Abstract
To save unnecessary repetition, throughout this work, unless the contrary is explicitly stated, all inverse semigroups are countable, all locally compact Hausdorff spaces have a countable basis, all Hilbert spaces are separable and all representations of *-algebras on Hilbert spaces are assumed non-degenerate.1
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So for us, a representation of a *-algebra A on a Hilbert space H is a *-homomorphism T from A into B(H) such that the span of the vectors T(a)(ξ), where a ∈ A and ξ ∈ H, is dense in H.
In the book, bold face (as in 4.5) will be used for section references to distinguish them from equation references.
An account of noncommutative geometry with particular relevance to physical theories is given in the book by Landi ([151]).
Often the symbol s is used for the source map. However, like Renault ([230]), we have preferred to use d for this map, reserving s for an inverse semigroup element.
A discussion of Ehresmann’s work on ordered groupoids is given by Mark Lawson in his book [157].
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© 1999 Springer Science+Business Media New York
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Paterson, A.L.T. (1999). Introduction. 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_1
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DOI: https://doi.org/10.1007/978-1-4612-1774-9_1
Publisher Name: Birkhäuser, Boston, MA
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Online ISBN: 978-1-4612-1774-9
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