arXiv subjects
Publications and source records attributed to Gabor Szabo.
We show that two simple, separable, nuclear and $\mathcal{Z}_0$-stable $\mathrm{C}^\ast$-algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the UCT and can be viewed as a tracial stably projectionless analog of the homotopy rigidity theorem for Kirchberg algebras.
We investigate the notion of tracial $\mathcal Z$-stability beyond unital C*-algebras, and we prove that this notion is equivalent to $\mathcal Z$-stability in the class of separable simple nuclear C*-algebras.
Ion acceleration resulting from the interaction of 11 fs laser pulses of ~35 mJ energy with ultrahigh contrast (<10^-10), and 10^19 W/cm^2 peak intensity with foil targets made of various materials and thicknesses at normal (0-degree) and 45-degree laser incidence is investigated. The maximum energy of the protons accelerated from both the rear and front sides of the target was above 1 MeV. A conversion efficiency from laser pulse energy to proton beam is estimated to be as high as ~1.4 % at 45-degree laser incidence using a 51 nm-thick Al target. The excellent laser contrast indicates the predominance of vacuum heating via the Brunels effect as an absorption mechanism involving a tiny pre-plasma of natural origin due to the Gaussian temporal laser pulse shape. Experimental results are in reasonable agreement with theoretical estimates where proton acceleration from the target rear into the forward direction is well explained by a TNSA-like mechanism, while proton acceleration from the target front into the backward direction can be explained by the formation of a charged cavity in a tiny pre-plasma. The exploding Coulomb field from the charged cavity also serves as a source for forward-accelerated ions at thick targets.
We provide the rigorous foundations for a categorical approach to the classification of C*-dynamics up to cocycle conjugacy. Given a locally compact group $G$, we consider a category of (twisted) $G$-C*-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of $G$-C*-algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying C*-algebras. This reduces a given classification problem for C*-dynamics to the prevalence of certain uniqueness and existence theorems, and may provide a useful alternative to the Evans--Kishimoto intertwining argument in future research.
We revisit Matui-Sato's notion of property (SI) for C*-algebras and C*-dynamics. More specifically, we generalize the known framework to the case of C*-algebras with possibly unbounded traces. The novelty of this approach lies in the equivariant context, where none of the previous work allows one to (directly) apply such methods to actions of amenable groups on highly non-unital C*-algebras, in particular to establish equivariant Jiang-Su stability. Our main result is an extension of an observation by Sato: For any countable amenable group $Γ$ and any non-elementary separable simple nuclear C*-algebra $A$ with strict comparison, every $Γ$-action on $A$ has equivariant property (SI). A more general statement involving relative property (SI) for inclusions into ultraproducts is proved as well. As a consequence we show that if $A$ also has finitely many rays of extremal traces, then every $Γ$-action on $A$ is equivariantly Jiang-Su stable. We moreover provide applications of the main result to the context of strongly outer actions, such as a generalization of Nawata's classification of strongly outer automorphisms on the (stabilized) Razak-Jacelon algebra.
We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.
Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C$^*$-algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliott's program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property $Γ$, which under minimality confirms the Toms-Winter conjecture for such crossed products by the aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.
We introduce and study strongly self-absorbing actions of locally compact groups on C*-algebras. This is an equivariant generalization of a strongly self-absorbing C*-algebra to the setting of C*-dynamical systems. The main result is the following equivariant McDuff-type absorption theorem: A cocycle action $(α,u): G\curvearrowright A$ on a separable C*-algebra is cocycle conjugate to its tensorial stabilization with a strongly self-absorbing action $γ: G\curvearrowright\mathcal{D}$, if and only if there exists an equivariant and unital $*$-homomorphism from $\mathcal{D}$ into the central sequence algebra of $A$. We also discuss some non-trivial examples of strongly self-absorbing actions.
