We show that the following properties of the C^*-algebras in a class P are inherited by simple unital C^*-algebras in the class of asymptotically tracially in P: (1) n-comparison, (2) α-comparison (1 ≤ α < ∞).

We prove that a surjective map (on the positive cones of unital C^*-algebras) preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra. We represent weighted geometric mean preserving bijective maps on the positive cones of prime C^*-algebras in terms of Jordan *-isomorphisms of the algebras. We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space, answering an open question arisen by Dye. Finally, we give a description for Fuglede-Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak's work on this topic.

In this paper, we study some properties of weighted composition operators on a class of weighted Bergman spaces \begin{document}$ A_\varphi^p $\end{document} with 0 < p ≤ ∞ and \begin{document}$ \varphi \in \mathcal{W}_0 $\end{document}. Also, we completely characterize the q-Carleson measure for \begin{document}$ A_\varphi^p $\end{document} in terms of the averaging function and the generalized Berezin transform with 0 < q < ∞. As applications, the boundedness and compactness of weighted composition operators acting from one Bergman space \begin{document}$ A_\varphi^p $\end{document} to another \begin{document}$ A_\varphi^q $\end{document} are equivalently described and the Schatten class property of the weighted composition operator acting on \begin{document}$ A_\varphi^2 $\end{document} are given. Our main results are expressed in terms of certain Berezin type integral transforms.

Quantum uncertainty relations are mathematical inequalities that describe the lower bound of products of standard deviations of observables (i.e., bounded or unbounded self-adjoint operators). By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for $k$ observables, of which the formulation depends on the even or odd quality of $k$. This universal uncertainty relation is tight at least for the cases $k=2$ and $k=3$. For two observables, the uncertainty relation is a simpler reformulation of Schrödinger's uncertainty principle, which is also tighter than Heisenberg's and Robertson's uncertainty relations.

The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F. Defining primes in a non-abelian monoid is highly non-trivial, which relies on a concept called "castling". Three types of castlings are essential to grasp the arithmetics. The divisor function τ on Thompson's monoid S satisfies τ(uv) ≤ τ(u)τ(v) for any u, v ∈ S. Then the limit τ₀(u)=lim_n→∞(τ(u_n))^1/n exists. The quantity Ç(S)=sup_1≠u∈S τ₀(u)/τ(u) describes the complexity for castlings in S. We show that Ç(S)=1. Moreover, the Möbius function on S is calculated. And the Liouville function Ω on S is studied.

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.

Recently, Gehér and Šemrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries. In this paper, we study the surjective L²-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.