Catalog

Repositories

Latest version of each Zenodo concept by Logica Artificialis. A row shows only the files on that record, and a GitHub address only when the manuscript names one.

4 repositories

• 10.5281/zenodo.238333062026

  Finite configurations and nuclear quotients of Thompson's group F

  Martino Lupini · Thompson groups, C*-algebras, Operator algebras

  Let F be Thompson’s group, let P = F′, and let Ug denote the canonical prefix-table unitaries in the Cuntz algebra O₂. We prove that the finite ordinary tensor images Bn = C*(Uf⊗n : f ∈ F) and Cn = C*(Up⊗n : p ∈ P) are nuclear. A representation has finite separated derived capacity when products of sufficiently many displacement operators with separated interior supports vanish. Capacity at most d gives an actual marked quotient of B2d+2, or of C2d for a nontrivial P-representation. Infinite capacity is equivalent to admitting the regular representation as an actual quotient. Consequently, if F is nonamenable, every nonzero nuclear quotient of its full group C*-algebra, and likewise of C*(P), is a quotient of one of these finite canonical images. The finite nuclearity and norm estimates are unconditional; nonamenability enters only in passing from nuclearity to finite capacity. The result gives a countable cofinal family of nuclear quotients, with a degree depending on the quotient.

  PDF · LaTeX · GitHub named

• 10.5281/zenodo.238320582026

  Charmenability of Thompson's group F

  Martino Lupini · Thompson groups, C*-algebras, Amenability

  We prove that Thompson's group F and its commutator subgroup F′ are charmenable. The main assertion is that every nonempty compact convex conjugation-invariant family of normalized positive-definite functions on either group contains a trace. Its proof uses commuting groups supported on disjoint intervals and a stationary averaging argument on their joint von Neumann algebra. It is independent of nonamenability of F. The character condition in charmenability follows from the classification of finite factor representations by Dudko and Medynets. We also prove that F is not charfinite and that F′ is charfinite precisely when F is nonamenable. Every nonzero simple quotient of either full group C*-algebra is scalar or is the corresponding reduced group algebra, and every nonzero nuclear quotient admits a multiplicative character.https://github.com/eulogica/charmenability-of-thompsons-group-f

  No files · GitHub named

• 10.5281/zenodo.238280862026

  Normal-semigroup generators for the bi-order cones of Thompson's group F′

  Martino Lupini · Thompson groups, Orderability, Groups

  The commutator subgroup F′ of Thompson's group has four bi-invariant total orders, determined by the slope at the first or last point of support. We prove that the positive cone of each order is generated by two elements as a normal semigroup: every positive element is a nonempty product of conjugates of those two elements. An explicit pair f, g works for all four cones, using the four choices of signs (f, g), (f⁻¹, g⁻¹), (f⁻¹, g), and (f, g⁻¹). This answers the finite-generation question posed by Navas and Rivas in 2008, including their stronger requirement that the same pair yield both prescribed partitions of F′\{1}. Two generators are necessary for each cone.

  PDF · LaTeX · GitHub named

• 10.5281/zenodo.234199962026

  Thompson's group F′ is an O*-group: Positive conjugacy semigroups in Thompson's group F′

  Martino Lupini · Thompson groups, Orderability, Groups

  We give an explicit countable simple example for the Mal’cev–Fuchs question whether a simple O*-group exists, posed in 1963 and 1965. Every bi-invariant partial order on the commutator subgroup F′ of Thompson’s group F extends to a bi-invariant total order. Every partial positive cone lies in one of four cones determined by slopes at the leftmost or rightmost support endpoint. The central ingredient is a factorization theorem for right-moving one-bump maps with the same support: equality of their two endpoint slopes is equivalent to a sufficiently large power of one being a product of the corresponding number of conjugates of the other, supported on that interval. The proof combines bounded drift in orbit coordinates with a uniform correction bound from the commutator width of F′.

  PDF · LaTeX · GitHub named