Abstract
In this section we give a brief and largely self-contained account of the results on inverse semigroups that will be required in the sequel. The results we need from the algebraic theory of semigroups are well-known, and are contained in the standard textbooks (such as [50, 51, 133, 202]). However, for the convenience of the reader and for the purpose of establishing notation, an account concentrating on what we will need later from that theory is desirable.
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Notes
This also follows directly from Proposition 4.3.1 later.
This is the terminology used by Muhly in [179]. When X is an arbitrary set, Vaisman ([263, p. 138]) calls the trivial groupoid X×X the banal groupoid. Weinstein([273]) comments that it is also called the coarse groupoid.
More precisely, we have f(xt)g(t−1) = h(xt) where h(t) = f(t)g(t−1x). Note that the pointwise product h may not belong to Cc(G) but it will belong to Bc(G), and we can use the Bc(G)-version of (iii) of Definition 2.2.2 with h in place of f.
Renault ([230, p.50]) requires representations of Cc(G) to be continuous in the inductive limit topology on Cc(G) rather than the I-norm but the fact that we can use the I-norm here is a simple folk-lore result that uses the appropriate version of [230, Ch.2, Proposition 1.4].
Renault’s definition of locally compact groupoid does not require the existence of a left Haar system. In Definition 2.2.3, we also do not need to assume a priori the existence of a left Haar system, such systems coming automatically from Proposition 2.2.5 below.
The (useful) example of Appendix C gives a counterexample.
In fact we only need to require that G0 is a Hausdorff submanifold and that every Gu is Hausdorff.
This is observed by Brylinkski and Nistor in [40, p.342].
The terminology smooth groupoid is sometimes used in place of Lie groupoids. (See, for example, the book [56, p.101] of Connes.) The terminology Lie groupoids was used in the important paper of Coste, Dazord and Weinstein ([63]), and seems to becoming established — see, for example, the paper of Weinstein ([274, p.15]) and the book of Vaisman ([263, p.140]). Indeed, it is natural to use the terminology Lie groupoids to refer to the groupoid version of Lie groups. References for the definition of a Lie groupoid in the Hausdorff case are [84, 85, 86, 208, 209, 210, 211, 164]. Kirill Mackenzie in his book ([168, p.84]) refers to these groupoids as differentiable groupoids. He uses the expression Lie groupoids to refer to a special class of what is called in the present book Lie groupoids. In [63, 263], the authors do not assume a Lie groupoid to be Hausdorff and, as in Definition 2.3.1 below, require G0 to be Hausdorff. They do not require the Gu to be Hausdorff. We do require that condition as well in order to ensure that every Lie groupoid is a locally compact groupoid (Corollary 2.3.1), thus making available for Lie groupoids the representation theory for locally compact groupoids.
Of course, this definition also applies for a general (non-Hausdorff) manifold.
Another natural example of a Lie algebroid is that of the cotangent bundle T* M of a Poisson manifold. For a discussion of the Lie algebroids arising in the study of Poisson geometry, the reader is referred to [274]. The Lie algebroid A(G) is also important in the study of index theory for elliptic pseudodifferential operators on a Lie groupoid G, and is used in the construction of a natural asymptotic morphism associated with G ([191, 199, 198]).
In fact, Aof and Brown ([7]), developing ideas of Pradines ([208]), have shown that there is a version of the holonomy groupoid for any topological groupoid.
The Cuntz groupoid, described in Chapter 1, is similar in form to the holonomy groupoid, and can be regarded as a 0-dimensional analogue of the holonomy groupoid.
I am grateful to Alan Weinstein for helpful information about the history of pseudogroups and papers concerning them.
See the first chapter of the book by Mark Lawson ([157]) for a recent discussion of pseudogroups. I am grateful to Dr. Lawson for allowing me to consult this part of his book.
The recognition of the importance of the pseudogroup concept is also implicit in the claim by J. H. C. Whitehead ([229, p.9]) that the work on pseudogroups may be Elie Cartan’s best.
In [56], R* is replaced by (0,1]. This is convenient for deformations. On the other hand, it gives a Lie groupoid with boundary rather than simply a Lie groupoid.
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© 1999 Springer Science+Business Media New York
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Paterson, A.L.T. (1999). Inverse Semigroups and Locally Compact Groupoids. In: Groupoids, Inverse Semigroups, and their Operator Algebras. Progress in Mathematics, vol 170. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1774-9_2
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