Subject / 32 records

Operator algebras

C*-algebras, von Neumann algebras, invariants, representations, and classification.

Classification: Second Brain

32 records

• 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.

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• 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

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• 10.5281/zenodo.236830422026

  Nonmeagre ideals from traces on corona algebras

  Martino Lupini · C*-algebras

  We show that a tracial state on the corona of a separable nonunital C*-algebra gives a proper nonmeagre ideal in its multiplier algebra. Here nonmeagreness is the block condition introduced by Vignati and Yilmaz: every good approximate identity and every disjoint sequence of finite intervals admit an infinite block sum in the ideal. More generally, an ideal containing the original algebra is nonmeagre whenever its quotient has only countable families of pairwise orthogonal nonzero positive elements. Applying this criterion to Goodearl's simple AF example answers Question 4.13 of Vignati and Yilmaz affirmatively in ZFC. Goodearl's examples also yield a simple AF algebra with uncountably many maximal proper nonmeagre multiplier ideals.

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• 10.5281/zenodo.236828572026

  Local units and multiplier algebras of AF corona ideals

  Martino Lupini · Mathematics

  Let A be a simple unital separable infinite-dimensional AF algebra, fix a tracial state τ, and put D = A ⊗ K. Let I be the closed trace ideal of M(D) associated with τ, and set J = I/D and Q = M(D)/D. We prove that, under the Continuum Hypothesis, the canonical inclusion Q ⊂ M(J) is proper. More precisely, there is a positive contraction m ∈ M(J) such that dist(m,Q) = 1/2 and |M(J)| = dens‖·‖ M(J) = 2^c, where c = 2^ℵ₀. The construction produces globally compatible multiplier actions on an exhaustive family of projection corners. Countable Riesz separation suffices for the limit-stage interpolation. In ZFC, every countable subset of J has a projection unit, with a lift of arbitrarily small finite trace, and every multiplier agrees on that subset on both sides with an element of J.

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• 10.5281/zenodo.236824122026

  The noncommutative weak extension principle under OCA

  Martino Lupini · C*-algebras, Descriptive set theory

  We prove that the Open Colouring Axiom implies the noncommutative weak extension principle. Every *-homomorphism between coronas of separable nonunital C*-algebras therefore splits, by a commuting projection, into a part with Borel graph and a part with nonmeagre kernel. This removes Martin's Axiom from Vignati and Yilmaz's theorem and answers their proposed weakening for Theorem A. In particular, all such homomorphisms with stable source are Borel under the Open Colouring Axiom alone. We also obtain Borel product lifts for bounded linear maps between reduced products of separable Banach spaces and separable C*-algebras which respect the canonical coordinate restrictions.

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• 10.5281/zenodo.234404152026

  Lipschitz positive decompositions of von Neumann preduals

  Martino Lupini · Mathematics

  We prove that the selfadjoint predual of a complex von Neumann algebra admits a Lipschitz positive decomposition if and only if the algebra is subhomogeneous. In particular, infinite-dimensional selfadjoint trace class gives a negative answer to the Lipschitz decomposition problem, confirming Messerschmidt's 2018 conjecture. For a nonzero subhomogeneous algebra of maximal matrix degree $d$, the least Lipschitz constant of a positive majorant $q(x)\geq\pm x$ is attained and equals the corresponding constant $c_d$ for $d\times d$ trace class. Among nonzero von Neumann algebras, the optimal constant equals one exactly for the abelian algebras. The finite-dimensional constants satisfy $$c\log\log N\leq c_N\leq C\log N$$ for all sufficiently large $N$, with absolute constants $c,C>0$. The lower bound applies to arbitrary positive majorants; the upper bound comes from the classical absolute-value estimate.

