p-embeddings

This is the sequel to Bhattacharjee et al. (in press) [3] where the notion of a p  -extension of commutative rings was investigated: a unital extension of commutative rings, say R?S$R?S$, is a p  -extension if for every s∈S$s\in S$ there is an r∈R$r\in R$ such that rS=sS$rS=sS$. In this article we apply the theory of p  -extensions to rings of continuous functions. We show that this concept lays between the concepts of C^?$C^{?}$-embeddings and z-embeddings.     

Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities

For Ω$\Omega$, an open bounded subset of R^N${\mathbb{R}}^N$ with smooth boundary and 1<p<∞$1<p<\infty$, we establish W^1,p(Ω)$W^{1,p}\left(\Omega \right)$a priori bounds and prove the compactness of solution sets to differential inequalities of the form

|divA(x,∇u)|≤F(x,u,∇u),$\left|\text{div}A(x,\nabla u)\right|\le F(x,u,\nabla u)\text{,}$

which are bounded in L^∞(Ω)$L^{\infty }\left(\Omega \right)$. The main point in this work is that the nonlinear term F$F$ may depend on ∇u$\nabla u$ and may grow as fast as a power of order p$p$ in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein–Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.     

Nonlinear maps preserving Jordan *-products

Let η$\eta$ be a non-zero scalar. In this paper, we investigate a bijective map ?$?$ between two von Neumann algebras, one of which has no central abelian projections, satisfying ?(AB+ηBA^∗)=?(A)?(B)+η?(B)?(A)^∗$?(AB+\eta B{A}^{\ast })=?\left(A\right)?\left(B\right)+\eta ?\left(B\right)?{\left(A\right)}^{\ast }$ for all A,B$A,B$ in the domain. It is showed that ?$?$ is a linear *-isomorphism if η$\eta$ is not real and ?$?$ is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η$\eta$ is real.     

MF actions and K-theoretic dynamics

We study the interplay of C^?$C^{?}$-dynamics and K  -theory. Notions of chain recurrence for transformations groups (X,Γ)$(X,\Gamma )$ and MF actions for non-commutative C^?$C^{?}$-dynamical systems (A,Γ,α)$(A,\Gamma ,\alpha )$ are translated into K-theoretical language, where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates. We are particularly interested in actions of free groups on AF algebras, in which case we prove that a K-theoretic coboundary condition determines whether or not the reduced crossed product is a matricial field (MF) algebra. One upshot is the equivalence of stable finiteness and being MF for these reduced crossed product algebras.     

Orthogonally additive polynomials on Fourier algebras

We show that an n-homogeneous polynomial P   on the Fourier algebra A(G)$A(G)$ of a locally compact group G   can be represented in the form P(f)=〈T,fⁿ〉$P(f)=〈T,f^n〉$(f∈A(G))$(f\in A(G))$ for some T   in the group von Neumann algebra VN(G)$\mathit{VN}(G)$ of G if and only if it is orthogonally additive and completely bounded.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.