Erratum to “Can $ℬ(ℓ^{p})$ ever be amenable?” (Studia Math. 188 (2008), 151-174)

Matthew Daws; Volker Runde

Displaying similar documents to “Erratum to “Can $ℬ (ℓ^{p})$ ever be amenable?” (Studia Math. 188 (2008), 151-174)”

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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We say that an ideal I on $_{κ} λ$ is semiproper if the corresponding poset $ℙ_{I}$ is semiproper. In this paper we investigate properties of semiproper ideals on $_{κ} λ$ .

On norm closed ideals in $L (ℓ_{p}, ℓ_{q})$

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L (ℓ_{p} \oplus ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L (ℓ_{p}, ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L (ℓ_{p}, ℓ_{q})$ , including one that has not been studied before. The proofs use various methods...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_{m}(A)$ of maximal invertible ideals. In this note we study some properties of $I_{m}(A)$ and we prove that, if $I(A)$ is a free group on $I_{m}(A)$ , then $A$ is a locally factorial Krull domain.

Squarefree monomial ideals with maximal depth

Ahad Rahimi (2020)

Czechoslovak Mathematical Journal

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Let $(R, 𝔪)$ be a Noetherian local ring and $M$ a finitely generated $R$ -module. We say $M$ has maximal depth if there is an associated prime $𝔭$ of $M$ such that depth $M = dim R / 𝔭$ . In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.

On quasi $n$ -ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$ -ideals and the class of $(2, n)$ -ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$ -ideal if $\sqrt{I}$ is an $n$ -ideal of $R .$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$ -ideals, the quasi primary ideals, the $(2, n)$ -ideals and the $p r$ -ideals. Moreover, we use the quasi $n$ -ideals to characterize...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Effective Nullstellensatz for arbitrary ideals

János Kollár (1999)

Journal of the European Mathematical Society

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Let $f_{i}$ be polynomials in $n$ variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials $g_{i}$ such that $\sum g_{i} f_{i} = 1$ . The effective versions of this result bound the degrees of the $g_{i}$ in terms of the degrees of the $f_{j}$ . The aim of this paper is to generalize this to the case when the $f_{i}$ are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.

Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub, Mehrdad Nasernejad (2021)

Czechoslovak Mathematical Journal

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We provide a construction of monomial ideals in $R = K [x, y]$ such that $μ (I^{2}) < μ (I)$ , where $μ$ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$ , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $μ (I^{k})$ that generalize...

$(δ, 2)$ -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity, let $ℐ (ℛ)$ be the set of all ideals of $R$ and $δ : ℐ (ℛ) \to ℐ (ℛ)$ an expansion of ideals of $R$ defined by $I \mapsto δ (I)$ . We introduce the concept of $(δ, 2)$ -primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(δ, 2)$ -primary ideal if whenever $a, b \in R$ and $a b \in I$ , then $a^{2} \in I$ or $b^{2} \in δ (I)$ . Our purpose is to extend the concept of $2$ -ideals to $(δ, 2)$ -primary ideals of commutative rings. Then we investigate the basic properties of $(δ, 2)$ -primary ideals and also discuss the relations among $(δ, 2)$ -primary, $δ$ -primary...

Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

Lingli Zeng, Jizhu Nan (2016)

Czechoslovak Mathematical Journal

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Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$ th root of unity and $char K \neq p$ . Suppose that a classical group $G$ acts on the $F$ -vector space $V$ . Then it can induce the actions on the vector space $V \oplus V$ and on the group algebra $K [V \oplus V]$ , respectively. In this paper we determine the structure of $G$ -invariant ideals of the group algebra $K [V \oplus V]$ , and establish the relationship between the invariant ideals of $K [V]$ and the vector invariant ideals of $K [V \oplus V]$ , if $G$ is a unitary group or orthogonal...

