On Generic Well-posedness of Restricted Chebyshev Center Problems in Banach Spaces

Let B （resp. K, BC,KC） denote the set of all nonempty bounded （resp. compact, bounded convex, compact convex） closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let B° stand for the set of all F ∈B such that the problem （F, G） is well-posed. We proved that, if X is strictly convex and Kadec, the set KC ∩ B° is a dense Gδ-subset of KC ／ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set B ／B° （resp. K ／ B°, BC ／B°, KC ／ B°） is a-porous in B （resp. K,BC, KC）. Moreover, we prove that for most （in the sense of the Baire category） closed bounded subsets G of X, the set K ／ B° is dense and uncountable in K.     

An Extended Ockham Algebra with Endomorphism Kernel Property

An algebraic structure A is said to have the endomorphism kernel property if every congruence on A, other than the universal congruence, is the kernel of an endomorphism on A. In this paper, we consider the EKP （that is, endomorphism kernel property） for an extended Ockham algebra A. In particular, we describe the structure of the finite symmetric extended de Morgan algebras having EKP.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.     

Universal Similarity Factorization Equalities over Generalized Clifford Algebras

For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate.     

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.     

The Existence of Maximal w^＊-closed Submodules in Nest Algebra Modules

Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for U to have a maximal w^＊-closed submodule.     

The Galois Correspondence in Field Algebra of G-spin Model

Suppose that G is a finite group and D（G） the double algebra of G. For a given subgroup H of G, there is a sub-Hopf algebra D（G; H） of D（G）. This paper gives the concrete construction of a D（G; H）-invariant subspace AH in field algebra of G-spin model and proves that if H is a normal subgroup of G, then AH is Galois closed.     

Additive Mappings Preserving Rank-one Idempotents

A complete characterization of additive mappings φ ： B（X）→B（Y）, which map rank-one idempotents to themselves or to zero, is given.     

On Maximal Abelian Self-adjoint Subalgebras of Factors of Type Ⅱ1

In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.     

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

The Density of Linear Symplectic Cocycles with Simple Lyapunov Spectrum in YIC（X, SL（2, R））

Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V, there exists a Zn∈G such that Zn belong to Vn and 0 〈 m（Zn） 〈 1/n. Let T be an ergodic automorphism of （X, G） preserving m, and A belong to the space of linear measurable symplectic cocycles     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

Multiplicative Mappings of Rings

Let R and F be arbitrary associative rings. A mapping φ of R onto F is called a multiplicative isomorphism if φ is bijective and satisfies φ（xy） = φ（x）φ（y） for all x, y ∈ R. In this short note, we establish a condition on R, in the case where R may not contain any non-zero idempotents, that assures that φ is additive, which generalizes the famous Martindale＇s result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.