• Dr. Arun Dayal

In this paper we extend some properties of probability measures, in its non-commutative frame work under the assumption of convolution semigroups and the associated semigoup of operators on the L2 space of normalised Haar measure, when G is the real line. Using Peter-Weyl theory, we extend some properties of pseudo differential operators on compact groups. We show that the Hunt semigroup and its generator are pseudo differential operators in the sense of Ruzhansky and Turunen. We show here that the generator has the same Sobolev regularity as the Laplacian. We derive the transition kernel for convolution semigroups of cenetral measures. We derive a necessary and sufficient condition for the semigroup to be trace-class for any positive time which is equivalent to that the corresponding probability measure has a squareintegrable density.

Convolution semigroups, Central measure, Symmetric measure, Gaussian measure, Fourier transform, Brownian motion.

"NON - COMMUTATIVE REPRESENTATION OF SEMI GROUP MEASURES", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.4, Issue 9, page no.665-670, September-2017, Available :http://www.jetir.org/papers/JETIR1709096.pdf

"NON - COMMUTATIVE REPRESENTATION OF SEMI GROUP MEASURES", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.4, Issue 9, page no. pp665-670, September-2017, Available at : http://www.jetir.org/papers/JETIR1709096.pdf