*(English)*Zbl 1030.46069

From the text: We continue the study of the phenomena of amenability for Fell bundles over discrete groups, initiated in [*R. Exel*, J. Reine Angew. Math. 492, 41-73 (1997; Zbl 0881.46046)]. By definition, a Fell bundle is said to be amenable if the left regular representation of its cross-sectional ${C}^{*}$-algebra is faithful. This property is also equivalent to the faithfulness of the standard conditional expectation. The starting point for our work is Theorem 6.7 of the paper cited above, where it is shown that a certain grading of the Cuntz-Krieger algebra gives rise to an amenable Fell bundle over a free group. Our main goal is to further pursue the argument leading to this result, in order to obtain a large class of amenable Fell bundles.

We show that a Fell bundle $\mathbb{B}={\left\{{B}_{t}\right\}}_{t\in \mathbb{F}}$, over an arbitrary free group $\mathbb{F}$, is amenable, whenever it is orthogonal (in the sense that ${B}_{x}^{*}{B}_{y}=0$, if $x$ and $y$ are distinct generators of $\mathbb{F})$ and semi-saturated (in the sense that ${B}_{ts}$ coincides with the closed linear span of ${B}_{t}{B}_{s}$, when the multiplication “$ts$” involves no cancelation).