arXiv subjects

Anna Duwenig

Publications and source records attributed to Anna Duwenig.

16 recordsLinked to original sources

arXiv2026

Cartan subalgebras in self-similar graph $C^*$-algebras

For a self-similar graph $(G, E)$, we find a distinguished subgroupoid of the associated path groupoid $\mathcal{G}_{G,E}$ -- the symmetric cycline subgroupoid $\mathcal{S}_{\text{sym}}$. If the acting group $G$ is abelian, we show that $\mathcal{S}_{\text{sym}}$ is open, abelian, and normal. For $G=\mathbb{Z}$, we describe the dual bundle $\hat{\mathcal{S}}_{\text{sym}}$ of $\mathcal{S}_{\text{sym}}$ which can be used to provide a different groupoid model for the self-similar graph $C^*$-algebra $\mathcal{O}_{\mathbb{Z}, E}\cong C^*_r(\mathcal{G}_{\mathbb{Z},E})$. For a large class of self-similar graphs $(\mathbb{Z}, E)$, we further prove that $\mathcal{S}_{\text{sym}}$ is maximal among open abelian subgroupoids of $\mathrm{Iso}(\mathcal{G}_{\mathbb{Z},E})^{\circ}$ and closed in $\mathcal{G}_{\mathbb{Z},E}$, so that it gives rise to a Cartan subalgebra of $\mathcal{O}_{\mathbb{Z}, E}$. This result seems new even for genuine actions. Our proofs heavily rely on careful studies of dynamical behaviours of cycline triples of $(\mathbb{Z}, E)$ and on a dynamical-flavour classification for the vertices of $E$. Some results hold in more general settings and may be of independent interest.

math.OA↗

arXiv2025

Non-traditional C-diagonals in twisted groupoid C-algebras

We identify which conditions on an open normal subgroupoid of a LCH étale groupoid with twist are necessary and sufficient for the subgroupoid's reduced twisted C*-algebra to be a C*-diagonal in the ambient groupoid C*-algebra. We do so by first giving an explicit description of the Weyl groupoid and Weyl twist associated to any non-traditional Cartan subalgebra, that is, a Cartan subalgebra that is induced from a non-trivial open normal subgroupoid, as studied in [DWZ2025]. We then combine this description with Kumjian-Renault theory to establish the necessary and sufficient conditions to get a C*-diagonal.

math.OA↗

arXiv2025

The Operator Algebras Mentor Network: Impact of Community-Based Mentoring

This paper aims to determine if membership within the Operator Algebras Mentor Network (OAMN) is beneficial to its members. The OAMN is an international mentoring initiative that offers support in small groups to women and minority genders in the particularly male-dominated field of operator algebras (OA) in mathematics. Expected advantages of membership include raising awareness of the lack of gender diversity in this field, providing advice to mentees by mentors (e.g., pertaining to career or work/life balance), broadening one's network in OA, etc. A questionnaire was sent to OAMN members and a control group of non-members at similar institutions and similar positions to collect their experience with the mentoring initiative and perception of gender dynamics within the OA discipline, together with basic demographics. The initial analysis of the data collected shows that mentoring junior women and other minority genders in the area has a positive effect on mentees' networking ability, self-promotion, and raising awareness of gender issues within OA as a whole.

math.OA↗

arXiv2025

Non-traditional Cartan subalgebras in twisted groupoid C*-algebras

Well-known work of Renault shows that if $\mathcal{E}$ is a twist over a second countable, effective, étale groupoid $G$, then there is a naturally associated Cartan subalgebra of the reduced twisted groupoid C*-algebra $C^*_{r}(G; E)$, and that every Cartan subalgebra of a separable C*-algebra arises in this way. However twisted C*-algebras of non-effective groupoids $G$ can also possess Cartan subalgebras: In work by the first author together with Gillaspy, Norton, Reznikoff, and Wright, sufficient conditions on a subgroupoid $S$ of $G$ were found that ensure that $S$ gives rise to a Cartan subalgebra in the cocycle-twisted C*-algebra of $G$. In this paper, we extend these results to general twists $\mathcal{E}$, and we refine the conditions on the subgroupoid for $C^*_{r}(S;\mathcal{E}_S)$ to be a Cartan subalgebra of $C^*_{r}(G;\mathcal{E})$.

math.OA↗

arXiv2024

Norm upper-semicontinuity of functions supported on open abelian isotropy in étale groupoids (a corrigendum to "Reconstruction of groupoids and C*-rigidity of dynamical systems," Adv. Math 390 (2021), 107923)