Different types of concave plasmonic nanoresonators have been optimized to achieve superradiantly enhanced emission of SiV color centers in diamond. Comparative study has been performed to consider advantages of different N number of SiV color centers, different diamond-silver (bare) and diamond-silver-diamond (coated) core-shell nanoresonator types, as well as of spherical and ellipsoidal geometry. The complete fluorescence enhancement (qualified by Px factor) monitoring and the cQE corrected quantum efficiency weighted PxcQE objective function optimization promotes to design bad-cavities for plasmonic Dicke effect. The switching into a collective Dicke state via optimized nanoresonators results in a radiated power proportional to N^2, which manifest itself in an enhancement proportional to N both of the excitation and emission rates. Accordingly, enhancement proportional to N^2 of the Px factor and PxcQE has been reached both via four and six SiV color centers arranged in symmetrical square and hexagonal patterns inside all types of inspected nanoresonators. Coated spherical and bare ellipsoidal nanoresonators result in stronger non-cooperative fluorescence enhancement, while superradiance is better achieved via bare spherical nanoresonators independently of SiV color centers number, and via coated (bare) ellipsoidal nanoresonators seeded by four (six) SiV color centers. Indistinguishable superradiant state of four color centers and line-width narrowing is achieved via bare nanoresonators. Six color centers seeded bare spherical (ellipsoidal) nanoresonators result in larger fluorescence enhancement and more significantly overridden superradiance thresholds, while having slightly more (less) pronounced bad-cavity characteristics. Both phenomena are simultaneously optimized in ellipsoidal bare nanoresonators embedding six color centers with a slightly larger detuning.
In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing C*-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $\mathcal D$ is a strongly self-absorbing C*-algebra, and $α: G\curvearrowright A$ is an action on a separable, $\mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $α$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $\mathcal D$. This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some non-trivial applications. Most notably it is shown that for any $k\geq 1$ and on any strongly self-absorbing Kirchberg algebra, there exists a unique $\mathbb R^k$-action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.
In this paper we consider a bootstrap class $\mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$\mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Γ\in\mathfrak C$ and any strongly self-absorbing C*-algebra $\mathcal D$: (1) There is a unique strongly outer $Γ$-action on $\mathcal D$ up to (very strong) cocycle conjugacy. (2) If $α: Γ\curvearrowright A$ is a strongly outer action on a separable, unital, nuclear, simple, $\mathcal D$-stable C*-algebra with at most one trace, then it absorbs every $Γ$-action on $\mathcal D$ up to (very strong) cocycle conjugacy. In fact we establish more general relative versions of these two results for actions of amenable groups that have a predetermined quotient in the class $\mathfrak C$. For the monotracial case, the proof comprises an application of Matui--Sato's equivariant property (SI) as a key method.
We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $\mathcal{O}_\infty$ and $\mathcal{O}_2$. This generalizes results known for finite groups and poly-$\mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a precursor to a general Kirchberg-Phillips-type classification theory: If two outer actions of an amenable group on a unital Kirchberg algebra are equivariantly homotopy equivalent, then they are conjugate. This marks the first C*-dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups.
We prove that every $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,\mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}\times Y,σ\times S)$, where $D\in\mathbb{N}$, $L\in\mathbb{N}\cup\{0\}$ and $σ$ is the shift on the Hilbert cube $([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}$; in particular, when $(Y,\mathbb{Z}^{k},S)$ is an irrational $\mathbb{Z}^{k}$-rotation on the $k$-torus, $(X,\mathbb{Z}^{k},T)$ embeds in $(([0,1]^{2^kD+1})^{\mathbb{Z}^k},σ)$, which is compared to a previous result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we give a complete and detailed proof of Takens' embedding theorem with a continuous observable for $\mathbb{Z}$-actions and deduce the analogous result for $\mathbb{Z}^{k}$-actions. Lastly, we show that the Lindenstrauss--Tsukamoto conjecture for $\mathbb{Z}$-actions holds generically, discuss an analogous conjecture for $\mathbb{Z}^{k}$-actions appearing in a forthcoming paper by the first two authors and Tsukamoto and verify it for $\mathbb{Z}^{k}$-actions on finite dimensional spaces.
