Fell bundle - Knowlpedia

Let $G$ be a locally compact group. A Fell bundle $\mathcal B=(B_s)_{s\in G}$ is a Banach bundle over $G$ with continuous operations

B_s\times B_t\longrightarrow B_{st},\qquad B_s\longrightarrow B_{s^{-1}},\quad b\longmapsto b^*,

that are associative and involutive, satisfy $\lVert bc\rVert\leq\lVert b\rVert\lVert c\rVert$ , $\lVert b^*b\rVert=\lVert b\rVert^2$ , and make $b^*b$ positive in the $C^*$ -algebra $B_e$ . The bundle axioms also require the usual fiberwise linearity and continuity. Thus each $B_s$ behaves as a homogeneous component, while $B_e$ is the unit fiber. Multiplication and involution link different fibers without identifying them, so the total space need not itself be a $C^*$ -algebra.

Cross-sectional algebras

Compactly supported continuous sections form a convolution $*$ -algebra. After choosing a left Haar measure, its operations are

(f*g)(s)=\int_G f(t)g(t^{-1}s)\,dt,\qquad f^*(s)=\Delta(s^{-1})f(s^{-1})^*,

where $\Delta$ is the modular function. Completing this algebra in the universal norm gives the full cross-sectional algebra $C^*(\mathcal B)$ ; the regular representation gives $C_r^*(\mathcal B)$ . These constructions extend full and reduced crossed products.

Examples and saturation

For a $C^*$ -dynamical system $(A,G,\alpha)$ , take $B_s=A$ as Banach spaces and define

(a,s)(b,t)=(a\alpha_s(b),st),\qquad (a,s)^*=(\alpha_{s^{-1}}(a^*),s^{-1}).

Its cross-sectional algebras are the ordinary crossed products of the action. Fell bundles also encode partial actions and graded $C^*$ -algebras.

A Fell bundle is saturated when $\overline{\operatorname{span}}(B_sB_t)=B_{st}$ for every $s,t$ . Saturation is an additional property in the convention used here. Some older sources build it into “ $C^*$ -algebraic bundle,” so hypotheses should be compared before importing a theorem.

References

Ruy Exel, Partial Dynamical Systems, Fell Bundles and Applications, Mathematical Surveys and Monographs 224, American Mathematical Society, 2017. AMS record. Relevant: Chapters 16–20 on Fell bundles and their cross-sectional algebras.
J. M. G. Fell and Robert S. Doran, Representations of -Algebras, Locally Compact Groups, and Banach -Algebraic Bundles, Volumes 1–2, Academic Press, 1988. Volume 1 publisher record. Relevant: the axioms and representation theory of Banach $*$ -algebraic bundles.