Equivariant K-theory of compactifications of algebraic groups

In this paper we describe the G × G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl group. We further give an explicit basis of K(X) over K(G/B), and also determine the structure constants with respect to this basis.     

Arens regularity ofK(X)

LetX be a reflexive Banach space andK(X) the operator algebra of compact linear operatorsu:XX. In this note we prove the following two results: a) Any decomposable bilinear operator fromK(X)×K(X) intoK(X) is Arens regular; b) ifX has an unconditional basis then any bounded bilinear operator fromK(X)×K(X) intoK(X) is Arens regular.     

关于有界线性算子的非紧测度的若干性质

利用非紧测度在商空间B(X,Y)/K(X,Y)上构造了一个范数,指出这一范数可由B(X,Y)上的某一范数生成.并且X或Y是Hilbert空间时,B(X,Y)/K(X,Y)上赋于这种范数时是完备的.另外,还建立了非紧测度与熵数,近似数之间的一些联系.     

K(o)the-Bochner空间的凸性

本文利用K(o)the函数空间的性质以及K(o)the函数空间与K(o)the-Bochner空间的关系,讨论了K(o)the-Bochner空间E(X)的凸性,主要结果如下:(a)给出E(X)的端点的充分条件,得到了E(X)严格凸的判据,相应地推广了Lp(μ,X)以及LΦ(X)的结果;(b)讨论了E(X)的弱局部一致凸和局部完全k-凸;(c)刻画了E(X)的强凸,给出了E(X)强凸的充要条件.     

Agreeable domains

An integral domain D with quotient field K is defined to be agreeable if for each fractional ideal F of D[X] with F C K[X] there exists 0 = s ε D with sF C D[X]. D is agreeable ? D satisfies property (*) (for 0 ^ f(X) G K[X], there exists 0 = s ε D so that f(X)g(X) ε D[X] for g(X) ε K[X] implies that sg(X) ε D[X]) &; D[X] is an almost principal domain, i.e., for each nonzero ideal I of D[X] with IK[X] = K[X], there exists f(X) ε I and 0 = s ε D with sI C (f(X)). If D is Noetherian or integrally closed, then D is agreeable. A number of other characterizations of agreeable domains are given as are a number of stability properties. For example, if D is agreeable, so is ?^αD_{P α} and for a pair of domains D?D′ with a [DD:′]≠0, D is agreeable?D′ is agreeable. Results on agreeable domains are used to give an alternative treatment of Querre's characterization of divisorial ideals in integrally closed polynomial rings. Finally, the various characterizations of D being agreeable are considered for polynomial rings in several variables.     

关于K—极凸Banach空间

引进Ｋ－极凸Ｂａｎａｃｈ空间，证明了ＸＫ－极凸当且仅当Ｘ自反、Ｋ－严格凸且有（Ｈ）性质，得到了Ｋ－极凸空间的一些性质，并讨论了Ｋ－极凸与Ｋ－Ｋ－强光滑、Ｋ－一致凸及完全Ｋ－凸的关系。     

Permanence properties of C(X,E)

Let C(X,E) be the vector space of all continuous functions on a completely regular Hausdorff space X with values in a locally convex space E, equipped with the compact-open topology. In this note it will be shown that for many classes of locally convex spaces K C(X,E) lies in K if and only if C(X)=C(X,¦K) (¦K=¦R or C) and E belong to K. This is valid for the classes K. of all metrizable, normed, (DF)-, -locally topological, separable, quasi-complete, complete, nuclear, Schwartz, semi-Montel, Montel, semi-reflexive, reflexive and quasi-normable locally convex spaces, respectively. But in general C(X,E) is not quasi-barrelled, barrelled and bornological, respectively, if C(X) and E belong to the same class. We shall give sufficient conditions for C(X,E) to be quasi-barrelled and barrelled, respectively.Herrn Professor Gottfried Köthe zum 75. Geburtstag gewidmet     

Models of Curves and Finite Covers

Let K be a discrete valuation field with ring of integers O _K .Letf : X ! Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over O K of X and of Y. Three such relationships are listed below. Consider a Galois cover f : X ! Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K,thenX achieves semi-stable reduction over some explicit tame extension of K.B/.WhenK is strictly henselian, we determine the minimal extension L=K with the property that X L has semi-stable reduction. Let f : X ! Y be a finite morphism, with g.Y/ > 2. We show that if X has a stable model X over O _K ,thenY has a stable model Y over O _K , and the morphism f extends to a morphism X ! Y. ! Y. Finally, given any finite morphism f : X ! Y, is it possible to choose suitable regular models X and Y of X and Y over O _K such that f extends to a finite morphism X ! Y ?As wasshown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situ-ations, with f a cyclic cover of any order > 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.     

AUTOMORPHISMS OF SL (2, K) OVER SKEW FIELDS

In this paper, the author proves the following resu: It Let K be a skew field and A be an automorphism of SL(2, K). Then there exists A∈GL(2, K), an automorphism σ or an anti-automorphism τ of K, such that A is of theform AX=AX~σA~(-1) for all X∈SL(2, K)or AX=A(X~τ~2)~(-1)A~(-1) for all X∈SL(2, K),where X~σ, X~τ are the matrices obtained by applying σ, τ on X respee tively and X' is thetranspose of X.     

