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Aluthge transform is a well-known mapping defined on bounded linear operators. Especially, the convergence property of its iteration has been studied by many authors. In this paper, we discuss the problem for the induced Aluthge transforms which is a generalization of the Aluthge transform defined in 2021. We give the polar decomposition of the induced Aluthge transformations of centered operators and show its iteration converges to a normal operator. In particular, if $T$ is an invertible centered matrix, then iteration of any induced Aluthge transformations converges. Using the canonical standard form of matrix algebras we show that the iteration of any induced Aluthge transformations with respect to the weighted arithmetic mean and the power mean converge. Those observation are extended to the $C^*$-algebra of compact operators on an infinite dimensional Hilbert space, and as an application we show the stability of $\mathcal{AN}$ and $\mathcal{AM}$ properties under the iteration of the induced Aluthge transformations. We also provide concrete forms of their limit points for centered matrices and several examples. Moreover, we discuss the limit point of the induced Aluthge transformation with respect to the power mean in the injective $II_1$-factor $\mathcal{M}$ and determine the form of its limit for some centered operators in $\mathcal{M}$., Comment: 32 pages, to appear in Trans. Amer. Math. Soc

In this note, we provide several examples of inclusions of unital $C\sp*$-algebras of index finite-type with the Rokhlin property lacking in literature and more importantly we construct the example which does not belong to the fixed point algebra model which means that for the inclusion $P \subset A$ there is neither finite group action $ G \curvearrowright A$ such that $P=A^{G}$ nor finite dimensional $C\sp*$- Hopf algebra action $H \curvearrowright A$ such that $P=A^{H}$., Comment: 12pages, 1 figure

We demonstrate that the absorption of femtosecond x-ray pulses can excite quasi-spherical high-wavevector coherent acoustic phonon wavepackets using an all x-ray pump and probe scattering experiment. The time- and momentum-resolved diffuse scattering signal is consistent with strain pulses induced by the rapid electron cascade dynamics following photoionization at uncorrelated excitation centers. We quantify key parameters of this process, including the localization size of the strain wavepacket and the energy absorption efficiency, which are determined by the photoelectron and Auger electron cascade dynamics, as well as the electron-phonon interaction. In particular, we obtain the localization size of the observed strain wave packet to be 1.5 and 2.5 nm for bulk SrTiO$_3$ and KTaO$_3$ single crystals, even though there are no nanoscale structures or light-intensity patterns that would ordinarily be required to generate acoustic waves of wavelengths much shorter than the penetration depth. Whereas in GaAs and GaP we do not observe a signal above background. The results provide crucial information on x-ray matter interactions, which sheds light on the mechanism of x-ray energy deposition, and the study of high wavevector acoustic phonons and thermal transport at the nanoscale.

X-ray laser-induced structural changes in silicon undergoing femtosecond melting have been investigated by using an x-ray pump-x-ray probe technique. The experimental results for different initial sample temperatures reveal that the onset time and the speed of the atomic disordering are independent of the initial temperature, suggesting that equilibrium atomic motion in the initial state does not play a pivotal role in the x-ray-induced ultrafast melting. By comparing the observed time-dependence of the atomic disordering and the dedicated theoretical simulations, we interpret that the energy transfer from the excited electrons to ions via electron-ion coupling (thermal effect) as well as a strong modification of the interatomic potential due to electron excitations (non-thermal effect) trigger the ultrafast atomic disordering. Our finding of the interplay of thermal and non-thermal effects in the x-ray-induced melting demonstrates that accurate modeling of intense x-ray interactions with matter is essential to ensure a correct interpretation of experiments using intense x-ray laser pulses.

X-ray diffraction of silicon irradiated with tightly focused femtosecond x-ray pulses (photon energy: 11.5 keV, pulse duration: 6 fs) was measured at various x-ray intensities up to $4.6\times10^{19}$ W/cm$^2$. The measurement reveals that the diffraction intensity is highly suppressed when the x-ray intensity reaches of the order of $10^{19}$ W/cm$^2$. With a dedicated simulation, we confirm the observed reduction of the diffraction intensity is attributed to the femtosecond change in individual atomic scattering factors due to the ultrafast creation of highly ionized atoms through photoionization, Auger decay, and subsequent collisional ionization. We anticipate that this ultrafast reduction of atomic scattering factor will be a basis for new x-ray nonlinear techniques, such as pulse shortening and contrast variation x-ray scattering.

Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. This study demonstrates that the set of functions satisfying this inequality includes additional elements and provides illustrative examples to support this claim. Furthermore, we present a characterization of matrix decreasing functions based on a matrix version of the above inequality.

We systematically synthesized perovskite-type oxides Sr1-xCaxCoO3 containing unusually high valence Co4+ ions by a high pressure technique, and investigated the effect of systematic lattice change on the magnetic and electronic properties. As the Ca content x exceeds about 0.6, the structure changes from cubic to orthorhombic, which is supported by the first-principles calculations of enthalpy. Upon the orthorhombic distortion, the ground state remains to be apparently ferromagnetic with a slight drop of the Curie temperature. Importantly, the compounds with x larger than 0.8 show antiferromagnetic behavior with positive Weiss temperatures and nonlinear magnetization curves at lowest temperature, implying that the ground state is noncollinear antiferromagnetic or helimagnetic. Considering the incoherent metallic behavior and the suppression of the electronic specific heat at high x region, the possible emergence of a helimagnetic state in Sr1-xCaxCoO3 is discussed in terms of the band-width narrowing and the double-exchange mechanism with the negative charge transfer energy as well as the spin frustration owing to the next-nearest neighbor interaction., Comment: 13 pages, 4 figures

We report ultrafast x-ray scattering experiments of the quasi-1D charge density wave (CDW) material (TaSe$_4$)$_2$I following photoexcitation with femtosecond infrared laser pulses. From the time-dependent diffraction signal at the CDW sidebands we identify an amplitude mode derived primarily from the transverse acoustic component of the CDW static distortion. The dynamics of this acoustic amplitude mode are described well by a model of a displacive excitation, which we interpret as mediated through a coupling to the optical phonon component associated with the tetramerization of the Ta chains.

Coherent nonlinear spectroscopies and imaging in the X-ray domain provide direct insight into the coupled motions of electrons and nuclei with resolution on the electronic length and time scale. The experimental realization of such techniques will strongly benefit from access to intense, coherent pairs of femtosecond X-ray pulses. We have observed phase-stable X-ray pulse pairs containing more thank 3 x 10e7 photons at 5.9 keV (2.1 Angstrom) with about 1 fs duration and 2-5 fs separation. The highly directional pulse pairs are manifested by interference fringes in the superfluorescent and seeded stimulated manganese K-alpha emission induced by an X-ray free-electron laser. The fringes constitute the time-frequency X-ray analogue of the Young double-slit interference allowing for frequency-domain X-ray measurements with attosecond time resolution., Comment: 39 pages, 13 figures, to be published

We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2., Comment: 15 pages. We rewrite Section 4 and Section 5 of the first version

We present a general characterization of k-positivity for a positive map in terms of the estimation of the Ky Fan norm of the matrix constructed from the Kraus operators of the associated completely positive map. Combining this with the result given by Takasaki and Tomiyama we construct a family of positive maps between matrix algebras of different dimensions depending on a parameter. The estimate bounds on the parameter to obtain the $k$-positivity are better than those derived from the spectral conditions considered by Chru\'sci\'nski and Kossakowski. We further look with special attention at the case where we give the precise bound for the regions of decomposability., Comment: 10 pages

To shorten the duration of x-ray pulses, we present a nonlinear optical technique using atoms with core-hole vacancies (core-hole atoms) generated by inner-shell photoionization. The weak Coulomb screening in the core-hole atoms results in decreased absorption at photon energies immediately above the absorption edge. By employing this phenomenon, referred to as saturable absorption, we successfully reduce the duration of x-ray free-electron laser pulses (photon energy: 9.000 keV, duration: 6-7 fs, fluence: 2.0-3.5$\times$10$^5$ J/cm$^2$) by $\sim$35%. This finding that core-hole atoms are applicable to nonlinear x-ray optics is an essential stepping stone for extending nonlinear technologies commonplace at optical wavelengths to the hard x-ray region.

