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Borel groupoid (Definition)

Definitions

  • Borel function
    Definition 0.1   A function $f_B: (X; \mathcal{B}) \to (X; \mathcal{C}$) of Borel spaces is defined to be a Borel function if the inverse image of every Borel set under $f_B ^{-1}$ is also a Borel set.
  • Borel groupoid
    Definition 0.2   Let ${\mathbb{G}}$ be a groupoid and ${\mathbb{G}}^{(2)}$ a subset of ${\mathbb{G}}\times {\mathbb{G}}$– the set of its composable pairs. A Borel groupoid is defined as a groupoid ${\mathbb{G}}_B$ such that ${\mathbb{G}}_B^{(2)}$ is a Borel set in the product structure on ${\mathbb{G}}_B \times {\mathbb{G}}_B$, and also such that the functions $(x,y) \mapsto xy$ from ${\mathbb{G}}_B^{(2)}$ to ${\mathbb{G}}_B$, and $x \mapsto x^{-1}$ from ${\mathbb{G}}_B$ to ${\mathbb{G}}_B$ are all (measurable) Borel functions (ref. [1]).

Analytic Borel space

${\mathbb{G}}_B$ becomes an analytic groupoid if its Borel structure is analytic.

A Borel space $(X; \mathcal{B})$ is called analytic if it is countably separated, and also if it is the image of a Borel function from a standard Borel space.

Bibliography

1
M.R. Buneci. 2006., Groupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1, p.75 .



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Cross-references: standard Borel space, Borel space, groupoid, Borel set, function, Borel function
There are 4 references to this object.

This is version 2 of Borel groupoid, born on 2009-02-04, modified 2010-04-28.
Object id is 486, canonical name is BorelGroupoid.
Accessed 490 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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