Closed ideals of a quasianalytic Fréchet algebra

In this note, we examine the structure of closed ideals of a quasianalytic weighted Beurling algebra A\cal {A}. This algebra is contained in C^￥ (G){\cal C}^\infty (\mit\Gamma) and contains the set A^￥ (D)A^\infty (D). Like in a previous article (see [6]), we use division properties and we give a characterization of closed ideals I such that I?A^￥ 1 { 0}I\cap A^\infty\! \ne \{ 0\} . Then, we use a factorization property proved in [2], which allows us to describe all the closed ideals of A\cal {A}.     

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.     

The coherence of complemented ideals in the space of real analytic functions

We characterize when an ideal of the algebra ${A(\mathbb{R}^d)}We characterize when an ideal of the algebra A(\mathbbR^d){A(\mathbb{R}^d)} of real analytic functions on \mathbbR^d{\mathbb{R}^d} which is determined by the germ at \mathbb R^d{\mathbb {R}^d} of a complex analytic set V is complemented under the assumption that either V is homogeneous or V?\mathbbR^d{V\cap \mathbb{R}^d} is compact. The characterization is given in terms of properties of the real singularities of V. In particular, for an arbitrary complex analytic variety V complementedness of the corresponding ideal in A(\mathbbR^d){A(\mathbb{R}^d)} implies that the real part of V is coherent. We also describe the closed ideals of A(\mathbbR^d){A(\mathbb{R}^d)} as sections of coherent sheaves.     

On the closed Lie ideals of certainC *-algebras

In this article we determine the closed Lie ideals of a uniformly hyperfiniteC ^*-algebra, and of the tensor product of such an algebra withC(X), the space of continuous functions on a compact, Hausdorff space. This is done by localizing the Lie ideals in algebras of the form , where is an algebra over a field of characteristic not equal to 2.This research is partially supported by NSERC (Canada)     

The $${{\rm GL}(2,\mathbb{C})}$$ McKay correspondence

In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}, it means that the endomorphism ring of the special CM \mathbbC{\mathbb{C}} [[x, y]] ^G -modules can be used to build the dual graph of the minimal resolution of \mathbbC²/G{\mathbb{C}^{2}/G}, extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of SL(2,\mathbbC){{\rm SL}(2,\mathbb{C})} to all finite subgroups of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}.     

Derivations into duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra A. LetI be a closed two-sided ideal inA, we sayA is I-weakly amenable if the first cohomology group ofA with coefficients in the dual space I* is zero; i.e.,H ¹(A, I*) = {0}, and,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study theI-weak amenability of a Banach algebraA for some special closed two-sided idealI.     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

Ideals of some configurations of lines in $${\mathbb{P}^n_\mathbf{K}}$$ : generators and macaulayness

We consider ideals arising from the intersection of hyperplanes of the projective space \mathbbPⁿ{\mathbb{P}^n} belonging to a partition. We determinate their generators and we prove that they are Cohen-Macaulay.     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

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.     

Squared cycles in monomial relations algebras

Let be an algebraically closed field. Consider a finite dimensional monomial relations algebra of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern in the syzygies of a specific factor module of the corresponding indecomposable projective module is found, allowing us to conclude that the square of any cycle must lie in the ideal I.     

Finite Groups with $$\mathfrak F$$ -Subnormal Subgroups

Mathematical Notes - Codes in the dihedral group algebra $$\mathbb{F}_q{D_{2n}}$$, i.e., left ideals in this algebra, are studied. A generating idempotent is constructed for every code in...     

Ideals of a C *-algebra generated by an operator algebra

In this paper, we consider ideals of a C ^*-algebra C^*(B){C^*(\mathcal{B})} generated by an operator algebra B{\mathcal{B}} . A closed ideal J í C^*(B){J\subseteq C^*(\mathcal{B})} is called a K-boundary ideal if the restriction of the quotient map on B{\mathcal{B}} has a completely bounded inverse with cb-norm equal to K ⁻¹. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from C^*(B){C^*(\mathcal{B})} onto B{\mathcal{B}} which is a projection on B{\mathcal{B}} is completely bounded. Moreover, we prove that Kadison’s similarity problem is decided on one particular C ^*-algebra which is a completion of the *-double of M₂(\mathbbC){M_2(\mathbb{C})} .     

Cyclic codes over $${{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}$$

In this work, we focus on cyclic codes over the ring \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287, 2007) to characterize the ring (\mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂)/(xⁿ-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over \mathbbF₂+u\mathbbF₂+v\mathbbF₂+uv\mathbbF₂{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general.     

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

Embeddings of general curves in projective spaces: the range of the quadrics

Let C ì \mathbbP^r C \subset {\mathbb{P}^r} be a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either h⁰( \mathbbP^r,I_C(2) ) = 0 {h^0}\left( {{\mathbb{P}^r},{\mathcal{I}_C}(2)} \right) = 0 or h¹( \mathbbP^r,I_C(2) ) = 0 {h^1}\left( {{\mathbb{P}^r},{\mathcal{I}_C}(2)} \right) = 0 (a problem called the maximal rank conjecture in the range of quadrics).     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

Upper Bounds for $${\mathbb R}$$-Linear Resolvents

We discuss upper bounds for the resolvent of an \mathbbR{\mathbb{R}}-linear operator in \mathbbC^d{\mathbb{C}^d}.