We consider étale Hausdorff groupoids in which the interior of the isotropy is abelian. We prove that the norms of the images under regular representations, of elements of the reduced groupoid $C^*$-algebra whose supports are contained in the interior of the isotropy vary upper semicontinuously. This corrects an error in [T.M. Carlsen, E. Ruiz, A. Sims and M. Tomforde, "Reconstruction of groupoids and C*-rigidity of dynamical systems," Adv. Math 390 (2021), 107923].

math.OA↗

arXiv2024

The Zappa-Szép product of twisted groupoids

We define and study the external and the internal Zappa-Szép product of twists over groupoids. We determine when a pair $(Σ_{1},Σ_{2})$ of twists over a matched pair $(\mathcal{G}_{1},\mathcal{G}_{2})$ of groupoids gives rise to a Zappa-Szép twist $Σ$ over the Zappa-Szép product $\mathcal{G}_{1}\bowtie\mathcal{G}_{2}$. We prove that the resulting (reduced and full) twisted groupoid C*-algebra of the Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ is a C*-blend of its subalgebras corresponding to the subtwists $Σ_{i}\to \mathcal{G}_{i}$. Using Kumjian-Renault theory, we then prove a converse: Any C*-blend in which the intersection of the three algebras is a Cartan subalgebra in all of them, arises as the reduced twisted groupoid C*-algebras from such a Zappa-Szép twist $Σ\to \mathcal{G}_{1}\bowtie\mathcal{G}_{2}$ of two twists $Σ_{1}\to \mathcal{G}_{1}$ and $Σ_{2}\to \mathcal{G}_{2}$.

math.OA↗

arXiv2024

Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids

We describe how to recover a Lie structure on a twist over a Hausdorff étale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff étale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We use these results in the setting of twists over étale groupoids to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections.

math.OA↗

arXiv2024

Alexandrov groupoids and the nuclear dimension of twisted groupoid $\mathrm{C}^*$-algebras

We consider a twist $E$ over an étale groupoid $G$. When $G$ is principal, we prove that the nuclear dimension of the reduced twisted groupoid $\mathrm{C}^*$-algebra is bounded by a number depending on the dynamic asymptotic dimension of $G$ and the topological covering dimension of its unit space. This generalizes an analogous theorem by Guentner, Willett, and Yu for the $\mathrm{C}^*$-algebra of $G$. Our proof uses a reduction to the unital case where $G$ has compact unit space, via a construction of ``groupoid unitizations'' $\widetilde{G}$ and $\widetilde{E}$ of $G$ and $E$ such that $\widetilde{E}$ is a twist over $\widetilde{G}$. The construction of $\widetilde G$ is for r-discrete (hence étale) groupoids $G$ which are not necessarily principal. When $G$ is étale, the dynamic asymptotic dimension of $G$ and $\widetilde{G}$ coincide. We show that the minimal unitizations of the full and reduced twisted groupoid $\mathrm{C}^*$-algebras of the twist over $G$ are isomorphic to the twisted groupoid $\mathrm{C}^*$-algebras of the twist over $\widetilde{G}$. We apply our result about the nuclear dimension of the twisted groupoid $\mathrm{C}^*$-algebra to obtain a similar bound on the nuclear dimension of the $\mathrm{C}^*$-algebra of an étale groupoid with closed orbits and abelian stability subgroups that vary continuously.

math.OA↗

arXiv2023

The Imprimitivity Fell Bundle

Given a full right-Hilbert C*-module $\mathbf{X}$ over a C*-algebra $A$, the set $\mathbb{K}_{A}(\mathbf{X})$ of $A$-compact operators on $\mathbf{X}$ is the (up to isomorphism) unique C*-algebra that is strongly Morita equivalent to the coefficient algebra $A$ via $\mathbf{X}$. As bimodule, $\mathbb{K}_{A}(\mathbf{X})$ can also be thought of as the balanced tensor product $\mathbf{X}\otimes_{A} \mathbf{X}^{\mathrm{op}}$, and so the latter naturally becomes a C*-algebra. We generalize both of these facts to the world of Fell bundles over groupoids: Suppose $\mathscr{B}$ is a Fell bundle over a groupoid $\mathcal{H}$ and $\mathscr{M}$ an upper semi-continuous Banach bundle over a principal right $\mathcal{H}$-space $X$. If $\mathscr{M}$ carries a right-action of $\mathscr{B}$ and a sufficiently nice $\mathscr{B}$-valued inner product, then its imprimitivity Fell bundle $\mathbb{K}_{\mathscr{B}}(\mathscr{M})=\mathscr{M}\otimes_{\mathscr{B}} \mathscr{M}^{\mathrm{op}}$ is a Fell bundle over the imprimitivity groupoid of $X$, and it is the unique Fell bundle that is equivalent to $\mathscr{B}$ via $\mathscr{M}$. We show that $\mathbb{K}_{\mathscr{B}}(\mathscr{M})$ generalizes the 'higher order' compact operators of Abadie and Ferraro in the case of saturated bundles over groups, and that the theorem recovers results such as Kumjian's Stabilization trick.