We report the observation of a novel nonlinear effect in the hard x-ray range. Upon illuminating Fe and Cu metal foils with intense x-ray pulses tuned near their respective K edges, photons at nearly twice the incoming photon energy are emitted. The signal rises quadratically with the incoming intensity, consistent with two-photon excitation. The spectrum of emitted high-energy photons comprises multiple Raman lines that disperse with the incident photon energy. Upon reaching the double K-shell ionization threshold, the signal strength undergoes a marked rise. Above this threshold, the lines cease dispersing, turning into orescence lines with energies much greater than obtainable by single electron transitions, and additional Raman lines appear. We attribute these processes to electron-correlation mediated multielectron transitions involving double-core hole excitation and various two-electron de-excitation processes to a final state involving one or more L and M core-holes., Comment: 6 pages, 4 figures

With their brilliance and temporal structure, X-ray free-electron laser can unveil atomic-scale details of ultrafast phenomena. Recent progress in split-and-delay optics (SDO), which produces two X-ray pulses with time-delays, offers bright prospects for observing dynamics at the atomic-scale. However, their insufficient pulse energy has limited its application either to phenomena with longer correlation length or to measurement with a fixed delay-time. Here we show that the combination of the SDO and self-seeding of X-rays increases the pulse energy and makes it possible to observe the atomic-scale dynamics in a timescale of picoseconds. We show that the speckle contrast in scattering from water depends on the delay-time as expected. Our results demonstrate the capability of measurement using the SDO with seeded X-rays for resolving the dynamics in temporal and spatial scales that are not accessible by other techniques, opening opportunities for studying the atomic-level dynamics., Comment: 22 pages, 5 figures

A nanofocusing optical system referred to as $\textit{100 exa}$ for an X-ray free-electron laser (XFEL) was developed to generate an extremely high intensity of 100 EW/cm$^2$ (10$^2$$^0$ W/cm$^2$) using total reflection mirrors. The system is based on Kirkpatrick-Baez geometry, with 250 mm long elliptically figured mirrors optimized for the SPring-8 Angstrom Compact Free-Electron Laser (SACLA) XFEL facility. The nano-precision surface employed is coated with rhodium and offers a high reflectivity of 80%, with a photon energy of up to 12 keV, under total reflection conditions. Incident X-rays on the optics are reflected with a large spatial acceptance of over 900 $\mu$m. The focused beam is 210 nm $\times$ 120 nm (full width at half maximum) and was evaluated at a photon energy of 10 keV. The optics developed for $\textit{100 exa}$ efficiently achieved an intensity of 1 $\times$ 10$^2$$^0$ W/cm$^2$ with a pulse duration of 7 fs and a pulse energy of 150 $\mu$J (25% of the pulse energy generated at the light source). The experimental chamber, which can provide varied stage arrangements and sample conditions, including vacuum environments and atmospheric pressure helium, was set up with the focusing optics to meet the experimental requirements.

Split-pulse x-ray photon correlation spectroscopy has been proposed as one of the unique capabilities made possible with the x-ray free electron lasers. It enables characterization of atomic scale structural dynamics that dictates the macroscopic properties of various disordered material systems. Central to the experimental concept are x-ray optics that are capable of splitting individual coherent femtosecond x-ray pulse into two distinct pulses, introduce an adjustable time delay between them, and then recombine the two pulses at the sample position such that they generate two coherent scattering patterns in rapid succession. Recent developments in such optics showed that, while true 'amplitude splitting' optics at hard x-ray wavelengths remains a technical challenge, wavefront and wavelength splitting are both feasible, able to deliver two micron sized focused beams to the sample with sufficient relative stability. Here, we however show that the conventional approach to speckle visibility spectroscopy using these beam splitting techniques can be problematic, even leading to a decoupling of speckle visibility and material dynamics. In response, we discuss the details of the experimental approaches and data analysis protocols for addressing issues caused by subtle beam dissimilarities for both wavefront and wavelength splitting setups. We also show that in some scattering geometries, the Q-space mismatch can be resolved by using two beams of slightly different incidence angle and slightly different wavelengths at the same time. Instead of measuring the visibility of weak speckle patterns, the time correlation in sample structure is encoded in the 'side band' of the spatial autocorrelation of the summed speckle patterns, and can be retrieved straightforwardly from the experimental data. We demonstrate this with a numerical simulation.

