arXiv subjects
Publications and source records attributed to David Gao.
We prove the existence of strongly converging unitary representations in various new settings of countable groups, in particular, arising as extensions by exact groups: semidirect products with amenable groups; generalized wreath products with abelian base; graph wreath products; various free-by-cyclic groups including Gersten's group; and general Bernoulli shift crossed products on $C^*$-algebras.
We develop a general theory of selflessness for C*-correspondences, with several applications. Specializing this theory to completely positive maps, in particular to conditional expectations, gives rise to a novel notion of relative selflessness. Among applications, the machinery developed here yields new examples of selfless C*-algebras, for instance among minimal tensor products and reduced crossed products, new examples of MF C*-algebras arising as reduced amalgamated free products, a conceptually new proof of Kirchberg's $\mathcal{O}_{\infty}$-absorption theorem, and the lack of ``phantom'' traces on ultrapowers.
The network transport layer is increasingly implemented in the NIC hardware to meet the performance demands of modern workloads, but this has made it difficult to evolve or deploy new transport protocols. Existing approaches either fix protocol logic in the data-path or build protocol-specific assumptions into the architecture that limit the range of protocols that can be supported on a single hardware substrate. We present PITA, a protocol-independent transport architecture that enables full data-path programmability while sustaining line-rate performance. PITA eliminates protocol-specific assumptions by structuring the data-path around a uniform abstraction over events, state, and instructions, and rethinks core components, including scheduling, packet generation, and data reassembly, to operate on this abstraction. We evaluate PITA along key dimensions reflecting the goals of its protocol-agnostic datapath design. Specifically, we show that PITA supports diverse protocol semantics by showing it can implement TCP and \roce on the same data path and preserve their distinct end-to-end behavior. Through targeted microbenchmarks and synthesis on Alveo U250 cards, we show that PITA's redesigned components sustain high performance under demanding conditions, with modest hardware overhead and meeting timing at 250MHz.
We initiate the study of characters of surface groups and their corresponding tracial representations. We show that any tracial representation can be approximated arbitrarily well in the Wasserstein topology by factorial tracial representations with spectral gap. In particular, we deduce that the space of traces of a surface group is the Poulsen simplex, thereby resolving positively a question posed by Orovitz, Slutsky, and the third author.
We prove a new "Toeplitz exactness" theorem for strong convergence. This is a machine to upgrade strong convergence in the general setting of $C^\ast$-correspondences, and has several applications.
We find a general family of selfless inclusions in reduced amalgamated free products of C*-algebras. Apart from generalizing prior works due to McClanahan, Ivanov and Omland, our work yields a few other applications. We present a short new approach to construct HNN extensions of C*-algebras and find several new examples of selflessness using this. This generalizes results of Ueda, Ivanov and de la Harpe-Preaux. As another application our work yields a short proof of selflessness for arbitrary graph products of C*-algebras over graphs of more than 2 vertices and diameter greater than 3.
A countable group $G$ is said to be \emph{matricial field} (MF) if it admits a strongly converging sequence of approximate homomorphisms into matrices; i.e, the norms of polynomials converge to those in the left regular representation. $G$ is \emph{purely MF} (PMF) if these maps are actual homomorphisms, and $G$ is further \emph{purely finite field} (PFF) if the image of each homomorphism is finite. By developing a new operator algebraic approach to these problems, we are able to prove the following result bringing several new examples into the fold. Suppose $G$ is a MF (resp., PMF, PFF) group and $H<G$ is separable (i.e., $H=\cap_{i\in \mathbb{N}}H_i$ where $H_i<G$ are finite index subgroups) and $K$ is a residually finite MF (resp., PMF, PFF) group. If either $G$ or $K$ is exact, then the amalgamated free product $G*_{H}(H\times K)$ is MF (resp., PMF, PFF). Our work has several applications, we list some below: 1. The Brown--Douglas--Fillmore semigroups of many new examples of reduced group $C^*$-algebras are shown to be not groups. 2. Arbitrary group doubles $G*_HG$ of MF (resp., PMF, PFF) over separable subgroups $H$ are MF (resp., PMF, PFF). Moreover, $G*H$ is PFF whenever $G,H$ are PFF, and either $G$ or $H$ is exact. 3. Arbitrary graph products of residually finite exact MF (resp., PMF, PFF) groups are MF (resp., PMF, PFF), yielding a significant generalization of the breakthrough work of M. Magee and J. Thomas. 4. The open problem of proving PFF for fundamental groups of closed hyperbolic 3-manifolds is resolved. This has geometric significance in the theory of minimal surfaces via A. Song's approach.
Using a new approach involving embedding spaces in II$_1$ factors with plenty of freely independent Haar unitaries, we prove that $C^\ast_r(\mathbb{F}_n)\ncong C^\ast_r(\mathbb{F}_m)$ for $n \neq m$. This recovers the seminal result of Pimsner and Voiculescu with a short new proof.
We are able to explicitly compute the bimodule structure of von Neumann algebra inclusions in handle constructions, which arise as inductive limits of iterated amalgamated free products not elementarily equivalent to $L(\mathbb{F}_2)$. Our computation is achieved via identifying delicate normal form decompositions in amalgamated free products built in an iterated fashion. Using these techniques, we are able to show that the handles constructions are always full, without any need to appeal to Property (T) phenomena which was essential in all previous works. Furthermore our bimodule machinery works in the setting of arbitrary von Neumann algebras equipped with faithful normal states, yielding examples of full $\mathrm{III}_1$ factors via handle constructions.
