Zero Jordan product determined Banach algebras

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $\varphi\colon A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\varphi(a,b)=\sigma(ab+ba)$ for some continuous linear map $\sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications

Maps preserving zeros of a polynomial

Let \A be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps \phi:\A\to \A that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where \A=M_n(F). We also consider quite general algebras \A, but only for specific polynomials $f$.Comment: 11 pages, accepted for publication in Linear Algebra App

Quantum Monte Carlo simulations for estimating FOREX markets: A speculative attacks experience

The foreign exchange markets, renowned as the largest financial markets globally, also stand out as one of the most intricate due to their substantial volatility, nonlinearity, and irregular nature. Owing to these challenging attributes, various research endeavors have been undertaken to effectively forecast future currency prices in foreign exchange with precision. The studies performed have built models utilizing statistical methods, being the Monte Carlo algorithm the most popular. In this study, we propose to apply Auxiliary-Field Quantum Monte Carlo to increase the precision of the FOREX markets models from different sample sizes to test simulations in different stress contexts. Our findings reveal that the implementation of Auxiliary-Field Quantum Monte Carlo significantly enhances the accuracy of these models, as evidenced by the minimal error and consistent estimations achieved in the FOREX market. This research holds valuable implications for both the general public and financial institutions, empowering them to effectively anticipate significant volatility in exchange rate trends and the associated risks. These insights provide crucial guidance for future decision-making processes

The Kadison problem on a class of commutative Banach algebras with closed cone

summary:The main result of the paper characterizes continuous local derivations on a class of commutative Banach algebra $A$ that all of its squares are positive and satisfying the following property: Every continuous bilinear map $\Phi$ from $A\times A$ into an arbitrary Banach space $B$ such that $\Phi(a,b)=0$ whenever $ab=0$, satisfies the condition $\Phi (ab,c)=\Phi(a,bc)$ for all $a,b,c\in A$

Forecasting Stock Market Crashes via Real-Time Recession Probabilities: A Quantum Computing Approach

A fast and precise prediction of stock market crashes is an important aspect of economic growth, fiscal and monetary system because it facilitates the government the application of suitable policies. Many works have examined the behaviour of the fall of stock markets and have built models to predict them. Nevertheless, there are limitations to the available research, and the literature calls for more investigation on the topic, as currently the accuracy of the models remains low and they have only been extended for the largest economies. This study provides a comparison of Quantum forecast methods stock market declines and, therefore, a new prediction model of stock market crashes via real-time recession probabilities with the power to accurately estimate future global stock market downturn scenarios. A 104-country sample has been used, allowing the sample compositions to take into account the regional diversity of the alert warning indicators. To obtain a robust model, several alternative techniques have been employed on the sample under study, being Quantum Boltzmann Machines, which have obtained very good prediction results due to their ability to remember features and develop long-term dependencies from time series and sequential data. Our model has large policy implications for the appropriate macroeconomic policy response to downside risks, offering tools to help achieve financial stability at the international level

Estimating DSGE Models using Multilevel Sequential Monte Carlo in Approximate Bayesian Computation

21-25Dynamic Stochastic General Equilibrium (DSGE) models allow for probabilistic estimations with the aim of formulating macroeconomic policies and monitoring them. In this study, we propose to apply the Sequential Monte Carlo Multilevel algorithm and Approximate Bayesian Computation (MLSMC-ABC) to increase the robustness of DSGE models built for small samples and with irregular data. Our results indicate that MLSMC-ABC improves the estimation of these models in two aspects. Firstly, the accuracy levels of the existing models are increased, and secondly, the cost of the resources used is reduced due to the need for shorter execution time