Let $\sigma$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $OM_+^1$ the set of normalized positive operator monotone functions on $(0, \infty)$. In this paper, we study class of $\sigma$-subpreserving functions $f\in OM_+^1$ satisfying $$f(A\sigma B) \le f(A)\sigma f(B)$$ for all positive operators $A$ and $B$. We provide some criteria for $f$ to be trivial, i.e., $f(t)=1$ or $f(t)=t$. We also establish characterizations of $\sigma$-preserving functions $f$ satisfying $$f(A\sigma B) = f(A)\sigma f(B)$$ for all positive operators $A$ and $B$. In particular, when $\lim_{t\rightarrow 0} (1\sigma t) =0$, the function $f$ preserves $\sigma$ if and only if $f$ and $1\sigma t$ are representing functions for weighted harmonic means.

The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$, $C_i$ be unital C*-algebras and let $\alpha_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $\beta$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, such that $ \alpha_1 \otimes \alpha_2 \geq \beta$, factorizes as $\beta = \gamma \otimes \alpha_2$ for some positive map $\gamma$. In particular we show that when $\alpha_i \colon A_i \rightarrow B(\mathcal H_i)$ are completely positive (CP) maps for some Hilbert spaces $\mathcal H_i$ $(i=1,2)$, and $\alpha_2$ is a pure CP map and $\beta$ is a CP map so that $\alpha_1 \otimes \alpha_2 - \beta$ is also CP, then $\beta = \gamma \otimes \alpha_2$ for some CP map $\gamma$. We show that a similar result holds in the context of positive linear maps when $A_2 = C_2 = B(\mathcal H)$ and $\alpha_2 = id$. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map $\tau$ from a unital C*-algebra $A$ to a C*-algebra $C$, if $\tau \otimes id_k$ is decomposable for some $k \geq 2$, where $id_k$ is the identity map on the algebra $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $\tau $ is completely positive., Comment: 4 pages

We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense., Comment: 16 pages

We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an application we construct subspaces of $\C^m \otimes \C^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $k$ and which has a basis of Schmidt rank $k$ for $k=2,3,4$., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1804.04204. We fix the gap in the proof and put the detailed. The final version will be appeared in the journal

We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$ to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a manner reminiscent of several tracial approximation properties. We are particularly interested how regularity properties in the Elliott classification program pass from $B$ to $A$. Among them, we show that the strict comparison property and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting., Comment: This is a finalized version. It will appear in J. Topology and Analysis. 16 pages

The geometric discord $\mathcal{D}$ of a state is a measure of the quantumness of the state and the negativity $\mathcal{N}$ is a measure of the entanglement of a state. It was proved by D. Girolami and G. Adesso that for states on $\mathbb{C}^2\otimes\mathbb{C}^2$, the geometric discord is always greater than or equal to the square of the negativity and conjectured that this holds in general. S. Rana and P. Parashar showed that this relation does not hold for all states on $\mathbb{C}^2\otimes\mathbb{C}^n$ for $n>2$. We provide several analytic families of states on $\mathbb{C}^2\otimes\mathbb{C}^3$ violating this relation. Certain upper and lower bounds for $\mathcal{N}^2-\mathcal{D}$ are obtained for states on $\mathbb{C}^m\otimes\mathbb{C}^n$ for any $m, n\in\mathbb{N}$., Comment: Bounds for the difference of the square of the negativity and the geometric discord are improved. 9 pages, 4 figures

