Groupoids, Inverse Semigroups, and their Operator Algebras by Alan Paterson

By Alan Paterson

In fresh years, it has develop into more and more transparent that there are very important connections referring to 3 ideas -- groupoids, inverse semigroups, and operator algebras. there was loads of growth during this region over the past twenty years, and this booklet provides a cautious, up to date and fairly large account of the subject material.

After an introductory first bankruptcy, the second one bankruptcy offers a self-contained account of inverse semigroups, in the community compact and r-discrete groupoids, and Lie groupoids. The part on Lie groupoids in bankruptcy 2 incorporates a specific dialogue of groupoids really very important in noncommutative geometry, together with the holonomy groupoids of a foliated manifold and the tangent groupoid of a manifold. The illustration theories of in the community compact and r-discrete groupoids are constructed within the 3rd bankruptcy, and it really is proven that the C*-algebras of r-discrete groupoids are the covariance C*-algebras for inverse semigroup activities on in the community compact Hausdorff areas. a last bankruptcy affiliates a common r-discrete groupoid with any inverse semigroup. Six next appendices deal with issues relating to these lined within the textual content.

The ebook should still attract a large choice mathematicians and graduate scholars in fields comparable to operator algebras, research on groupoids, semigroup thought, and noncommutative geometry. it is going to even be of curiosity to mathematicians drawn to tilings and theoretical physicists whose concentration is modeling quasicrystals with tilings. An attempt has been made to make the ebook lucid and 'user friendly"; hence it's going to be obtainable to any reader with a uncomplicated history in degree thought and practical analysis.

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1 The set E separates the elements of £1 (E). Proof. Suppose that the preceding statement is false. Then there exists non-zero 9 E £l(E) such that e(g) = 0 for all e E E. We will derive a contradiction. We can suppose that 9 is of the form 2:::1 are r where the er's are distinct, {a r } E £1 and a1 =I- O. We first show 1 that o:(g) = 0 for all 0: E X. Indeed, let 0: E X and F = {e E E : o:(e) = I}. Then F is a subsemilattice of E which is a net under the semilattice ordering. By taking products of e's in F, we can obtain an element e with e E F coinciding with 0: on any preassigned finite number of elements of E.

N} on which Sn acted as well, giving a localization (Z, Sn) which in turn gave the Cuntz groupoid Cn. For S in general, E(S) is not concretely represented by strings. What we do instead, however, is look at the space X = X(S) of non-zero semicharacters on E(S) which, either in the pointwise topology or equivalently, as the space of non-zero multiplicative linear functionals on the commutative Banach algebra £l(E(S)), is a locally compact Hausdorff space. 2 to obtain an r-discrete groupoid. Unfortunately, the pair (X, S) is not usually a localization since the domains of the partial homeomorphisms associated with S need not form a basis for X.

Is dense in 'H.. (The latter requirement is equivalent to the condition: if ~ E 11. ) Given such a representation 11", the operator 1I"(e) is a projection if e E E(S). Note that if e :::; I in E(S), then el = e, and applying 11" to this equality gives 1I"(e) :::; 11"(1), so that 11" is increasing on E(S). Further, for each s E S, the operator 1I"(s) is a partial isometry on £2 (S) since 11"( s)11" (s*) = 11"( ss*) is a projection. It follows that 11" takes S into an inverse semigroup of partial isometries on a Hilbert space (or equivalently, into a *-semigroup of partial isometries on a Hilbert space).

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