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Fourier and Fourier-Stieltjes algebra of Fell bundles over discrete groups
Operator Algebras · Submitted 19 Apr 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For a Fell bundle $\mathcal{B}=\left\{B_{s}\right\}_{s \in G}$ over a discrete group $G$, we use representations theory of $\mathcal{B}$ to construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$ of $\mathcal B$.
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For a Fell bundle $\mathcal{B}=\left\{B_{s}\right\}_{s \in G}$ over a discrete group $G$, we use representations theory of $\mathcal{B}$ to construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$ of $\mathcal B$. When $\mathcal B$ is saturated we show $B(\mathcal{B})$ is canonically isomorphic to the dual space of the cross sectional $C^{*}$-algebra $C^{*}(\mathcal{B})$ of $\mathcal{B}$. When there is a compatible family of co-multiplications on the fibers we show that $B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach algebras. This holds in particular if either the fiber $B_e$ at identity is a Hopf $C^*$-algebra or $\mathcal{B}$ is the Fell bundle of a $C^*$-dynamical system. When $A(\mathcal{B})$ is a Banach algebra with bounded approximate identity, we show that $B(\mathcal{B})$ is the multiplier algebra of $A(\mathcal{B})$. We prove a Leptin type theorem by showing that amenability of $G$ implies the existence of bounded approximate identity for $A(\mathcal{B})$ for bundles coming from a $C^*$-dynamical system $(A,G,\gamma)$. The converse is left as an open problem.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For a Fell bundle $\mathcal{B}=\left\{B_{s}\right\}_{s \in G}$ over a discrete group $G$, we use representations theory of $\mathcal{B}$ to construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$ of $\mathcal B$. | contribution 0.95 | 0.95 | 0.12 | asserted 1.00 |
| s1 | When $\mathcal B$ is saturated we show $B(\mathcal{B})$ is canonically isomorphic to the dual space of the cross sectional $C^{*}$-algebra $C^{*}(\mathcal{B})$ of $\mathcal{B}$. | result 0.99 | 0.95 | 0.21 | asserted 1.00 |
| s2 | When there is a compatible family of co-multiplications on the fibers we show that $B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach algebras. | result 0.97 | 0.94 | 0.14 | asserted 1.00 |
| s3 | This holds in particular if either the fiber $B_e$ at identity is a Hopf $C^*$-algebra or $\mathcal{B}$ is the Fell bundle of a $C^*$-dynamical system. | result 0.76 | 0.80 | 0.07 | asserted 0.99 |
| s4 | When $A(\mathcal{B})$ is a Banach algebra with bounded approximate identity, we show that $B(\mathcal{B})$ is the multiplier algebra of $A(\mathcal{B})$. | result 0.99 | 0.94 | 0.15 | asserted 1.00 |
| s5 | We prove a Leptin type theorem by showing that amenability of $G$ implies the existence of bounded approximate identity for $A(\mathcal{B})$ for bundles coming from a $C^*$-dynamical system $(A,G,\gamma)$. | result 0.94 | 0.94 | 0.17 | asserted 1.00 |
| s6 | The converse is left as an open problem. | limitation 0.56 | 0.30 | 0.04 | asserted 0.71 |
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For a Fell bundle $\mathcal{B}=\left\{B_{s}\right\}_{s \in G}$ over a discrete group $G$, we use representations theory of $\mathcal{B}$ to construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$ of $\mathcal B$. When $\mathcal B$ is saturated we show $B(\mathcal{B})$ is canonically isomorphic to the dual space of the cross sectional $C^{*}$-algebra $C^{*}(\mathcal{B})$ of $\mathcal{B}$. When there is a compatible family of co-multiplications on the fibers we show that $B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach algebras. This holds in particular if either the fiber $B_e$ at identity is a Hopf $C^*$-algebra or $\mathcal{B}$ is the Fell bundle of a $C^*$-dynamical system. When $A(\mathcal{B})$ is a Banach algebra with bounded approximate identity, we show that $B(\mathcal{B})$ is the multiplier algebra of $A(\mathcal{B})$. We prove a Leptin type theorem by showing that amenability of $G$ implies the existence of bounded approximate identity for $A(\mathcal{B})$ for bundles coming from a $C^*$-dynamical system $(A,G,\gamma)$. The converse is left as an open problem.
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We refer the reader for more details to [ref] .
On Nuclearity of $C^*$-algebras of Fell bundles over \'etale groupoids · multiplier algebra and bounded approximate identities · 0.92
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