 

S. Sundar
Representations of the weak Weyl commutation relation
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Published: 
November 18, 2022. 
Keywords: 
Weak Weyl relations, Semigroups of isometries, Morita equivalence. 
Subject [2010]: 
Primary 46L05 ; Secondary 81S05. 


Abstract
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:={U_{χ}: χ ∈ G^{*}} be a strongly continuous group of unitaries
and let V:={V_{a}: a ∈ P} be a strongly continuous semigroup of isometries. We call (U,V) a weak Weyl
pair if U_{χ}V_{a}=χ(a)V_{a}U_{χ}
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 {V_{a}V_{a}^{*}: 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.


Acknowledgements
N/A


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

