Ideal amenability of Banach algebras on locally compact groups

In this paper we study the ideal amenability of Banach algebras. LetA be a Banach algebra and letI be a closed two-sided ideal inA, A isI-weakly amenable ifH ¹(A,I ^*) = {0}. Further,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenability of Banach algebras on locally compact groups.     

Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.     

(Super) module amenability, module topological centre and semigroup algebras

For a Banach algebra $\mathcal{A}For a Banach algebra A\mathcal{A} which is also an \mathfrakA\mathfrak{A}-bimodule, we study relations between module amenability of closed subalgebras of A"\mathcal{A}', which contains A\mathcal{A}, and module Arens regularity of A\mathcal{A} and the role of the module topological centre in module amenability of A"\mathcal{A}'. Then we apply these results to A=l¹(S)\mathcal{A}=l^{1}(S) and \mathfrakA=l¹(E)\mathfrak{A}=l^{1}(E) for an inverse semigroup S with subsemigroup E of idempotents. We also show that l ¹(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group \fracS ? \frac{S}{\approx}, is amenable. Moreover, we define super module amenability and show that l ¹(S) is super module amenable if and only if \fracS ? \frac{S}{\approx} is finite.     

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If \frak A {\frak A} is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \frak A {\frak A} (i) the quotient \frak A/L {\frak A}/L has the approximation property and (ii) the canonical map from \frak A \check? L^{^} {\frak A} \check{\otimes} L^\perp to (\frak A / L) \check? L^{^} ({\frak A} / L) \check{\otimes} L^\perp is open, then \frak A {\frak A} is finite-dimensional. As an application, we show that, if \frak A {\frak A} is an amenable Banach algebra whose underlying Banach space is an \scr L^p {\scr L}^p -space with p ? (1,￥) p\in (1,\infty) such that for each maximal left ideal L the quotient \frak A/L {\frak A}/L has the approximation property, then \frak A {\frak A} is finite-dimensional.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Graded rings and essential ideals

LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentA _e and essential ideals of the smash productA#G ^*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.     

Countability properties of the ideal space of a Banach algebra and d-algebras

The space Id(A) of closed two-sided ideals of a Banach algebra A carries the weak topology. IfA is separable Id(A) is expected to satisfy topological countability conditions. Some results concerning first and second countability are achieved.     

A short proof of a theorem on complex lindenstrauss spaces

A short proof is given of a theorem of Lima which asserts that a Banach spaceA with the property that every family of four balls inA with the weak intersection property has a non-empty intersection, is a Lindenstrauss space, i.e.A* is isometric to anL ¹(μ)-space.     

Dimension de l’anneau des polynomes a valeurs entieres

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.      

Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I_>, which consists of all elements of A such that v(a)>v(I), for all Rees valuations v of I. The ideal I_> plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition Af and is contained in every reduction of I. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.     

Ideal Structure and Stable Rank of $${\mathbb {C} e+ \ell^2(I)}$$ with the Hadamard Product

Let I be any index set. We consider the Banach algebra \mathbb C e+ l²(I){\mathbb {C} e+ \ell^2(I)} with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For I=\mathbb N{I=\mathbb {N}} we also characterize all prime z-ideals in this Banach algebra.     

On the generalized Lie structure of associative algebras

We study the structure of Lie algebras in the category ^H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in ^H MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain. Dedicated to the memory of S. A. Amitsur Supported by a Fulbright grant. Supported by NSF grant DMS-9203375.     

Two-Sided Closed Ideals of Certain Algebras of Unbounded Operators

This paper investigates the two-sided uniformly closed ideals of the maximal Op*-algebra L⁺(D) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non-trivial two-sided closed ideals of L⁺(D) is well-ordered by inclusion and the α-th closed ideal ??_α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then ??_α. An element A in L⁺(D) belongs to the minimal closed ideal ??₀ if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.     

The consistency strength of “every stationary set reflects”

The consistency strength of a regular cardinal so that every stationary set reflects is the same as that of a regular cardinal with a normal idealI so that everyI-positive set reflects in aI-positive set. We call such a cardinal areflection cardinal and such an ideal areflection ideal. The consistency strength is also the same as the existence of a regular cardinal κ so that every κ-free (abelian) group is κ⁺-free. In L, the first reflection cardinal is greater than the first greatly Mahlo cardinal and less than the first weakly compact cardinal (if any). Research supported by NSERC grant # A8948. Publication # 367. Research partially supported by the BSF.     

Topological and bornological characterisations of ideals in von Neumann algebras: II

SupposeM is a von Neumann algebra on a Hilbert spaceH andI is any norm closed ideal inM. We extend to this setting the well known fact that the compact operators on a Hilbert space are precisely those whose restrictions to the closed unit ball are weak to norm continuous.     

Regularity of ideals and their radicals

In this paper we compare the regularityreg I of a homogeneous idealI⊂K[x ₁,...,x _n] with that of its radical. We prove that regI ≥ reg ifR/I is a BuchsbaumR-module or ifI is a monomial ideal. We also prove the same result when defines a non-singular curve inP ³ under some additional hypotheses     

A semilattice structure for Arens products

Let (A,?) be a Banach algebra. Then for n∈?, A ⁽²ⁿ⁾ has 2 ⁿ Arens products. In this paper we study the relations between the Arens products on A ⁽²ⁿ⁾. Moreover, if P _n (A) denotes the set of all Arens products on A ⁽²ⁿ⁾, for n∈?, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)?{?} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.     

Approximate and pseudo-amenability of various classes of Banach algebras

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of ?¹-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.     

Approximate amenability is not bounded approximate amenability

We obtain a new criterion sufficient for approximate amenability of Banach algebras, and an associated criterion sufficient for approximate amenability of _c0$c_0$-direct sums of approximately amenable Banach algebras. We use these criteria to give examples of Banach algebras which are approximately amenable, but not boundedly approximately amenable. Thus we answer a question which was open since the year 2004. This is, so to speak, the complementary result to one of our earlier ones in the paper [F. Ghahramani, C.J. Read, Approximate identities in approximate amenability, J. Funct. Anal. 262 (9) (2012) 3929–3945], where we gave examples of boundedly approximately amenable Banach algebras that are not boundedly approximately contractible.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.