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Francesco D'Andrea

Publications and source records attributed to Francesco D'Andrea.

At least 19 recordsLinked to original sources
arXiv2026

Quantum CW-complexes in a Waldhausen category for unital C*-algebras

Using the ring structure of the K-groups of finite CW-complexes, Atiyah and Todd unravelled the K-theory of complex projective spaces $CP^n$. Herein, in the realm of noncommutative topology, we develop a new framework of finite quantum CW-complexes using the language of Waldhausen categories, which allows us to enrich the class of standard morphisms between unital C*-algebras by adding inverses of *-homomorphisms that are isomorphisms in K-theory. Our concept of quantum CW-complexes subsumes earlier constructions and enjoys a plethora of examples. Moreover, the framework allows us to reduce problems concerning the multipushout quantum complex projective space $CP^n_H$ to the much more approachable setting of the Vaksman-Soibelman quantum complex projective space $CP^n_q$ enjoying the availability of graph-algebraic methods. In particular, these methods permit us to transport the ring structure from $K^0(CP^n)$ to $K^0(CP_q^n)$. Finally, we adapt the formalism of Waldhausen categories to determine a natural set of free generators of $K^0(CP_H^n)$ from a natural set of free generators of $K^0(CP_q^n)$, and to transport the ring structure from $K^0(CP_q^n)$ to $K^0(CP_H^n)$.

math.KT↗
arXiv2026

Relation morphisms of directed graphs

Associating graph algebras to directed graphs leads to both covariant and contravariant functors from suitable categories of graphs to the category k-Alg of algebras and algebra homomorphisms. As both functors are often used at the same time, finding a new category of graphs that allows a "common denominator" functor unifying the covariant and contravariant constructions is a fundamental problem. Herein, we solve this problem by first introducing the relation category of graphs RG, and then determining the concept of admissible graph relations that yields a subcategory of RG admitting a contravariant functor to k-Alg simultaneously generalizing the aforementioned covariant and contravariant functors. Although we focus on Leavitt path algebras and graph C*-algebras, on the way we unravel functors to k-Alg given by path algebras, Cohn path algebras and Toeplitz graph C*-algebras from suitable subcategories of RG. Better still, we illustrate relation morphisms of graphs by naturally occurring examples, including Cuntz algebras, quantum spheres and quantum balls.

math.RA↗
arXiv2026

On amplified graph C*-algebras as cores of Cuntz-Krieger algebras

Given a finite directed acyclic graph $R$, we construct from it two graphs $E_R$ and $F_R$, one by adding a loop at every vertex of $R$ and one by replacing every arrow of $R$ by countably infinitely many arrows. We show that the graph C*-algebra $C^*(F_R)$ is isomorphic to the AF core of $C^*(E_R)$. Examples include C*-algebras of a quantum flag manifolds and quantum teardrops. We discuss in detail the quantum Grassmannian $Gr_q(2,4)$ and use our description as AF core to study its CW-structure.

math.OA↗
arXiv2026

The graph groupoid of a quantum sphere

Quantum spheres are among the most studied examples of compact quantum spaces, described by C*-algebras which are Cuntz-Krieger algebras of a directed graph, as proved by Hong and Szymański in 2002. About five years earlier, in 1997, Sheu proved that the C*-algebra of a quantum sphere is a groupoid C*-algebra. Here we show that the path groupoid of the directed graph of Hong and Szymański is isomorphic to the groupoid discovered by Sheu.

math.OA↗
arXiv2026

Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes

We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case.

math.KT↗
arXiv2025

Pullback of quantum principal bundles

We introduce an abstract framework of Cartesian squares beyond the context of fiber products, and use it to extend the notion of pullback from classical to compact quantum principal bundles. Based only on our abstract notion of a Cartesian square, we extend key concepts of Equivariant Topology, such as the pullback of a family of group actions, orbit spaces, slices and global sections, change of base and structure group, free actions, and the groupoid of compact principal bundles. Finally, we embed the thus extended Equivariant Topology inside the 2-category of Grothendieck categories in such a way that our notion of a Cartesian square becomes the appropriate Beck-Chevalley condition.

math.KT↗
arXiv2025

Ordine privo di periodicità: il fascino matematico delle tassellazioni

This is a review (in Italian) on aperiodic tilings of the plane intended for a general audience. First, we recall some basic results about lattices and periodic tilings. Then, we move on to one-dimensional (domino) tilings and Wang tilings. We present a beautiful proof of the existence of an aperiodic set of Wang prototiles due to J. Kari. Next, we discuss Penrose tilings and their properties. Finally, we briefly present the recent discovery by D. Smith and his collaborators of an aperiodic monotile.

math.HO↗
arXiv2025

On the K-theory of the AF core of a graph C*-algebra

In this paper, we study multiplicative structures on the K-theory of the core $A:=C^*(E)^{U(1)}$ of the C*-algebra $C^*(E)$ of a directed graph $E$. In the first part of the paper, we study embeddings $E\to E\times E$ that induce a *-homomorphism $A\otimes A\to A$. Through Künneth formula, any such a *-homomorphism induces a ring structure on $K_*(A)$. In the second part, we give conditions on $E$ such that $K_*(A)$ is generate by "noncommutative line bundles" (invertible bimodules). The same conditions guarantee the existence of a homomorphism of abelian groups $K_0(A)\to\mathbb{Z}[λ]/(\det(λΓ-1))$ (where $Γ$ is the adjacency matrix of $E$) that is compatible with the tensor product of line bundles. Examples include the C*-algebra $C(\mathbb{C}P^{n-1}_q)$ of a quantum projective space, the $UHF(n^\infty)$ algebra, and the C*-algebra of the space parameterizing Penrose tilings. For the first algebra, as a corollary we recover some identities that classically follow from the ring structure of $K^0(\mathbb{C}P^{n-1})$, and that were proved by Arici, Brain and Landi in the quantum case. Incidentally, we observe that the C*-algebra of Penrose tilings is the AF core of the Cuntz algebra $\mathcal{O}_2$, if the latter is realized using the appropriate graph.

