant, and proceeded to prove seminal results concerning one parameter
E0- semigroups using their product systems. Later, product systems (and E0-
semigroups) over semigroups other than [0, ∞) were studied, especially those
over convex cones. In this talk, will discuss some results on multiparameter product systems
(product systems over closed, convex cones in R
d
). This talk is divided into
two parts. In the first part, we discuss multiparameter decomposable product
systems. Roughly, a product system is said to be decomposable, if it possesses
enough left coherent sections to generate the product system. We will first
describe the structure of a multiparameter decomposable product system. Then,
we will illustrate these structural results in a particular case.

CAR flows associated to isometric representations of cones form basic exam-
ples of E0- semigroups. In the second part of the talk, we characterise isometric
representations of a cone, , with commuting range projections, whose asso-
ciated CAR flow is type I. Moreover, we compute the index and gauge group
(certain invariants associated to a product system/ E0- semigroups) for this
class of CAR flows.

This talk is based on the work with S. Sundar, and is the thesis defense talk.

The Zoom link is given below.

zoom.us/j/98206472628

Meeting ID: 982 0647 2628
Passcode: 560304