A universal property for groupoid C*-algebras. II. Fell bundles

Abstract

We define possibly unsaturated, upper semicontinuous Fell bundles over Hausdorff, locally compact groupoids and establish a universal property for representations of their full section C*-algebras on Hilbert modules over arbitrary C*-algebras. Based on this, we prove that the full section C*-algebra is functorial and exact, and we define a quasi-orbit space and a quasi-orbit map. We deduce and extend Renault's Integration and Disintegration Theorems to general Fell bundles using our universal property.

Figures (5)

• Figure 1: A pair $(g,h)\in G^2$ of composable arrows generates a commutative triangle of arrows in $G$. We number the edges so that the one opposite the vertex $v_i(g,h)$ is $d_i(g,h)$ for $i=0,1,2$.
• Figure 2: Two parallel isomorphisms of correspondences built from $U$. Each triangle or quadrilateral means an isomorphism of $\mathrm C^*$-correspondences. The three quadrilaterals containing $U$ are copies of \ref{['eq:U_diagram']}. The triangles without label mean the canonical isomorphisms of $\mathrm C^*$-correspondences in \ref{['eq:compose_measure_family']}. The triangle marked $\mu$ also involves the multiplication map $\mu\colon d_2^*\mathcal{A} \otimes_{v_1^*\mathsf{A}} d_0^*\mathcal{A} \hookrightarrow d_1^*\mathcal{A}$ that acts on fibres as in \ref{['eq:multiplication_isometries']}.
• Figure 3: The unitary $d_1^*(U)$ as the composite of $\mathrm{id} \otimes U$ with two canonical isomorphisms $\gamma_1$ and $\gamma_2$ from Lemma \ref{['lem:compose_corr_from_measure-family']}.
• Figure 4: The first column defines a unitary $U'_h(g)$ for all $(g,h)\in G^2$ with $gh,g\in X_1$. Here the unlabelled unitaries are built from the isomorphisms in Lemma \ref{['lem:equality_of_products']} and the equality $I\cdot J = J\cdot I$ for two ideals in a $\mathrm C^*$-algebra. The two horizontal isometries are induced by the multiplication in the Fell bundle. The unitary $\bar{U}_h$ is required to make the whole diagram commute for $\tilde{\alpha}_{r(h)}$-almost all $g \in G_{r(h)}$. This condition uniquely determines $\bar{U}_h$ whenever it exists.
• Figure 5: The gist of the proof of Lemma \ref{['lem:doubleprime']}. The data is a chain of three composable arrows $x\xrightarrow{k} y\xrightarrow{h} z\xrightarrow{g} w$. The unlabelled arrows in the diagram are induced by the multiplication in the Fell bundle, tensored with the identity on the appropriate fibre of $\mathcal{H}$. When marked with "$\cong$", then they are invertible by a repeated application of Lemma \ref{['lem:equality_of_products']}. Otherwise, they are only isometries. The arrows marked with $U_l$ or $U_l^*$ for an arrow $l$ in $G$ act by a tensor product of an identity map with the arrow $U_l$ or $U_l^*$ acting on the last two or three tensor factors.

Theorems & Definitions (232)

• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5
• Lemma 2.6
• proof