arXiv subjects

Roberto Hernández Palomares

Publications and source records attributed to Roberto Hernández Palomares.

11 recordsLinked to original sources

arXiv2026

Complex Hadamard Matrices - Quantum Symmetries, Equivalence and Non-Local Games

We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local ``Hadamard equivalence'' game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak--Sołtan--Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.

math.QA↗

arXiv2026

Property (SP) and Inclusions of C*-algebras

As a generalization of Gabe and Neagu's inclusions of real rank zero, we introduce a notion of Property (SP) for inclusions of C*-algebras. We begin the systematic study of SP-inclusions, establishing a topological description in the commutative setting and permanence under many constructions. We also extend known results about the permanence of Property (SP) for C*-algebras under classical symmetries to the setting of quantum symmetries; we show that Property (SP) for (simple, separable) C*-algebras is preserved under irreducible C*-discrete inclusions. Finally, we resolve (in the negative) a question raised by Gabe and Neagu about inclusions coming from Furstenburg boundaries. Namely, we show the existence of a countable, discrete, non-amenable group $G$ such that $C_r^*(G) \subseteq C(\partial_F G) \rtimes_r G$ is not an SP-inclusion (and therefore not an inclusion of real rank zero).

math.OA↗

arXiv2026

Remarks on C*-discrete inclusions

We obtain a Galois correspondence between the lattice of intermediate C*-discrete subalgebras intermediate to a given irreducible C*-discrete inclusion, and characterize these as targets of compatible expectations under a traciality condition. Furthermore, we define freeness for actions of tensor categories on C*-algebras, and show simplicity is preserved under taking reduced crossed products. Finally, we show under certain constraints that A-valued semicircular systems give rise to C*-discrete inclusions, and thus are crossed products by an action of a tensor category. Along the way, we show the set of single algebraic generators of a dualizable bimodule forms an open subset.

math.OA↗

arXiv2026

Properly Outer Actions of Tensor Categories on C$^*$-algebras

We discuss proper outerness for finite index endomorphisms and finite index bimodules of simple C$^*$-algebras, extending recent similar results by Izumi concerning the purely infinite setting. Our main result is that proper outerness holds automatically for finite index outer endomorphisms of simple C$^*$-algebras. Consequently, freeness for outer actions of unitary tensor categories on simple C$^*$-algebras is also shown to hold automatically. As applications, we obtain structural results about potentially infinite index irreducible discrete inclusions of C$^*$-algebras, such as C$^*$-irreducibility.

math.OA↗

arXiv2026

Quantum graphs and spin models

We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from deforming graphs using higher-idempotent splittings of quantum isomorphisms for which we prove that the relevant algebraic, combinatorial, and topological properties of the original graphs are preserved along with the quantum automorphism group. We also obtain exotic examples of highly regular quantum graphs using the quantum Fourier transform and a method of iterated convolution. Our examples include quantum versions of the strongly regular $9$-Paley, $16$-Clebsch and the Higman-Sims graphs, yielding new models for their regularity parameters. As applications, we construct a compact quantum group that is monoidally equivalent to $SO_q(5)$ at the square of the golden ratio, whose dual is infinite with property (T), and exhibit a highly-regular quantum graph with no classical analogue. Finally, we introduce quantum spin models, construct explicit examples and make contact with quantum Hadamard matrices.

math.OA↗

arXiv2025

Inclusions of Operator Algebras from Tensor Categories: beyond irreducibility

We derive faithful inclusions of C*-algebras from a coend-type construction in unitary tensor categories. This gives rise to different potential notions of discreteness for an inclusion in the non-irreducible case, and provides a unified framework that encloses the theory of compact quantum group actions. We also provide examples coming from semi-circular systems and from factorization homology. In the irreducible case, we establish conditions under which the C*-discrete and W*-discrete conditions are equivalent.

math.OA↗

arXiv2025

Discrete inclusions from Cuntz-Pimsner algebras

We show that the core inclusion arising from a Cuntz-Pimsner algebra generated by a full, faithful and dualizable correspondence is C*-discrete, and express it as a crossed-product by an action of a unitary tensor category. In particular, we show the inclusion of the UHF subalgebra of the Cuntz algebra arising as the fixed-point subalgebra under the gauge symmetry, is irreducible and C*-discrete. We describe the dualizable bimodules appearing under this inclusion, including their semisimple decompositions and fusion rules, their Watatani indices and Pimsner-Popa bases, as well as their sets of cyclic algebraic generators.

math.OA↗

arXiv2024

Quantum graphs, subfactors and tensor categories I

We develop an equivariant theory of graphs with respect to quantum symmetries and present a detailed exposition of various examples. We portray unitary tensor categories as a unifying framework encompassing all finite classical simple graphs, (quantum) Cayley graphs of finite (quantum) groupoids, and all finite-dimensional quantum graphs. We model a quantum set by a finite-index inclusion of C*-algebras and use the quantum Fourier transform to obtain all possible adjacency operators. In particular, we show every finite-index subfactor can be regarded as a complete quantum graph and describe how to find all its subgraphs. As applications, we prove a version of Frucht's Theorem for finite quantum groupoids, and introduce a version of path spaces for quantum graphs.

math.OA↗

arXiv2024

Discrete Inclusions of C*-algebras

We introduce the category of C*-discrete inclusions of C*-algebras $A\subset B$ with a faithful conditional expectation $E:B\twoheadrightarrow A$. This class includes many examples such as finite Watatani index inclusions, and also abundant infinite index inclusions like crossed products by outer actions of discrete (quantum) groups and unitary tensor categories. We prove irreducible $(A'\cap B= \mathbb{C}1)$ C*-discrete inclusions are precisely crossed products by outer actions of unitary tensor categories and certain C*-algebra objects.

math.OA↗

arXiv2024

Q-system completion for C* 2-categories

A Q-system in a C* 2-category is a unitary version of a separable Frobenius algebra object and can be viewed as a unitary version of a higher idempotent. We define a higher unitary idempotent completion for C* 2-categories called Q-system completion and study its properties. We show that the C* 2-category of right correspondences of unital C*-algebras is Q-system complete by constructing an inverse realization $†$ 2-functor. We use this result to construct induced actions of group theoretical unitary fusion categories on continuous trace C*-algebras with connected spectra.

math.OA↗

arXiv2024

K-theoretic classification of inductive limit actions of fusion categories on AF-algebras

We introduce a K-theoretic invariant for actions of unitary fusion categories on unital C*-algebras. We show that for inductive limits of finite dimensional actions of fusion categories on unital AF-algebras, this is a complete invariant. In particular, this gives a complete invariant for inductive limit actions of finite groups on AF-algebras. We apply our results to obtain a classification of finite depth, strongly AF-inclusions of unital AF-algebras.

math.OA↗