Subject / 7 records

Groupoids and Cartan pairs

Weyl groupoids, cohomological lengths, exactness, assembly, and Cartan invariants.

Classification: Second Brain

7 records

• 10.5281/zenodo.238276522026

  Failure of rational injectivity for minimal principal étale groupoids

  Martino Lupini · Orderability, Groupoids, Groups

  Applying OpenAI's torsion-free group counterexample, we construct a second-countable locally compact Hausdorff minimal principal ample groupoid with Cantor unit space whose ordinary reduced Baum--Connes assembly map has an infinite-order kernel class in degree zero. Its rationalization is therefore not injective, with the natural coefficient algebra of continuous functions on the unit space. The groupoid is a transformation groupoid of a finitely generated torsion-free group and is compactly generated. We use the invariant-measure trace class of Antonini, Azzali and Skandalis to preserve an ordinary assembly kernel class under extension of coefficients to continuous functions. Elek's free minimal measure-preserving Cantor action supplies the required dynamics. The standard compact-action comparison identifies group assembly with coefficients and transformation-groupoid assembly. Thus minimality and principality do not restore the natural-coefficient reduced Baum--Connes assertion.

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• 10.5281/zenodo.233625992026

  Exactness and amenability at infinity for locally compact groupoids

  Martino Lupini · Groupoids, Amenability, Groups

  For a second-countable locally compact Hausdorff étale groupoid G, we prove the equivalence of exactness of Cr*(G), exactness of reduced crossed products, and strong amenability at infinity. These conditions are also equivalent to exactness of any one twisted reduced algebra. The amenable action can be chosen with a proper open surjective anchor admitting a continuous section. This answers Anantharaman–Delaroche’s 2016 exactness questions and the étale case of LaLonde’s 2017 question about exact transformation groupoids. Consequently, exactness together with canonical weak containment implies amenability; for compact unit space, exactness is equivalent to amenability of the Furstenberg boundary action. We also prove permanence of strong amenability at infinity under continuous strongly surjective homomorphisms between second-countable groupoids with Haar systems, when the kernel and target have this property. No Haar system on the kernel is required. Finally, we construct an inner-exact étale group bundle with exact fibers and nonexact reduced algebra, supplying the example asked for by Anantharaman–Delaroche in 2016 and answering LaLonde’s 2017 group-bundle question negatively. Its fibers are hyperbolic groups converging strongly to a free group, while their Cayley graphs fail uniform property A.

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• 10.5281/zenodo.233485852026

  Dynamical spectra of unital C*-algebras: AF realizations and Cartan pairs

  Martino Lupini · C*-algebras, Groupoids, Descriptive set theory

  For a separable unital C*-algebra, the dynamical length records the descriptive complexity of its inner derivations and inner automorphisms. For a Cartan pair, it also records two cohomological lengths of the Weyl groupoid and the length of the automorphism group preserving the Cartan subalgebra. We determine the spectra of AF algebras, standard AF pairs, C*-diagonals in AF algebras, and arbitrary Cartan pairs whose ambient algebra is AF. Nonconstant tuples form a finite list at the bottom of the scale. The positive constant coordinates are exactly {β+1 : β<ω₁} ∪ {α+1/2, α+1/2+ε : 0<α<ω₁}, where the half-step symbols are formal indices in the scale. We construct the fractional realizations through zero-dimensional incidence systems and prove that every positive derivation length of a separable unital C*-algebra has a standard AF-pair realization. Each positive value has a standard diagonal realization in a separable unital 2-subhomogeneous AF algebra satisfying Fell's condition. The absolute spectrum of all separable unital C*-algebras coincides with the AF spectrum, and the spectrum of Kumjian diagonal pairs coincides with that of standard AF pairs. General Cartan pairs have one additional low value. We also determine the spectra within the scattered, continuous-trace, simple, and homogeneous classes. The exclusion of the remaining fractional candidates is imported from a companion theorem.

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• 10.5281/zenodo.233423112026

  Dynamical length of Cartan pairs: comparison, rigidity, and invariance

  Martino Lupini · Orderability, C*-algebras, Groupoids

  A Cartan subalgebra permits a separable unital C*-algebra to be studied through its twisted étale Weyl groupoid. We combine five cohomological, derivation, and automorphism lengths into the dynamical length ℓ(A,D) of a Cartan pair. These lengths measure the accumulation of coboundaries and inner symmetries through the canonical filtration of phantom subgroups.

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• 10.5281/zenodo.226858692026

  Dynamical spectra of separable unital scattered $C^*$-algebras and Cartan pairs

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.226843792026

  Cohomological lengths of groupoids

  Martino Lupini · Groupoids

  No abstract on this deposit.

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• 10.5281/zenodo.226842522026

  Normalizer correction and relative derivation and automorphism lengths for Cartan C*-subalgebras

  Martino Lupini · C*-algebras

  No abstract on this deposit.

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