New York Journal of Mathematics
Volume 28 (2022), 1512-1530


S. Sundar

Representations of the weak Weyl commutation relation

view    print

Published: November 18, 2022.
Keywords: Weak Weyl relations, Semigroups of isometries, Morita equivalence.
Subject [2010]: Primary 46L05 ; Secondary 81S05.

Let G be a locally compact, second countable, Hausdorff abelian group with Pontryagin dual G*. Suppose P is a closed subsemigroup of G containing the identity element 0. We assume that P has dense interior and P generates G. Let U:={Uχ: χ ∈ G*} be a strongly continuous group of unitaries and let V:={Va: a ∈ P} be a strongly continuous semigroup of isometries. We call (U,V) a weak Weyl pair if UχVa=χ(a)VaUχ for every χ ∈ G* and for every a ∈ P.

We work out the representation theory (the factorial and the irreducible representations) of the above commutation relation under the assumption that {VaVa*: a ∈ P} is a commuting family of projections. Not only does this generalise the results of [4] and [5], our proof brings out the Morita equivalence that lies behind the results. For P=R+2, we demonstrate that if we drop the commutativity assumption on the range projections, then the representation theory of the weak Weyl commutation relation becomes very complicated.



Author information

S. Sundar:
Institute of Mathematical Sciences
A CI of Homi Bhabha National Institute
4th cross street, CIT Campus
Taramani, Chennai 600113, India