A channel-cut Si(111) crystal with a channel width of 90 $\mu$m was developed for achieving reflection self-seeding in hard X-ray free-electron lasers (XFELs). With the crystal, a monochromatic seed pulse is produced from a broadband XFEL pulse generated in the first-half undulators with an optical delay of 119 fs at 10 keV. The small optical delay allows a temporal overlap between the seed optical pulse and the electron bunch by using a small magnetic chicane for the electron beam at the middle of the undulator section. A peak reflectivity reached 67%, which is a reasonable value as compared with the theoretical one of 81%. By using this monochromator, a monochromatic seed pulse without broadband background in spectrum was obtained at SACLA with a conversion efficiency from a broadband XFEL pulse of $\sim 2 \times 10^{-2}$, which is $\sim 10$ times higher than that of transmission self-seeding using a thin diamond (400) monochromator., Comment: 16 pages, 5 figures

We introduce and study a notion of Rokhlin property for an inclusion of unital $C^*$-algebras which could have no projections like the Jiang-Su algebra. We also introduce a notion of approximate representability and show a duality between them. We demonstrate the importance of these notions by showing the permanence of the tracial $\mathcal{Z}$-absorbingness and the strict comparison property., Comment: This is our third joint paper. 32 pages. Minor changes and updated

Let $P \subset A$ be an inclusion of $\sigma$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$. (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$; (7)C*-algebras of decomposition rank at most $n$, where $n \in Z^+$; (8) Separable simple C*-algebras that are stably isomorphic to AF algebras; (9) Separable simple C*-algebras that are stably isomorphic to AI algebras; (10) Separable simple C*-algebras that are stably isomorphic to AT algebras; (11) Separable simple C*-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes; (12) Separable C*-algebras with strict comparison of positive elements. In particular, when $\alpha : G \rightarrow \rm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the Rokhlin property in the sense of Nawata, the properties $(1) \sim (12)$ are inherited to the fixed point algebra $A^\alpha$ and the crossed product algebra $A \rtimes_\alpha G$ from $A$., Comment: 8 pages

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $\Phi_{\alpha, \beta,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,n}$ is positive for any natural number $n \geq 3$. We focus on the case of $n=3,$ that is, $\Phi_{\alpha, \beta,3}$ and study the properties of $2$-positivity, completely positivity and decomposability. In particular, we give values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,3}$ is $2$-positive and not completely positive., Comment: 14 pages, 4 figures

Following the results known in the case of a finite abelian group action on $C\sp*$-algebras we prove the following two theorems; 1. an inclusion $P\subset A$ of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property). 2. an inclusion $P\subset A$ of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). Moreover, we provide an alternate proof of Phillips' theorem about the relations between tracial Rokhlin action and tracially approximate representable dual action using a new conceptual framework suggested by authors., Comment: This is a second part of our joint research which is promised in the paper arXiv:1707.07377

Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $\alpha$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $\alpha$ fixes any tracial state of $A$. Then tsr$(A \rtimes_\alpha G) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$., Comment: 4 pages

We define a tracial analogue of the sequentially split $*$-homomorphism between $C^*$-algebras of Barlak and Szab\'{o} and show that several important approximation properties related to the classification theory of $C^*$-algebras pass from the target algebra to the domain algebra. Then we show that the tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$ gives rise to a tracial version of sequentially split $*$-homomorphism from $A\rtimes_{\alpha}G$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \to P$ of a finite Watatani index generates a tracial version of sequentially split map. By doing so, we provide a unified approach to permanence properties related to tracial Rokhlin property of operator algebras., Comment: A serious flaw in Definition 2.6 has been notified to the authors. We fix our definition and accordingly change statements in subsequent propositions and theorems. Moreover, a gap in the proof of Theorem 2.25 is fixed. We note our appreciation for such helpful comments in Acknowledgements section. Some typos are also caught. We hope that it is final

The GdFeO$_{3}$-type perovskite CaCoO$_{3}$ has been successfully synthesized by high-pressure oxygen annealing for the oxygen deficient perovskite. A detailed structural analysis based on synchrotron X-ray diffraction data and a thermogravimetric analysis show that the valence of Co is +4 and the sample is free from oxygen deficiency. This compound shows an antiferromagnetic ordering at 95 K, which has presumably helical spin arrangement, with keeping the incoherent metallic state down to the lowest temperature. This work demonstrates that the Co$^{4+}$-perovskite oxides exhibit a variety of magnetic phases by the band-width control through the lattice distortion., Comment: 5 pages, 5 figures