arXiv subjects

Giovanni Ferrer

Publications and source records attributed to Giovanni Ferrer.

10 recordsLinked to original sources

arXiv2026

Orthonormal bases for higher Hilbert spaces

In our previous article [arxiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the $\mathrm{C}^*$-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.

math.QA↗

arXiv2026

The many faces of higher Hilbert spaces

Finite-dimensional operator algebras can be viewed as $\mathrm{C}^*$, $\mathrm{W}^*$, or $\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the passage from Hermitian vector spaces to Hilbert spaces. Finally, we outline an inductive approach to defining higher Hilbert spaces in arbitrary dimension, suggesting an extension of these ideas beyond the 2-categorical setting.

math.QA↗

arXiv2026

The Homotopy 3-Type of Abelian C*-Algebras

We compute the homotopy groups at each unital abelian C*-algebra $C(T)$ in the Morita $3$-category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms of the topological data of the underlying compact Hausdorff space $T$. We also compute the actions of the first homotopy group on the second and third homotopy groups in terms of these topological invariants of $T$.

math.OA↗

arXiv2024

Manifestly unitary higher Hilbert spaces

Higher idempotent completion gives a formal inductive construction of the $n$-category of finite dimensional $n$-vector spaces starting with the complex numbers. We propose a manifestly unitary construction of low dimensional higher Hilbert spaces, formally constructing the $\mathrm{C}^*$-3-category of 3-Hilbert spaces from Baez's 2-Hilbert spaces, which itself forms a 3-Hilbert space. We prove that the forgetful functor from 3-Hilbert spaces to 3-vector spaces is fully faithful.

math.QA↗

arXiv2024

Foundations for operator algebraic tricategories

An operator algebraic tricategory is a higher categorical analogue of an operator algebra. For algebraic tricategories, Gordon, Power, and Street proved that every algebraic tricategory is equivalent to a Gray-category, a result later refined by Gurski. We adapt this result to the context of functional analysis, showing that every operator algebraic tricategory is equivalent to an operator Gray-category. We then categorify the Gelfand-Naimark theorem for operator algebras, inductively proving that every (small) operator algebraic tricategory is equivalent to a concrete operator Gray-category. We also provide several examples of interest for operator algebraic tricategories.

math.OA↗

arXiv2024

Dagger $n$-categories

Category theory provides a unified language for organizing composable operations in many disciplines. In disciplines where unitarity is fundamental -- such as functional analysis, quantum field theory, and quantum logic -- this language must also capture adjoints, leading to the notion of dagger categories. Higher category theory, which extends this framework to encode operations between operations, has recently become indispensable in both theoretical physics and pure mathematics. Finding a higher categorical analogue of a dagger category is therefore key to the foundations of quantum field theory. In this work, we present a coherent definition of \emph{dagger $(\infty,n)$-category} in terms of equivariance data trivialized on parts of the category. Our main example is the bordism $(\infty,n)$-category $\mathbf{Bord}_{n}^X$. This allows us to define (fully-local) \emph{reflection-positive topological quantum field theories} to be higher dagger functors out of $\mathbf{Bord}_{n}^X$.

math.CT↗

arXiv2023

3D Koch-type crystals

We consider the construction of a family $\{K_N\}$ of $3$-dimensional Koch-type surfaces, with a corresponding family of $3$-dimensional Koch-type ``snowflake analogues" $\{\mathcal{C}_N\}$, where $N>1$ are integers with $N \not\equiv 0 \,(\bmod\,\, 3)$. We first establish that the Koch surfaces $K_N$ are $s_N$-sets with respect to the $s_N$-dimensional Hausdorff measure, for $s_N=\log(N^2+2)/\log(N)$ the Hausdorff dimension of each Koch-type surface $K_N$. Using self-similarity, one deduces that the same result holds for each Koch-type crystal $\mathcal{C}_N$. We then develop lower and upper approximation monotonic sequences converging to the $s_N$-dimensional Hausdorff measure on each Koch-type surface $K_N$, and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set $\mathcal{C}_N$. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals $\mathcal{C}_N$, for $N>2$.

math.MG↗

arXiv2022

Gray-categories model algebraic tricategories

Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences. In this note, we adapt the technique of Gurski, Johnson, and Osorno to show the localization of GrayCat at the weak equivalences is equivalent to the category of algebraic tricategories and pseudo-natural equivalence classes of weak 3-functors.

math.CT↗

arXiv2019

Harmonic Gradients on Higher Dimensional Sierpinski Gaskets

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped with the standard Dirichlet form and measure $\mu$ we show there is a full $\mu$-measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$, and that this set is not all of $SG_N$. We also show there is a class of non-uniform measures on the usual Sierpinski Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere, in sharp contrast to the case with the standard measure.

math.CA↗

arXiv2019

Classifying Module Categories for Generalized Temperley-Lieb-Jones *-2-Categories

Generalized Temperley-Lieb-Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong *-pseudofunctors into the *-2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to *-equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami's classification of fiber functors on TLJ categories and DeCommer and Yamashita's classification of unitary modules for Rep(SUq(2)).

math.OA↗