Complemented ideals in the Fourier algebra of a locally compact group

In this paper we provide a necessary condition for a closed ideal in the Fourier algebra of a locally compact amenable group to be completely complemented. The classification of completely complemented ideals is completed in the case of an amenable discrete group. We also investigate the ideals possessing a bounded approximate identity.

     

On existence of finite universal Korovkin sets in Segal algebras

We prove that there exists a finite universal Korovkin set w.r.t positive operators for the centre of a Segal algebra on a compact groupG if and only ifG is metrizable. As a consequence it follows that a Segal algebra on a compact abelian group admits a finite universal Korovkin set w.r.t. positive operators iff the group is metrizable.     

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.     

Nilpotent ideals in a class of Banach algebras

We introduce the concepts of approximately complemented subspaces of normed spaces and approximately biprojective algebras. We prove that any approximately biprojective Banach algebra with left and right approximate identities does not have a nontrivial nilpotent ideal whose closure is approximately complemented.

     

Remarks on Segal algebras

Let B be an abstract Segal algebra in some Banach algebra A. There was some belief that in the commutative case A should be semi-simple, if B is, but this is not so (Section I). It is well known that a (proper) abstract Segal algebra does not have bounded right approximate units. It may however have a left unit. Pseudosymmetric Segal algebras in the sense of Reiter do not have bounded left approximate units (Section II). A nonfactorization proof is given for a class of algebras which contains most of the known examples of Segal algebras on abelian groups (Section III).     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.     

Metrically and topologically projective ideals of Banach algebras

In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.     

A Characterization of π-Complemented Algebras

π-complemented algebras are defined as those algebras (not necessarily associative or unital) such that each annihilator ideal is complemented by other annihilator ideal. Let A be a semiprime algebra. We prove that A is π-complemented if, and only if, every idempotent in the extended centroid of A lies in the centroid of A. We also show the existence of a smallest π-complemented subalgebra of the central closure of A containing A. In the case that A is a C*-algebra, this subalgebra turns out to be a norm dense *-subalgebra of the bounded central closure of A. It follows that a C*-algebra is boundedly centrally closed if, and only if, it is π-complemented.     

Metric n-Lie Algebras

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.     

套代数模中的插值问题

Let U be a weakly closed nest algebra module acting on a Hilbert space H. Given two operators X and Y in B(H), a necessary and sufficient condition for the existence of an operator T in U satisfying TX = Y is provided.     

HBa(D)空间的内插定理

研究了一类由H^p空间构成的Ba空间的内插性质，给出了H^Ba(D)空间的三个内插定理。     

When is the Range of a Multiplier on a Banach Algebra Closed?

In this paper we prove the Theorem: Let A be a Banach algebra with a bounded approximate identity (=BAI) such that every proper closed ideal of A is contained in a proper closed ideal with a BAI. Then a multiplier T:A → A has a closed range iff T factors as a product of an idempotent multiplier and an invertible multiplier.This work of the author is supported in part by the Turkish Academy of Sciences.     

量子代数模的张量积与不可分解模

令M是Z[v]的由v-1和奇素数p生成的理想,U是A=Z[v]M上相伴于对称Cartan矩阵的量子代数.k是特征为零的代数闭域,A→k(v(?)ξ)是环同态.U_k=U(?)_Ak,u_k是U_k的无穷小量子代数.令ξ是1的p次本原根.本文证明了:若有限维可积U_k模M,V中至少有一个是内射模,或者M,V中有一个模作为u_k模是平凡的,则有U_k模同构M(?)V≌V(?)M.我们还证明了:若有限维可积U_k模V作为u_k模是不可分解的,有限维可积U_k模M是不可分解的,且M|_(uk)是平凡的,则V(?)M是不可分解U_k模.令V和M是有限维可积U_k模,作为u_k模是同构的且具有单基座,本文证明V和M作为U_k模也是同构的.由此得到:不可分解内射u_k模提升为U_k模是唯一的.     

Ideals with bounded approximate identities in Fourier algebras

We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.     

A contribution to Segal algebras

Let A and B be Banach algebras. If B is an abstract Segal algebra in A, we have a bijective correspondence between the strictly irreducible representations of A and those of B. This gives a bijective correspondence for maximal modular left ideals. If A and B have approximate right units, we obtain a bijective correspondence for right resp. Two-sided ideals. For two-sided ideals this correspondence preserves the property of an ideal having approximate right units. This generalizes a Theorem by H. Reiter.It would be sufficient to require B to be a Banachspace.     

二宽度CSL代数的直和分解

本文给出了可分Ｈｉｌｂｅｒｔ空间上的二宽度ＣＳＬ代数分解成其对角与它的一个范数闭理想（即满足Ｒｉｎｇｒｏｓｅ条件的算子集）直和的充要条件是：此二宽度ＣＳＬ是由一个有限ＣＳＬ与一个有补ＣＳＬ生成的．     

Schroedinger Flows on Compact Hermitian Symmetric Spaces and Related Problems

In this note,we prove that the Schroedinger flow of maps from a closed riemann surface into a compact irreducible Hermitian symmetic space admits a global weak solution.Also,we show the existence of weak solutions to the initial value problem of Heisenberg model with Lie algebra values,which is closely related to the Schroedinger flow on compact Hermitian symmetric spaces.     

包含紧算子理想的Toeplitz算子代数的刻画

设G为一个torsion-free的离散群,（G,G＋）为一个拟序群,记T^G (G)为相应的Toeplitz算子代数,K(l^2(G 1)为l^2(G )上的紧算子全体,本文证明了K(l^2(G ))增包含于T^G (G)当且仅当下列两个条件时满足。（1）（G,G＋）为一个序群,（2）G中存在一个最小的正元。     

A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations

Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. We only consider finite-dimensional Lie algebras over a fields of characteristic zero.     

Nakayama Twisted Centers and Dual Bases of Frobenius Cellular Algebras

For a Frobenius cellular algebra, we prove that, if the left (right) dual basis of a cellular basis is again cellular, then the algebra is symmetric. Moreover, some ideals of the center are constructed by using the so-called Nakayama twisted center.