Euler class groups and 2-torsion elements

We exhibit an example of a smooth affine threefold A$A$ over a field of characteristic 0$0$ for which there exist non-trivial 2-torsion elements in the Euler class group E(A)$E\left(A\right)$ vanishing in the weak Euler class group _E0(A)$E_0\left(A\right)$. This gives a positive answer to a question of the first author and Raja Sridharan.     

The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.     

Polynomial numerical hulls of some normal matrices

In this note, polynomial numerical hulls of matrices of the form _A1⊕i_A2$A_1\oplus i{A}_2$, where _A1$A_1$ and _A2$A_2$ are Hermitian, are characterized.     

A note on residual variables of an affine fibration

In a recent paper [7], M.E. Kahoui has shown that if R   is a polynomial ring over C$\mathbb{C}$, A   an A³${\mathbb{A}}^3$-fibration over R, and W a residual variable of A then A   is stably polynomial over R[W]$R[W]$. In this article we show that the above result holds over any Noetherian domain R   provided the module of differentials _ΩR(A)${\Omega }_R(A)$ of the affine fibration A (which is necessarily a projective A-module by a theorem of Asanuma) is a stably free A-module.     

A discrete Fourier transform associated with the affine Hecke algebra

We introduce an explicit representation of the double affine Hecke algebra (of type _A1$A_1$) at q=1$q=1$ that gives rise to a periodic counterpart of a well-known Fourier transform associated with the affine Hecke algebra.     

Functions of perturbed n-tuples of commuting self-adjoint operators

Let (_A1,…,_An)$(A_1,\dots ,A_n)$ and (_B1,…,_Bn)$(B_1,\dots ,B_n)$ be n-tuples of commuting self-adjoint operators on Hilbert space. For functions f   on Rⁿ${\mathbb{R}}^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of f(_A1,…,_An)−f(_B1,…,_Bn)$f(A_1,\dots ,A_n)-f(B_1,\dots ,B_n)$ in terms of the corresponding norms of _Aj−_Bj$A_j-B_j$, 1?j?n$1?j?n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for n?3$n?3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary n  . We also get sharp estimates for quasicommutators f(_A1,…,_An)R−Rf(_B1,…,_Bn)$f(A_1,\dots ,A_n)R-Rf(B_1,\dots ,B_n)$ in terms of norms of _AjR−R_Bj$A_{j}R-R{B}_j$, 1?j?n$1?j?n$, for a bounded linear operator R.     

Uniform hypergraphs containing no grids

A hypergraph is called an r×r$r\times r$grid   if it is isomorphic to a pattern of r$r$ horizontal and r$r$ vertical lines, i.e., a family of sets {_A1,…,_Ar,_B1,…,_Br}$\{A_1,\dots ,A_r,B_1,\dots ,B_r\}$ such that _Ai∩_Aj=_Bi∩_Bj=0?$A_i\cap A_j=B_i\cap B_j=0?$ for 1≤i<j≤r$1\le i<j\le r$ and |_Ai∩_Bj|=1$|A_i\cap B_j|=1$ for 1≤i,j≤r$1\le i,j\le r$. Three sets _C1,_C2,_C3$C_1,C_2,C_3$ form a triangle   if they pairwise intersect in three distinct singletons, |_C1∩_C2|=|_C2∩_C3|=|_C3∩_C1|=1$|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1$, _C1∩_C2≠_C1∩_C3$C_1\cap C_2\ne C_1\cap C_3$. A hypergraph is linear  , if |E∩F|≤1$|E\cap F|\le 1$ holds for every pair of edges E≠F$E\ne F$.     

Operator norm attainment and inner product spaces

In this paper we prove that a finite dimensional real normed linear space X$\mathbb{X}$ is an inner product space iff for any linear operator T   on X$\mathbb{X}$, T   attains its norm at _e1,_e2∈_SX$e_1,e_2\in S_{\mathbb{X}}$ implies T   attains its norm at span{_e1,_e2}∩_SX$\mathit{span}\{e_1,e_2\}\cap S_{\mathbb{X}}$.     

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G   and 1<p<∞$1<p<\infty$ let _Ap(G)$A_p(G)$ be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)$A(G)$ (p=2$p=2$). It is shown that for any amenable group H  , a proper affine map α:Y⊂H→G$\alpha :Y\subset H\to G$ induces a p  -completely contractive algebra homomorphism _?α:_Ap(G)→_Ap(H)${?}_{\alpha }:A_p(G)\to A_p(H)$ by setting _?α(u)=u°α${?}_{\alpha }(u)=u°\alpha$ on Y   and _?α(u)=0${?}_{\alpha }(u)=0$ off of Y. Moreover, we show that if both G and H are amenable then any p  -completely contractive algebra homomorphism ?:_Ap(G)→_Ap(H)$?:A_p(G)\to A_p(H)$ is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2$p=2$) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.     

Perturbations of certain crossed product algebras by free groups

Given two von Neumann algebras M and N   acting on the same Hilbert space, d(M,N)$d(M,N)$ is defined to be the Hausdorff distance between their unit balls. The Kadison–Kastler problem asks whether two sufficiently close von Neumann algebras are spatially isomorphic. In this article, we show that if _P0$P_0$ is an injective von Neumann algebra with a cyclic tracial vector, G is a free group, α is a free action of G   on _P0$P_0$ and N   is a von Neumann algebra such that d(N,_P0?αG)<1/7×10⁻⁷$d(N,P_0{?}_{\alpha }G)<1/7\times {10}^{-7}$, then N   and _P0?αG$P_0{?}_{\alpha }G$ are spatially isomorphic. Suitable choices of the actions give the first examples of infinite noninjective factors for which this problem has a positive solution.     

S-Noetherian properties on amalgamated algebras along an ideal

Let A and B   be commutative rings with identity, f:A→B$f:A\to B$ a ring homomorphism and J an ideal of B  . Then the subring A?^fJ:={(a,f(a)+j)|a∈A and j∈J}$A{?}^{f}J:=\{(a,f(a)+j)|a\in A\text{and}j\in J\}$ of A×B$A\times B$ is called the amalgamation of A with B along with J with respect to f. In this paper, we investigate a general concept of the Noetherian property, called the S  -Noetherian property which was introduced by Anderson and Dumitrescu, on the ring A?^fJ$A{?}^{f}J$ for a multiplicative subset S   of A?^fJ$A{?}^{f}J$. As particular cases of the amalgamation, we also devote to study the transfers of the S  -Noetherian property to the constructions D+(_X1,…,_Xn)E[_X1,…,_Xn]$D+(X_1,\dots ,X_n)E[X_1,\dots ,X_n]$ and D+(_X1,…,_Xn)E?_X1,…,_Xn?$D+(X_1,\dots ,X_n)E?X_1,\dots ,X_n?$ and Nagata?s idealization.