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Semidirect product of groupoids and associated algebras

Pysiak L., Eckstein M., Heller M., Sasin W.

Demonstratio Mathematica

2014

Vol. 47, nr 2

289--299

One of the pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction, we define the semidirect product of groupoids Γ0 x Γ1 and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid Γ1 and prove that it is isomorphic to the convolutive algebra of the groupoid Γ0 x Γ1. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the Poincaré groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.

Convolution algebras for topological groupoids with locally compact fibres

Buneci M. R.

Opuscula Mathematica

2011

Vol. 31, no. 2

159-172

The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on G, we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the "left invariance" condition in the groupoid sense.