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• 10.5281/zenodo.234403022026

  Isometric canonical maps into free convex Banach lattices

  Martino Lupini · Mathematics

  For every $1<p<\infty$, we construct a three-dimensional real Banach lattice $X$ which is not $p$-convex with constant one, although its canonical positive map into the free $p$-convex Banach lattice with constant one is isometric. Dimension three is minimal. In every dimension $n\geq3$, such examples can be arbitrarily close to $\ell_p^n$ under the coordinate lattice isomorphism. An elementary example answers the necessity question raised by Marcel de Jeu and Xingni Jiang in 2025 for general convexity types; the three-dimensional construction gives a negative answer even for ordinary constant-one $p$-convexity. We characterize isometry by an exact factorization of dual functionals through two positive coordinates. For a Banach lattice, the criterion says that every sublattice of $X^*$ generated by two disjoint positive elements is $p'$-concave with constant one. The counterexample separates this two-generator condition from full concavity. At $p=\infty$, in contrast, isometry forces the original norm to be an AM-space norm.

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• 10.5281/zenodo.234301632026

  Completely positive coordinates for KK-theory and relative uniform stability

  Martino Lupini · C*-algebras, Amenability, Groups

  Let A be a unital nuclear C*-algebra, M a von Neumann algebra, and I a norm-closed ideal of M. We prove that a unital completely positive map Φ:A→M, multiplicative modulo I, admits a unital homomorphic correction with the same quotient map whenever

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• 10.5281/zenodo.234114932026

  Polarities, voltages, and capacitors: a categorical approach to hulls, envelopes, and completions

  Ivan Di Liberti, Martino Lupini · Orderability, C*-algebras

  This article provides a general framework in the context of category theory where one can recognize as particular instances of the same abstract construction several notions of completion, envelope, and hull, such as the completion of a Boolean algebra, the Dedekind–MacNeille completion of an ordered set, the multiplier ring of a ring, and the multiplier algebra of a C*-algebra.

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• 10.5281/zenodo.234077612026

  Lipschitz embeddings of symmetric operator-ideal balls into Hilbert space

  Martino Lupini · Operator algebras

  No abstract on this deposit.

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• 10.5281/zenodo.234055082026

  A universal Banach–Mazur stability radius for von Neumann algebras

  Martino Lupini · Mathematics

  We prove cohomological criteria for ordinary complex Banach–Mazur stability of von Neumann algebras, without separability, type, or density restrictions. A fixed algebra has a positive stability radius if its second bounded Hochschild cohomology vanishes and its third coboundary space is norm closed. For any class closed under countable bounded direct products, vanishing in degrees two and three gives one radius for the entire class. Assuming the degree-two and degree-three vanishing theorem in the September 2026 preprint of OpenAI, this yields one $\varepsilon>0$ such that, for all complex von Neumann algebras $M,N$,

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• 10.5281/zenodo.234046012026

  Recursive classification and admissible descent for preliminary C*-algebras with prescribed weighted spectrum

  Martino Lupini · C*-algebras

  We prove that compatible local separable preliminary C*-algebras on an open cover occur as the specified ideals of a global algebra exactly when their transported corona ranges are pairwise orthogonal. Here preliminary means that all irreducible representations are finite-dimensional. We fix a primitive spectrum X and positive integer representation dimensions w, with X second countable, locally compact, and sober and w satisfying the dimension-compatibility condition for preliminary spaces; realization of (X,w) is not assumed. The canonical homogeneous filtration is countable. Classical homogeneous classification and Busby extension theory then give an exact recursive classification by matrix bundles and attaching maps subject to a localized-kernel test for the prescribed topology. We organize these extension data by Čech cocycles satisfying corona orthogonality, modulo gauge and refinement, without reducing them to characteristic classes. A doubled-endpoint example exhibits a nonrealizable class in unrestricted nonabelian H¹. For constant weight n, admissibility is automatic and the invariant is the marked classical Fell–Tomiyama–Takesaki invariant Čech H¹(X,PU(n)). Geometric boundary models for two-stratum algebras allow singular multiplier fibres; explicit families yield attachment multiplicities and Fell ranks.

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• 10.5281/zenodo.234044292026

  Spectral ranks and dynamical lengths of preliminary C*-algebras

  Martino Lupini · C*-algebras, Descriptive set theory

  We classify the spectral ranks and automorphism lengths of nonzero separable preliminary C*-algebras, whose irreducible representations are all finite-dimensional. The spectral invariants are the Fell rank ρ and the Pedersen rank β of the irreducible-dimension function. Automorphism length q measures the transfinite descriptive complexity of automorphisms modulo those implemented by multiplier unitaries; its fractional values distinguish terminal Borel classes.

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• 10.5281/zenodo.233732062026

  A counterexample to the Global Glimm Problem

  Martino Lupini · C*-algebras

  We construct a unital, separable, nuclear C*-algebra whose hereditary subalgebras have no nonzero finite-dimensional representations, but which has no full square-zero element. This gives negative answers to the Global Glimm Problem, originating in Kirchberg and Rørdam’s work, and to Elliott and Rørdam’s full matrix-cone question. The algebra is a continuous field of CAR algebras over a countable product of finite quaternionic projective spaces. Every selfadjoint element is scalar in some fiber; consequently, there are no two orthogonal full positive elements and no full soft positive element. The construction realizes Dranishnikov–Levin’s equivariant obstruction in quaternionic bundles within the CAR-field framework of Hirshberg, Rørdam, and Winter.

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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.233411642026

  Solution to Dixmier's Problem about spectra of C*-algebras

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

  Dixmier asked in 1967 whether the spectra of simple separable non-type-I C*-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel σ-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for ω₂ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal.

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• 10.5281/zenodo.233387882026

  Automatic equivariant Jiang–Su stability in the nonunital tracial setting

  Martino Lupini · C*-algebras, Amenability, Groups

  Let A be a separable simple nuclear C*-algebra that absorbs the Jiang–Su algebra Z and admits a nonzero densely defined lower semicontinuous trace. Using the Gaussian construction of the October 2026 OpenAI preprint, we prove that every action α of a countable discrete amenable group on A is cocycle conjugate to α ⊗ idZ. This answers the general tracial part of Szabó’s 2019 conjecture on automatic equivariant Jiang–Su stability, extending the October 2026 unital theorem to algebras with possibly unbounded traces and to trace-scaling actions. The extension applies that construction in a tracial quotient of Kirchberg’s central sequence algebra and uses the nonunital absorption results of Szabó and Wouters. The transfer is expressed as a local equivariant Gamma criterion for nonseparable unital algebras admitting relative copies of Z. Its uniform trace-splitting estimates allow the original action and its full cone of traces to be retained. Combined with the established purely infinite case, the result gives automatic equivariant Jiang–Su stability for all separable simple nuclear Z-stable algebras.

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• 10.5281/zenodo.231695762026

  Polishability from orbit reducibility of coset equivalence relations

  Martino Lupini · Descriptive set theory, Groups

  We prove that a Borel subgroup H of a Polish group G is Polishable if and only if its coset equivalence relation is Borel reducible to the orbit equivalence relation of a Borel action of a Polish group. The proof combines Jankov–von Neumann uniformization with an invariant category criterion derived from Solecki's approximation theorems. Averaging the canonical category ideals of the target orbits over G supplies the invariant ideal needed for recognition. Consequently, a non-Polishable Borel subgroup has a coset relation that is not classifiable by countable structures.

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• 10.5281/zenodo.228398652026

  UCT-universal nuclear quotients of a binary wreath product

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.227646152026

  Universal minimal flows of automorphism groups\\of stable Kirchberg algebras

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227616302026

  Universal minimal flows for unital Kirchberg and AF algebras

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.227374132026

  A counterexample to the (approximate) Universal Coefficient Theorem for \texorpdfstring{$K$}{K}-homology of C*-algebras

  Martino Lupini · C*-algebras

  No abstract on this deposit.

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• 10.5281/zenodo.227224532026

  Extreme amenability of the automorphism group of the Jiang-Su algebra

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227221902026

  Extreme amenability of the automorphism group of the Razak-Jacelon algebra

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227189882026

  Universal minimal flows of Cuntz automorphism and unitary groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.226953502026

  Descriptive classification of classes of C*-algebras and locally compact groups

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

  No abstract on this deposit.

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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.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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• 10.5281/zenodo.226838592026

  Derivation length and automorphism length of unital C*-algebras

  Martino Lupini · C*-algebras

  No abstract on this deposit.

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• 10.5281/zenodo.226386402026

  C*-algebras of all successor ordinal derivation and automorphism lengths

  Martino Lupini · C*-algebras

  No abstract on this deposit.

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• 10.5281/zenodo.223344012026

  Bratteli graphs

  Martino Lupini · Mathematics

  No abstract on this deposit.

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