It is known that $ℬ\left({\ell }^p\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^p\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable so is ${\ell }^{\infty }(,\left(E\right))$ for any index set and for any infinite-dimensional ${ℒ}^p$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over...

Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

Let $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤=LieG$. Let $(e,h,f)$ be an $𝔰{𝔩}_2$-triple in $𝔤$ with $e$ being a long root vector in $𝔤$. Let $(·,·)$ be the $G$-invariant bilinear form on $𝔤$ with $(e,f)=1$ and let $\chi \in {𝔤}^{*}$ be such that $\chi \left(x\right)=(e,x)$ for all $x\in 𝔤$. Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...

Let $A\left(X\right)$ denote a subalgebra of $C\left(X\right)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A\left(X\right)$, we generalize the notion of $z_A^{\beta }$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C\left(X\right)$...

Let $I$ be an ideal in a commutative Noetherian ring $R$. Then the ideal $I$ has the strong persistence property if and only if $\left(I^{k+1}{:}_{R}I\right)=I^k$ for all $k$, and $I$ has the symbolic strong persistence property if and only if $\left(I^{(k+1)}{:}_R{I}^{\left(1\right)}\right)=I^{\left(k\right)}$ for all $k$, where $I^{\left(k\right)}$ denotes the $k$th symbolic power of $I$. We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

For 0 < γ ≤ 1, let $\Lambda {⁺}_{\gamma }$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{\gamma }(E,Q)$ be the closed ideal in $\Lambda {⁺}_{\gamma }$ consisting of those functions $f\in \Lambda {⁺}_{\gamma }$ for which (i) f = 0 on E, (ii) $|f\left(z\right)-f\left(w\right)|=o\left(\right|z-w{|}^{\gamma })$ as d(z,E),d(w,E) → 0, (iii) $f/Q\in \Lambda {⁺}_{\gamma }$. Also, for a closed ideal I in $\Lambda {⁺}_{\gamma }$, let $E_I$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions...

Let $X$ be a zero-dimensional space and $C_c\left(X\right)$ be the set of all continuous real valued functions on $X$ with countable image. In this article we denote by $C_c^K\left(X\right)$ (resp., $C_c^{\psi }\left(X\right))$ the set of all functions in $C_c\left(X\right)$ with compact (resp., pseudocompact) support. First, we observe that $C_c^K\left(X\right)=O_c^{{\beta }_{0}X\setminus X}$ (resp., $C_c^{\psi }\left(X\right)=M_c^{{\beta }_{0}X\setminus {\upsilon }_{0}X}$), where ${\beta }_{0}X$ is the Banaschewski compactification of $X$ and ${\upsilon }_{0}X$ is the $\mathbb{N}$-compactification of $X$. This implies that for an $\mathbb{N}$-compact space $X$, the intersection of all free maximal ideals in $C_c\left(X\right)$ is equal to $C_c^K\left(X\right)$, i.e., $M_c^{{\beta }_{0}X\setminus X}=C_c^K\left(X\right)$. By applying...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles...