We prove that no proper Segal algebra of a SIN group is approximately amenable.

The precise condition on a completely regular space $X$ for every character on $C\left( X \right)$ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained by relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience.

We examine spectral behavior of irreducible tuples that do not admit the boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $\left( {{T}_{1}},\,.\,.\,.\,,\,{{T}_{m}} \right)$must be the operator norm of $T_{1}^{*}\,{{T}_{1}}\,+\,.\,.\,.\,+\,T_{m}^{*}{{T}_{m}}$. We use this simple observation to ensure the boundary property for an irreducible, essentially normal, joint q-isometry, provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}\left[ {{z}_{1}},\,.\,.\,.\,,{{z}_{m}} \right]$-modules (of which the Drury–Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal.

Let $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${{S}^{n-1}}$. In this paper, using Fourier transform estimates and approximation to the operator ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\,\in \,\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$. The associated maximal operator is also considered.

Let $G$ be a group and $\mathbb{K}\,=\,\mathbb{C}\,\text{or}\,\mathbb{R}$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\,:\,G\,\to \,\mathbb{K}$ satisfying the inequality

$$\left| f\left( \sum\limits_{k=1}^{n}{{{x}_{k}}} \right)\,-\,\underset{k=1}{\overset{n}{\mathop{\Pi }}}\,f\left( {{x}_{k}} \right) \right|\,\,\le \phi \left( {{x}_{2}},\,.\,.\,.\,,{{x}_{n}} \right),\,\,\,\,\,\,\forall {{x}_{1}},\,.\,.\,.\,,{{x}_{n}}\,\in \,G,$$

Where $\phi :\,{{G}^{n-1}}\,\to \,[0,\,\infty )$. Also as a a distributional version of the above inequality we consider the stability of the functional equation

$$u\,\circ \,S\,-\,\overbrace{u\,\otimes \,.\,.\,.\,\otimes \,u}^{n-\text{times}}\,=\,0,$$

where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S\left( {{x}_{1}},\,.\,.\,.\,,{{x}_{n}} \right)\,=\,{{x}_{1}}\,+\,.\,.\,.\,+\,{{x}_{n}}$.

We prove that quasiconvex functions always admit descent trajectories bypassing all nonminimizing critical points.

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow that is shown to be equal to the index of the operator. This result is interpreted in terms of the K-theory of an associated mapping cone. It is then extended to connect Z2 indices of odd symmetric Fredholm operators to a Z2-valued spectral flow.

Bibasic sequences are used to study relative weak compactness and relative norm compactness of limited sets.

Let $G$ be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety $G/B$. The main theorem of this note gives an eõcient presentation of the equivariant cohomology ring $H_{S}^{*}$ (Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S\,\cong \,{{\mathbb{C}}^{*}}$ is a certain subgroup of a maximal torus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa–Harada–Masuda, which was only for Lie type $A$.

Let $\alpha :\,G\,\curvearrowright \,M$ be a spatial action of a countable abelian group on a “spatial” von Neumann algebra $M$ and let $S$ be its unital subsemigroup with $G\,=\,{{S}^{-1}}S$. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten $p$-class or the compact operators, of the ${{w}^{*}}$-semicrossed product of $M$ by $S$ when ${{M}^{'}}$ contains no non-zero compact operators. We also prove a weaker result when $M$ is a von Neumann algebra on a finite dimensional Hilbert space and $\left( G,\,S \right)\,=\,\left( \mathbb{Z},\,{{\mathbb{Z}}_{+}} \right)$, which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.

The prime vertex graph, $\Delta \left( X \right)$, and the common divisor graph, $\Gamma \left( X \right)$, are two graphs that have been defined on a set of positive integers $X$. Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.

It is well known that a discrete group that is both amenable and has Kazhdan’s Property $T$ must be finite. In this note we generalize this statement to the case of transformation groups. We show that if $G$ is a discrete amenable group acting on a compact Hausdorff space $X$, then the transformation group ${{C}^{*}}$-algebra ${{C}^{*}}\left( X,\,G \right)$ has Property $T$ if and only if both $X$ and $G$ are finite. Our approach does not rely on the use of tracial states on ${{C}^{*}}\left( X,\,G \right)$.

Inspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence

For the Bergman projection operator $P$ we prove that

$$\left\| P:\,{{L}^{1}}\left( B,\,d\lambda \right)\,\to \,{{B}_{1}} \right\|\,=\,\frac{\left( 2n\,+\,1 \right)!}{n!}.$$

Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of ${{\mathbb{C}}^{n}}$, and ${{B}_{1}}$ denotes the Besov space with an adequate semi-norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

We show that under some conditions a Gorenstein ring $R$ satisfies the Generalized Auslander–Reiten conjecture if and only if $R\left[ x \right]$ does. When $R$ is a local ring we prove the same result for some localizations of $R\left[ x \right]$.

We give a necessary and sufficient condition for a sequence to be a localization set for a determining average sampler.

Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon–Nikodým property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold and that $M\,=\,{{\ell }_{2}}$ is a counterexample.

Let $g\,\ge \,2$. A real number is said to be $g$-normal if its base $g$ expansion contains every finite sequence of digits with the expected limiting frequency. Let $\varphi$ denote Euler’s totient function, let $\sigma$ be the sum-of-divisors function, and let $\lambda$ be Carmichael’s lambda-function. We show that if $f$ is any function formed by composing $\varphi$, $\sigma$, or $\lambda$, then the number

$$0.f\left( 1 \right)f\left( 2 \right)f\left( 3 \right)\,.\,.\,.$$

obtained by concatenating the base $g$ digits of successive $f$-values is $g$-normal. We also prove the same result if the inputs 1,2,3.... are replaced with the primes 2, 3, 5.... The proof is an adaptation of a method introduced by Copeland and Erdõs in 1946 to prove the 10-normality of 0:235711131719...