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.     

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.     

Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

We prove that if a set S⊂Rⁿ$S\subset {\mathbb{R}}^n$ is Zariski closed at infinity, then the algebra of polynomials bounded on S cannot be finitely generated. It is a new proof of a fact already known to Plaumann and Scheiderer (2012) [1]. On the way we show that if the ring R[_ζ1,…,_ζk]⊂R[X]$\mathbb{R}[{\zeta }_1,\dots ,{\zeta }_k]\subset \mathbb{R}[X]$ contains the ideal (_ζ1,…,_ζk)R[X]$({\zeta }_1,\dots ,{\zeta }_k)\mathbb{R}[X]$, then the mapping (_ζ1,…,_ζk):Rⁿ→R^k$({\zeta }_1,\dots ,{\zeta }_k):{\mathbb{R}}^n\to {\mathbb{R}}^k$ is finite.     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Pointwise products of some Banach function spaces and factorization

The well-known factorization theorem of Lozanovski? may be written in the form L¹≡E⊙E^′$L^1\equiv E\odot E^{\prime }$, where ⊙ means the pointwise product of Banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E  , i.e., when F≡E⊙M(E,F)$F\equiv E\odot M(E,F)$, where M(E,F)$M(E,F)$ is the space of pointwise multipliers from E to F  . Properties of M(E,F)$M(E,F)$ were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E⊙F$E\odot F$. The formulas for pointwise product of Calderón–Lozanovski? _Eφ$E_{\phi }$-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant Banach function space may be factorized through a Marcinkiewicz space.     

Projective modules over overrings of polynomial rings and a question of Quillen

Let (R,m,K)$(R,\mathfrak{m},K)$ be a regular local ring containing a field k   such that either char k=0$k=0$ or char k=p$k=p$ and tr-deg K/_Fp?1$K/{\mathbb{F}}_p?1$. Let _g1,…,_gt$g_1,\dots ,g_t$ be regular parameters of R   which are linearly independent modulo m²${\mathfrak{m}}^2$. Let A=_Rg1?gt[_Y1,…,_Ym,_f1(_l1)⁻¹,…,_fn(_ln)⁻¹]$A=R_{g_1?g_t}[Y_1,\dots ,Y_m,f_1{(l_1)}^{-1},\dots ,f_n{(l_n)}^{-1}]$, where _fi(T)∈k[T]$f_i(T)\in k[T]$ and _li=_ai1Y1+?+_aimYm$l_i=a_{i1}{Y}_1+?+a_{im}{Y}_m$ with (_ai1,…,_aim)∈k^m−(0)$(a_{i1},\dots ,a_{im})\in k^m-(0)$. Then every projective A-module of rank ?t   is free. Laurent polynomial case _fi(_li)=_Yi$f_i(l_i)=Y_i$ of this result is due to Popescu.     

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action Γ?(X,μ)$\Gamma ?(X,\mu )$ of all Γ   in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra L^∞(X)?Γ${\mathrm{L}}^{\infty }(X)?\Gamma$ has L^∞(X)${\mathrm{L}}^{\infty }(X)$ as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in Popa and Vaes (in press) [38]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products _M1?BM2$M_1{?}_B{M}_2$ over a subalgebra B of type I.     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.