Convolution algebras for relational groupoids and reduction (original) (raw)

Groupoid Algebras for Transitive Groupoids

2004

For a measure groupoid (G, C) [3, Definition 2.3/ p.6]) and a Haar measure (ν, μ) with ν ∈ C ([3, Definition 3.11/p. 18]) P. Hahn has introduced two groupoid algebras, I (G, ν, μ) and II (G, ν, μ) ([4, Theorem 1.8/p. 42]). If (ν1, μ) and (ν2, μ) are Haar measures with ν1, ν2 ∈ C then I (G, ν1, μ) and I (G, ν2, μ) are isomorphic Banach *-algebras ([4, Remark 1.9/p. 42]). If (ν1, μ1) and (ν2, μ2) are Haar measures such that ν1 and ν2 are not equivalent measures, then the relation between I (G, ν1, μ1) and I (G, ν2, μ2) are not known. In the transitive case we shall introduce a groupoid algebra reflecting exactly the change of the Haar measure, in the sense that the algebras corresponding to the Haar measures (ν1, μ1) and (ν2, μ2) are ”naturally” isomorphic exactly when ν1 ̃ν2 and μ1 = μ2. In fact, only the multiplication of this algebra is sensitive to the change of the Haar measure.

Convolution algebras for topological groupoids with locally compact fibres

Opuscula Mathematica, 2011

The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on G, we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the "left invariance" condition in the groupoid sense.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

Journal of Functional Analysis, 1997

For locally compact groups, Fourier algebras and Fourier-Stieltjes algebras have proved to be useful dual objects. They encode the representation theory of the group via the positive de nite functions on the group: positive de nite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra W1, W2]. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a rst step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive de nite functions on G, is a Banach algebra when represented as an algebra of completely bounded maps on a C -algebra associated with G. This necesarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive de nite functions need not be complete. For groups, B(G) is isometric to the Banach space dual of C (G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from a C -algebra associated with G to a C -algebra associated with the equivalence relation induced by G. This paper adds weight to the clues in the earlier study of Fourier-Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored W5].

Inverse systems of groupoids, with applications to groupoid C⁎-algebras

Journal of Functional Analysis

We define what it means for a proper continuous morphism between groupoids to be Haar system preserving, and show that such a morphism induces (via pullback) a *-morphism between the corresponding convolution algebras. We proceed to provide a plethora of examples of Haar system preserving morphisms and discuss connections to noncommutative CW-complexes and interval algebras. We prove that an inverse system of groupoids with Haar system preserving bonding maps has a limit, and that we get a corresponding direct system of groupoid C *-algebras. An explicit construction of an inverse system of groupoids is used to approximate a σ-compact groupoid G by second countable groupoids; if G is equipped with a Haar system and 2-cocycle then so are the approximation groupoids, and the maps in the inverse system are Haar system preserving. As an application of this construction, we show how to easily extend the Maximal Equivalence Theorem of Jean Renault to σ-compact groupoids.

The Fourier-Stieltjes and Fourier algebras for locally compact groupoids

Contemporary Mathematics, 2003

The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of the operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification for this being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J. Renault), (2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A. Paterson). The present paper is expository in character. For motivational reasons, it starts with a description of the theory of B(G), A(G) in the locally compact group case, before discussing these three theories. Some open questions are also raised.

Morphisms of locally compact groupoids endowed with Haar systems

We shall generalize the notion of groupoid morphism given by Zakrzewski ( [19], [20]) to the setting of locally compact σ-compact Hausdorff groupoids endowed with Haar systems. To each groupoid Γ endowed with a Haar system λ we shall associate a C ∗-algebra C ∗ (Γ, λ), and we construct a covariant functor (Γ, λ)→C ∗ (Γ, λ) from the category of locally compact, σ-compact, Hausdorff groupoids endowed with Haar systems to the category of C ∗-algebras (in the sense of [18]). If Γ is second countable and measurewise amenable, then C ∗ (Γ, λ) coincides with the full and the reduced C ∗-algebras associated to Γ and λ.

1-Algebra of a Locally Compact Groupoid

ISRN Algebra, 2011

For a locally compact groupoid with a fixed Haar system and quasi-invariant measure , we introduce the notion of -measurability and construct the space 1(, , ) of absolutely integrable functions on and show that it is a Banach -algebra and a two-sided ideal in the algebra () of complex Radon measures on . We find correspondences between representations of on Hilbert bundles and certain class of nondegenerate representations of 1(, , ).