representation of locally compact groupoids

(Definition)

Definition 0.1. Let G_lc be a locally compact topological groupoid endowed with a Haar system

ν = {νu : u ∈ UGlc}.

A representation of G_lc together with its associated Haar system ν is defined as a triple

(μ, UGlc ∗ ℋ, L),

where:

μ is a quasi-invariant measure defined on U_{G_lc};
U_{G_lc} ∗ℋ is an analytical fibered Hilbert space, or Hilbert bundle, over U_{G_lc}; and
$L : UGlc − → Iso(UGlc ∗ ℋ )$
is a Borel groupoid morphism whose restriction to U_{G_lc} is the identification map. Thus the unit space
$UIso(UG ∗ℋ) lc$
is identified via L with U_{G_lc}.

For x ∈ G_lc, write

L(x) = [r(x), ^L(x),d(x )],

where

^L (x ) : ℋ(d (x )) −→ ℋ (r(x))

is a Hilbert-space isomorphism.

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Pending Errata and Addenda