#### Alladi Ramakrishnan Hall

#### Banach limits and Dilation theory

#### S Sundar

##### IMSc

*The basic question in dilation theory is to ask whether an irreversible dynamical system (a set with a semigroup action) can be embedded in a reversible dynamical system (where one has group actions). In my talk, the semigroups that I consider will be $N^k$ and $R_{+}^{k}$ and the set is usually a Hilbert space H. In this setting, we ask the following.*

1) Is it possible to dilate an action of $N^k$ on

H by isometries or contractions to an action

(unitary representation) of $Z^k$ on a bigger

Hilbert space K ?

2) What about 1) if we replace the discrete semigroup

$N^k$ by $R_{+}^{k}$?

We will see that Banach limits allow us to conclude that dilation results in continuous time (for $R_{+}^{k}$) come for free once we prove them in discrete time.

This talk will accessible to our graduate students, or to someone who has done a Master's course in functional analysis.

Done