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Dynamical complexity and \(K\)-theory of \(L^p\) operator crossed products. (English) Zbl 1480.19003

The author considers an analogue of the Baum-Connes assembly map in the setting of operator algebras of \(L^p\)-spaces, \(p\in[1,\infty)\).
The assembly map is formulated through the framework of Roe’s coarse geometric algebras [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0780.58043)], and the left-hand side is given in the form of Yu’s localization algebra [G. Yu, \(K\)-Theory 11, No. 4, 307–318 (1997; Zbl 0888.46047)].
Let \(\Gamma\) be countable discrete group and \(X\) a compact Hausdorff space equipped with a \(\Gamma\)-action. For \(p\) as above, it is proved the the \(\Gamma\)-assembly map with coeffiecient \(C(X)\) is an isomorphism, provided that the action groupoid \(\Gamma\ltimes X\) has finite dynamical complexity. The case \(p=2\) corresponds to the usual assembly map, and has been previously proved in [E. Guentner et al., “Dynamical complexity and controlled operator \(K\)-theory”, Preprint, arXiv:1609.02093].
The groupoid \(\Gamma\ltimes X\) is said to have finite dynamical complexity if it is contained in the smallest class of open subgroupoids that contains all relatively compact open subgroupoids and is closed under “decomposability”. Here, decomposability of an open subgroupoid \(G\) over a collection \(\mathcal{C}\) of open subgroupoids roughly means that, at any given scale, there is a cover of the unit space of \(G\) by two open sets such that the subgroupoids associated to the two open sets at that scale are both in \(\mathcal{C}\). This notion is inspired by dynamic asymptotic dimension [E. Guentner et al., Math. Ann. 367, No. 1–2, 785–829 (2017; Zbl 1380.37018)] and decomposition complexity [E. Guentner et al., Invent. Math. 189, No. 2, 315–357 (2012; Zbl 1257.57028)].
The proof is inspired by Yu’s proof of the coarse Baum-Connes conjecture for spaces with finite asymptotic dimension [G. Yu, Ann. Math. (2) 147, No. 2, 325–355 (1998; Zbl 0911.19001)]. The main tool in the proof is a controlled Mayer-Vietoris exact sequence, which is part of the framework of quantitative \(K\)-theory developed by H. Oyono-Oyono and G. Yu [Ann. Inst. Fourier 65, No. 2, 605–674 (2015; Zbl 1329.19009)]. In a previous paper [J. Funct. Anal. 274, No. 1, 278–340 (2018; Zbl 1386.19011)], the author extended such framework to a larger class of Banach algebras, so that it can be applied to the \(L^p\)-setting considered here.
The last section of the paper considers involutive versions of the \(L^P\)-algebras used in the main body, and raises some comparison questions in terms of \(K\)-theory and assembly maps.

MSC:

19K56 Index theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L85 Noncommutative topology
47L10 Algebras of operators on Banach spaces and other topological linear spaces
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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References:

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