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy. Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly $\cal Z$-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi-Matui, we obtain the following main result. If $G$ is a torsion-free abelian group and $\cal D$ is one of the known strongly self-absorbing C*-algebras, then strongly outer $G$-actions on $\cal D$ are unique up to (very strong) cocycle conjugacy. This is new even for $\mathbb{Z}^3$-actions on the Jiang-Su algebra.
Configuration of three different concave silver core-shell nanoresonators was numerically optimized to enhance the excitation and emission of embedded silicon vacancy (SiV) diamond color centers simultaneously. According to the tradeoff between the radiative rate enhancement and quantum efficiency (QE) conditional optimization was performed to ensure ~2-3-4 and 5-fold apparent cQE enhancement of SiV color centers with ~10% intrinsic QE. The enhancement spectra, as well as the near-field and charge distribution were inspected to uncover the physics underlying behind the optical responses. The conditionally optimized coupled systems were qualified by the product of the radiative rate enhancements at the excitation and emission, which is nominated as Px factor. The optimized spherical core-shell nanoresonator containing a centralized emitter is capable of enhancing considerably the emission via bonding dipolar resonance. The Px factor is 529-fold with 49.7% cQE at the emission. Decentralization of the emitter leads to appearance of higher order multipolar modes, which is not advantageous caused by their nonradiative nature. Transversal and longitudinal dipolar resonances of the optimized ellipsoidal core-shell resonator were tuned to the excitation and emission, respectively. The simultaneous enhancements result in 6.2x10^5 Px factor with 50.6% cQE at the emission. Rod-shaped concave core-shell nanoresonators exploit similarly transversal and longitudinal dipolar resonances, moreover they enhance the fluorescence more significantly due to their antenna-like geometry. Px factor of 8.34x10^5 enhancement is achievable while the cQE is 50.3% at the emission. The enhancement can result in 2.03x10^6-fold Px factor, when the criterion regarding the minimum QE is set to 20%.
This is a continuation of the study of strongly self-absorbing actions of locally compact groups on C*-algebras. Given a strongly self-absorbing action $γ: G\curvearrowright\mathcal{D}$, we establish permanence properties for the class of separable C*-dynamical systems absorbing $γ$ tensorially up to cocycle conjugacy. Generalizing results of both Toms-Winter and Dadarlat-Winter, it is proved that the desirable equivariant analogues of the classical permanence properties hold in this context. For the permanence with regard to equivariant extensions, we need to require a mild extra condition on $γ$, which replaces $K_1$-injectivity assumptions in the classical theory. This condition turns out to be automatic for equivariantly Jiang-Su absorbing C*-dynamical systems, yielding a large class of examples. It is left open whether this condition is redundant for all strongly self-absorbing actions, and we do consider examples that satisfy this condition but are not equivariantly Jiang-Su absorbing.
We introduce the concept of Rokhlin dimension for actions of residually finite groups on C*-algebras, extending previous notions of Rokhlin dimension for actions of finite groups and the integers, as introduced by Hirshberg, Winter and the third author. If the group has a box space of finite asymptotic dimension, then actions with finite Rokhlin dimension preserve the property of having finite nuclear dimension, when passing to the crossed product C*-algebra. A detailed study of the asymptotic dimension of box spaces shows that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a reasonably large class of examples. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group C*-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^m$-actions by the first author. We also provide some results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing C*-algebra.
We show that separable, nuclear and strongly purely infinite C*-algebras have finite nuclear dimension. In fact, the value is at most three. This exploits a deep structural result of Kirchberg and Rørdam on strongly purely infinite C*-algebras that are homotopic to zero in an ideal-system preserving way.