## 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}

## Keywords

Inverse Semigroup Partial Isometry Compact Hausdorff Space Inverse Subsemigroup Cuntz Algebra
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## Notes

- 1.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*.Google Scholar - 2.In the book, bold face (as in
**4.5**) will be used for section references to distinguish them from equation references.Google Scholar - 3.An account of noncommutative geometry with particular relevance to physical theories is given in the book by Landi ([151]).Google Scholar
- 4.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.Google Scholar - 5.A discussion of Ehresmann’s work on ordered groupoids is given by Mark Lawson in his book [157].Google Scholar

## Copyright information

© Springer Science+Business Media New York 1999