Bacteria commonly inhabit porous environments such as host tissues, soil, and marine sediments, where complex geometries constrain and redirect their motion. Although bacterial motility has been studied in porous media, the roles of cell length and pore shape in navigating these environments remain poorly understood. Here, we investigate how cell morphology and pore architecture jointly determine bacterial spreading behavior. Using genetically engineered E. coli with tunable cell length, we performed single-cell tracking in microfluidic devices that mimic ordered and disordered porous structures. We find that elongated bacteria traverse ordered pore networks more effectively than short cells, exhibiting straighter paths, greater directional persistence, and enhanced exploration efficiency. In contrast, in disordered porous media, elongated bacteria become trapped in dead-end regions for extended periods, resulting in markedly reduced navigational efficiency. Together, these results reveal how cell shape and environmental geometry interact to govern bacterial transport. Moreover, we suggest a new mechanism for separating antimicrobial-resistant (AMR) bacteria from elongated susceptible cells in designer porous media.
We show that there are $\mathrm{II}_1$ factors $M$ and elementary embeddings $M \to M^{\mathcal{U}}$ which do not lift to sequences of UCP maps, and in fact $M$ can be chosen from any given elementary equivalence class. Furthermore, under continuum hypothesis, we show that in the sense of cardinality "most" automorphisms of a ultrapower $M^{\mathcal{U}}$ of a separable $\mathrm{II}_1$ factor do not lift to a sequence of UCP maps $\varphi_n: M \to M$.
We find a subtle modification to the construction of Chifan-Ioana-Kunnawalkam Elayavalli, which yields a drastic simplification of the proof of the existence of two non Gamma non elementarily equivalent II$_1$ factors.
Layered oxides exhibit high ionic mobility and chemical flexibility, attracting interest as cathode materials for lithium-ion batteries and the pairing of hydrogen production and carbon capture. Recently, layered oxides emerged as highly tunable semiconductors. For example, by introducing anti-Frenkel defects, the electronic hopping conductance in hexagonal manganites was increased locally by orders of magnitude. Here, we demonstrate local acceptor and donor doping in Er(Mn,Ti)O$_3$, facilitated by the splitting of such anti-Frenkel defects under applied d.c. voltage. By combining density functional theory calculations, scanning probe microscopy, atom probe tomography, and scanning transmission electron microscopy, we show that the oxygen defects readily move through the layered crystal structure, leading to nano-sized interstitial-rich (p-type) and vacancy-rich (n-type) regions. The resulting pattern is comparable to dipolar npn-junctions and stable on the timescale of days. Our findings reveal the possibility of temporarily functionalizing oxide semiconductors at the nanoscale, giving additional opportunities for the field of oxide electronics and the development of transient electronics in general.
We prove that for every $n\geq 2$, the reduced group $C^*$-algebras of the countable free groups $C^*_r(\mathbb{F}_n)$ have strict comparison. Our method works in a general setting: for $G$ in a large family of non-amenable groups, including hyperbolic groups, free products, mapping class groups, right-angled Artin groups etc., we have $C^*_r(G)$ have strict comparison. This work also has several applications in the theory of $C^*$-algebras including: resolving Leonel Robert's selflessness problem for $C^*_r(G)$; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence into $C^*_r(G)$; full computations of the Cuntz semigroup of $C^*_r(G)$ and future directions in the $C^*$-classification program.
We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as {\L}o\'s's theorem and countable saturation, to this more general setting.
We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is preserved by taking several natural graph join operations. We prove that an action of a group on its Cayley graph is sofic if and only if the group is sofic. We show that arbitrary actions of amenable groups on graphs are sofic. Using a graph theoretic result of E. Hrushovski, we also show that arbitrary actions of free groups on graphs are sofic. Notably we show that arbitrary actions of sofic groups on graphs, with amenable stabilizers, are sofic, settling completely an open problem from \cite{gao2024soficity}. We also show that soficity is preserved by taking limits under a natural Gromov-Hausdorff topology, generalizing prior work of the first author \cite{gao2024actionslerfgroupssets}. Our work sheds light on a family of groups called graph wreath products, simultaneously generalizing graph products and generalized wreath products. Extending various prior results in this direction including soficity of generalized wreath products \cite{gao2024soficity}, B. Hayes and A. Sale \cite{HayesSale}, and soficity of graph products \cite{CHR, charlesworth2021matrix}, we show that graph wreath products are sofic if the action and acting groups are sofic. These results provide several new examples of sofic groups in a systematic manner.
In this note we study a family of graphs of groups over arbitrary base graphs where all vertex groups are isomorphic to a fixed countable sofic group $G$, and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are identical everywhere. We prove soficity for this family of groups under a flexible technical hypothesis for $H$ called $\sigma$-co-sofic. This proves soficity for group doubles $*_H G$, where $H<G$ is an arbitrary separable subgroup and $G$ is countable and sofic. This includes arbitrary finite index group doubles of sofic groups among various other examples.
We extract a precise internal description of the sequential commutation equivalence relation introduced in [KEP23] for tracial von Neumann algebras. As an application we prove that if a tracial von Neumann algebra $N$ is generated by unitaries $\{u_i\}_{i\in \mathbb{N}}$ such that $u_i\sim u_j$ (i.e, there exists a finite set of Haar unitaries $\{w_i\}_{i=1}^{n}$ in $N^\mathcal{U}$ such that $[u_i, w_1]= [w_k, w_{k+1}]=[w_n,u_j]=0$ for all $1\leq k< n$) then $N$ is singly generated. This generalizes and recovers several known single generation phenomena for II$_1$ factors in the literature with a unified proof.