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.     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.     

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.     

Remarks on a posteriori error estimation for finite element solutions

We utilize the classical hypercircle method and the lowest-order Raviart–Thomas H(div)$H(\mathrm{div})$ element to obtain a posteriori error estimates of the _P1$P_1$ finite element solutions for 2D Poisson's equation. A few other estimation methods are also discussed for comparison. We give some theoretical and numerical results to see the effectiveness of the methods.     

Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove _Lp$L_p$ Poincaré inequalities for the group measure space _L∞(_ΩH,γ)?G$L_{\infty }({\Omega }_H,\gamma )?G$, where both the group action and the Gaussian measure space (_ΩH,γ)$({\Omega }_H,\gamma )$ are associated with the representation α  . The idea of proof comes from Pisier?s method on the boundedness of Riesz transform and Lust-Piquard?s work on spin systems. Then we deduce a transportation type inequality from the _Lp$L_p$ Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel?s compact quantum metric spaces.     

A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k  -graph C^?$C^{?}$-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k  -graph C^?$C^{?}$-algebra, it is sufficient that the representation be injective on a distinguished abelian C^?$C^{?}$-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C^?$C^{?}$-algebra, each of which is the unique extension of a state on the distinguished abelian C^?$C^{?}$-subalgebra.     

Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)$N(A)$ be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A$d:A\to A$ is a derivation if and only if d(A)⊂N(A)$d(A)\subset N(A)$ and d(A²)={0}$d(A^2)=\{0\}$, where A²$A^2$ is the set of all products ab in A.     

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞)$(0,+\infty )$. We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L^1,2(0,∞)$L^{1,2}(0,\infty )$ space by employing a method recently introduced in [7] or [8].