Non-regularity for Banach function algebras

J. Feinstein, D. Somerset (2000)

Studia Mathematica

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Let A be a unital Banach function algebra with character space $Φ_{A}$ . For $x \in Φ_{A}$ , let $M_{x}$ and $J_{x}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_{x}$ is connected, and that if x does not belong to the Shilov boundary of A then the set $y \in Φ_{A} : M_{x} \supseteq J_{y}$ has an infinite connected subset. Various related results are given.

The group ring $𝕂 F$ of Richard Thompson’s Group $F$ has no minimal non-zero ideals

John Donnelly (2019)

Archivum Mathematicum

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We use a total order on Thompson’s group $F$ to show that the group ring $𝕂 F$ has no minimal non-zero ideals.

$α$ -ideals in $0$ -distributive posets

Khalid A. Mokbel (2015)

Mathematica Bohemica

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The concept of $α$ -ideals in posets is introduced. Several properties of $α$ -ideals in $0$ -distributive posets are studied. Characterization of prime ideals to be $α$ -ideals in $0$ -distributive posets is obtained in terms of minimality of ideals. Further, it is proved that if a prime ideal $I$ of a $0$ -distributive poset is non-dense, then $I$ is an $α$ -ideal. Moreover, it is shown that the set of all $α$ -ideals $α Id (P)$ of a poset $P$ with $0$ forms a complete lattice. A result analogous to separation theorem for...

Can $ℬ (ℓ^{p})$ ever be amenable?

Matthew Daws, Volker Runde (2008)

Studia Mathematica

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It is known that $ℬ (ℓ^{p})$ is not amenable for p = 1,2,∞, but whether or not $ℬ (ℓ^{p})$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ (ℓ^{p})$ is amenable for p ∈ (1,∞), then so are $ℓ^{\infty} (ℬ (ℓ^{p}))$ and $ℓ^{\infty} ((ℓ^{p}))$ . Moreover, if $ℓ^{\infty} ((ℓ^{p}))$ is amenable so is $ℓ^{\infty} (, (E))$ for any index set and for any infinite-dimensional $ℒ^{p}$ -space E; in particular, if $ℓ^{\infty} ((ℓ^{p}))$ is amenable for p ∈ (1,∞), then so is $ℓ^{\infty} ((ℓ^{p} \oplus ℓ ²))$ . We show that $ℓ^{\infty} ((ℓ^{p} \oplus ℓ ²))$ 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...

$0$ -ideals in $0$ -distributive posets

Khalid A. Mokbel (2016)

Mathematica Bohemica

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The concept of a $0$ -ideal in $0$ -distributive posets is introduced. Several properties of $0$ -ideals in $0$ -distributive posets are established. Further, the interrelationships between $0$ -ideals and $α$ -ideals in $0$ -distributive posets are investigated. Moreover, a characterization of prime ideals to be $0$ -ideals in $0$ -distributive posets is obtained in terms of non-dense ideals. It is shown that every $0$ -ideal of a $0$ -distributive meet semilattice is semiprime. Several counterexamples are discussed. ...

$G r$ - $(2, n)$ -ideals in graded commutative rings

Khaldoun Al-Zoubi, Shatha Alghueiri, Ece Y. Celikel (2020)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a group with identity $e$ and let $R$ be a $G$ -graded ring. In this paper, we introduce and study the concept of graded $(2, n)$ -ideals of $R$ . A proper graded ideal $I$ of $R$ is called a graded $(2, n)$ -ideal of $R$ if whenever $r s t \in I$ where $r, s, t \in h (R)$ , then either $r t \in I$ or $r s \in G r (0)$ or $s t \in G r (0)$ . We introduce several results concerning $g r$ - $(2, n)$ -ideals. For example, we give a characterization of graded $(2, n)$ -ideals and their homogeneous components. Also, the relations between graded $(2, n)$ -ideals and others that already exist, namely, the graded prime...

Displaying similar documents to “Erratum to “Can ℬ ( ℓ p ) ever be amenable?” (Studia Math. 188 (2008), 151-174)”

Displaying similar documents to “Erratum to “Can $ℬ (ℓ^{p})$ ever be amenable?” (Studia Math. 188 (2008), 151-174)”