math.KT↗
arXiv2025

Isomorphisms of quantum spheres

For $n\in\mathbb{N}$ and $q\in [0,1[$, the Vaksman-Soibelman quantum sphere $S^{2n+1}_q$ is described by an associative algebra $\mathcal{A}(S^{2n+1}_q)$ deforming the algebra of polynomial functions on the 2n+1 dimensional unit sphere. Its C*-enveloping algebra is known to be independent of the deformation parameter q. In contrast to what happens in the C*-algebraic setting, we show here that, for all $q,q'$ in the above range, $\mathcal{A}(S^{2n+1}_q)$ is isomorphic to $\mathcal{A}(S^{2n+1}_{q'})$ only if $q=q'$.

math.QA↗
arXiv2025

On characteristic classes of vector bundles over quantum spheres

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups $K_0 \to \mathbb{Z}[t]/(t^2)$ compatible with the tensor product of bimodules. Applications include the standard Podleś sphere $S^2_q$ and a quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on $S^4_q$ associated to the principal $SU_q(2)$-bundle $S^7_q \to S^4_q$ via irreducible corepresentations of $SU_q(2)$, and compute their characteristic classes.

math.QA↗
arXiv2024

Quantum spheres as graph C*-algebras: a review

In this survey, we discuss the description of Vaksman-Soibelman quantum spheres using graph C*-algebras, following the seminal work of Hong and Szymański. We give a slightly different proof of the isomorphism with a graph C*-algebra, borrowing the idea of Mikkelsen and Kaad of using conditional expectations to prove the desired result.

math.OA↗
arXiv2023

A guide to Penrose tilings

The aim of this book is to provide an elementary introduction, complete with detailed proofs, to the celebrated tilings of the plane discovered by Sir Roger Penrose in the `70s. The book covers many aspects of Penrose tilings, including the study of the space parameterizing Penrose tilings from the point of view of Connes' Noncommutative Geometry.

math.CO↗
arXiv2022

Tolerance Relations and Quantization

It is well known that "bad" quotient spaces (typically: non-Hausdorff) can be studied by associating to them the groupoid C*-algebra of an equivalence relation, that in the "nice" cases is Morita equivalent to the C*-algebra of continuous functions vanishing at infinity on the quotient space. It was recently proposed by A. Connes and W.D. van Suijlekom that a similar procedure for relations that are reflexive and symmetric but fail to be transitive (i.e. tolerance relations) leads to an operator system. In this paper we observe that such an operator system carries a natural product that, although in general non-associative, arises in a number of relevant examples. We relate this product to truncations of (C*-algebras of) topological spaces, discuss some geometric aspects and a connection with positive operator valued measures.

math.OA↗
arXiv2021

On the pseudo-manifold of quantum states

There are various statements in the physics literature about the stratification of quantum states, for example into orbits of a unitary group, and about generalized differentiable structures on it. Our aim is to clarify and make precise some of these statements. For A an arbitrary finite-dimensional C*-algebra and U(A) the group of unitary elements of A, we observe that the partition of the state space S(A) into U(A) orbits is not a decomposition and that the decomposition into orbit types is not a stratification (its pieces are not manifolds without boundary), while there is a natural Whitney stratification into matrices of fixed rank. For the latter, when A is a full matrix algebra, we give an explicit description of the pseudo-manifold structure (the conical neighborhood around any point). We then make some comments about the infinite-dimensional case.

math.OA↗
arXiv2020

Twisted reality and the second-order condition

An interesting feature of the finite-dimensional real spectral triple (A,H,D,J) of the Standard Model is that it satisfies a ``second-order'' condition: conjugation by J maps the Clifford algebra Cl_D(A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence Cl_D(A)-bimodule). This resembles a property of the canonical spectral triple of a closed oriented Riemannian manifold: there is a dense subspace of H which is a self-Morita equivalence Cl_D(A)-bimodule. In this paper we argue that on manifolds, in order for the self-Morita equivalence to be implemented by a reality operator J, one has to introduce a ``twist'' and weaken one of the axioms of real spectral triples. We then investigate how the above mentioned conditions behave under products of spectral triples.

math.QA↗
arXiv2020

A dual formula for the spectral distance in noncommutative geometry

In noncommutative geometry, Connes's spectral distance is an extended metric on the state space of a C*-algebra generalizing Kantorovich's dual formula of the Wasserstein distance of order 1 from optimal transport. It is expressed as a supremum. We present a dual formula - as an infimum - generalizing Beckmann's "dual of the dual" formulation of the Wasserstein distance. We then discuss some examples with matrix algebras, where such a dual formula may be useful to obtain upper bounds for the distance.

math.OA↗
arXiv2020

On the Notion of Noncommutative Submanifold

We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra $A$ is a quotient algebra $B$ such that all derivations of $B$ can be lifted to $A$. We will argue that in the case of smooth functions on manifolds every quotient algebra is a submanifold algebra, derive a topological obstruction when the algebras are deformation quantizations of symplectic manifolds, present some (commutative and noncommutative) examples and counterexamples.

math.QA↗