math.OA↗

arXiv2023

Imprimitivity theorems and self-similar actions on Fell bundles

We introduce the notion of self-similar actions of grouopids on other groupoids and Fell bundles. This leads to a new imprimitivity theorem arising from such dynamics, generalizing many earlier imprimitivity theorems involving group and groupoid actions.

math.OA↗

arXiv2022

Equivalence of Fell bundles is an equivalence relation

We introduce the notion of groupoid pre-equivalences and prove that they give rise to groupoid equivalences by taking certain quotients. Then, given an equivalence of Fell bundles $\mathscr{B}$ and $\mathscr{C}$ and another equivalence between $\mathscr{C}$ and $\mathscr{D}$, we construct an equivalence between $\mathscr{B}$ and $\mathscr{D}$ out of the tensor product bundle. As a consequence, we obtain that Fell bundle equivalence is indeed an equivalence relation.

math.OA↗

arXiv2022

Renault's $j$-map for Fell bundle $C^*$-algebras

If $p \colon \mathcal B\to G$ is a Fell bundle over an étale groupoid, then we show that there is an norm reducing injective linear map $j \colon C^*_r(G;\mathcal B)\to Γ_{0}(G;\mathcal B)$ generalizing the well know map $j \colon C^*_{r}(G)\to C_{0}(G)$ in the case of an étale groupoid.

math.OA↗

arXiv2021

Zappa-Szép product of a Fell bundle and a groupoid

We define the Zappa-Szép product of a Fell bundle by a groupoid, which turns out to be a Fell bundle over the Zappa-Szép product of the underlying groupoids. Under certain assumptions, every Fell bundle over the Zappa-Szép product of groupoids arises in this manner. We then study the representation associated with the Zappa-Szép product Fell bundle and show its relation to covariant representations. Finally, we study the associated universal C*-algebra, which turns out to be a C*-blend, generalizing an earlier result about the Zappa-Szép product of groupoid C*-algebras. In the case of discrete groups, the universal C*-algebra of a Fell bundle embeds injectively inside the universal C*-algebra of the Zappa-Szép product Fell bundle.

math.OA↗

arXiv2020

Cartan subalgebras for non-principal twisted groupoid $C^*$-algebras

Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, Σ)$ for any twist $Σ$ over $G$. However, there are many groupoids which are not topologically principal, yet their (twisted) $C^*$-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle $c$ on $G$ and a subgroupoid $S \subseteq G$ under which $C^*_r(S, c)$ is Cartan in $C^*_r(G, c)$. When $G$ is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

math.OA↗

arXiv2020

Transversals, duality, and irrational rotation

An early result of Noncommutative Geometry was Connes' observation in the 1980's that the Dirac-Dolbeault cycle for the $2$-torus $\mathbb{T}^2$, which induces a Poincaré self-duality for $\mathbb{T}^2$, can be 'quantized' to give a spectral triple and a K-homology class in $KK_0(A_θ\otimes A_θ, \mathbb{C})$ providing the co-unit for a Poincaré self-duality for the irrational rotation algebra $A_θ$ for any $θ\in \mathbb{R}\setminus \mathbb{Q}$. This spectral triple has been extensively studied since. Connes' proof, however, relied on a K-theory computation and does not supply a representative cycle for the unit of this duality. Since such representatives are vital in applications of duality, we supply such a cycle in unbounded form in this article. Our approach is to construct, for any non-trivial element $g$ of the modular group, a finitely generated projective module $\mathcal{L}_g$ over $A_θ\otimes A_θ$ by using a reduction-to-a-transversal argument of Muhly, Renault, and Williams, applied to a pair of Kronecker foliations along lines of slope $θ$ and $g(θ)$, using the fact that these flows are transverse to each other. We then compute Connes' dual of $[\mathcal{L}_g]$ for $g$ upper triangular, and prove that we obtain an invertible in $KK_0(A_θ, A_θ)$, represented by what one might regard as a noncommutative bundle of Dirac-Schrödinger operators. An application of $\mathbb{Z}$-equivariant Bott Periodicity proves that twisting the module by the family gives the requisite spectral cycle for the unit, thus proving self-duality for $A_θ$ with both unit and co-unit represented by spectral cycles.

math.KT↗

arXiv2019

Elliptic Operators and K-Homology

If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of square integrable sections of $S$ with the canonical left $C(M)$-action and the operator $χ(D)$ for $χ$ a normalizing function is a Fredholm module, and its $K$-homology class is independent of $χ$. In this expository article, we provide a detailed proof of this fact following the outline in the book "Analytic K-homology" by Higson and Roe.

math.KT↗