复合形在欧氏空间中的实现问题Ⅲ

<正> 设 K 是一个有限单形复合形,我们恒可视 K 为一充分高维数 N 的欧氏空间中的欧氏复合形,此时其所定空间将记为(?).在研究 K 是否可实现于某一确定维数 m 的欧氏盘间R~m 中的时候,我们曾引进下面的一些定义(见[1],记号略有不同):     

λ-重K_(1,4)-设计到λ-重K_(1,3)-最大填充的变形

令(X,B)为一个u阶的λ-重K_(1,4)-设计.对于每一个区组B=(a:b,c,d,e)∈B,若删去边{a,e},则得到一个K_(1,3)[a:b,c,d].令C为删去B中每一个区组的边{a,e}而得到的K_(1,4)的集合,F为被删去的边构成的集合.若F可以被重组成[λv(v-1)/24]个K_(1,3)的集合D,则(X,CUD)为一个v阶λ-重K_(1,3)-最大填充.称(X,C∪D)为λ-重K_(1,4-)设计(X,B)的变形.本文证明了v阶λ-重K_(1,4)-设计到u阶λ-重K_(1,3)-最大填充的变形存在的充要条件是λv(v-1)≡0(mod 8)且v≥5.     

强向量拟均衡问题解的存在性(英文)

设X是一个实的Hausdorff拓扑向量空间,Y是一个实的局部凸向量空间,C是Y中的闭凸锥,K（?）X是一个紧子集.F:X×X→Y是一个双向量函数,G:K→2K是一个集合值映射.我们考虑下面的强拟均衡问题:存在x∈G（x）,使得对任意的y∈G（x）,成立F（x,y）∈C.本文证明了当F是半连续时,上述问题解的存在性结论.     

广义共同逼近问题的适定性

设C是实Banach空间X中有界闭凸子集且0是C的内点，G是X中非空闭的有界相对弱紧子集.记K(X)为X的非空紧凸子集全体并赋Hausdorff距离，KG(X)为集合｛A∈K(X)；A∩G=｝的闭包.称广义共同逼近问题minC(A,G)是适定的是指它有唯一解(x0,z0)，且它的每个极小化序列均强收敛到(x0,z0).在C是严格凸和Kadec的假定下，证明了｛A∈K(X)；minC(A，G)是适定的｝含有KG(X)中稠Gδ子集，这本质地推广和延拓了包括De Blasi,Myjak and Papini［1］、Li［2］和De Blasi and Myjak［3］等人在内的近期相应结果.     

A NOTE ON SUBCATEGORIES OF REACHABLE MACHINES

Abstract

Given a category K. a state-behaviour process X: K → K and an image factorization system (E,M) for K, it is shown that under mild conditions the full subcategory with objects all reachable machines is coreflective. A corresponding result is obtained for coequalizer-reachable machines if X is only an input process.     

Total Valuation Rings of Ore Extensions

We consider extensions of a total valuation ring V of a skew field K to the Ore extension K(X;σ, δ) for an endomorphism σ of K and a σ-derivation δ. It is shown that there exists an extension R of V with X ∈ R, such that ${\overline X}$ is transcendental over V/J(V) if and only if (σ,δ) is compatible with V, where ${\overline X} = X + J(R^(1))$ . In the case V is invariant, it is established that there is an invariant extension R of V in K(X;σ,δ) such that ${\overline X}$ is transcendental if and only if σ(a)V = aV and δ(a) ∈ aV for all a ∈ K.     

When K+(n-4)L fails to be nef

Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. We study polarized pairs (X,L) for which K+(n−3)L is nef but K+(n−4)L fails to be nef. Supported by MURST funds     

Compactifications,realcompletions and spectral structures

Summary We give an intrinsic (set theoretical) method to obtain all realcompletions of a Tychonoff space X. It is based on the concept of a spectral structure. Every realcompletion of the space can be obtained as a space of terminal clusters relative to an appropriate spectral structure on X. Various applications of this concept are then given. For example we may characterize those spectral structures which yield the realcompletions between X and X, or which yield spaces that are realcomplete (that is, realcompact), or compact or pseudocompact or Lindelöf. We also determine the class K of compactifications K of X for which X will be real closed in K for every K K.Here we continue the use of the word realcompletion (see [2]) in place of real-compactification. As many investigations have shown and as the present article reinforces, the former is more appropriate than the latter.     

线性算子的摄动定理

该文利用Mbekhta M于1987年引入的两个子空间来研究线性算子的摄动. 证明了如下结论：设X=K(T)+W, 其中K(T), W均闭, dim［K(T)∩N(T)］< ∞. 若TWW, TW闭, 且存在闭子空间N, 使W=［W∩N(T)］N, 则： 当S∈B(X)可逆, ST= TS, SWW, 且‖S‖充分小时， T-S为上半Fredholm算子. 在上条件下, 若dimN<∞, K(T′)闭, 则T-S为Fredholm算子, 且R(T-S)=X.     

Topological dynamics of transformations induced on the space of probability measures

LetT be a continuous transformation of a compact metric spaceX. T induces in a natural way a transformationT _M on the spaceM (X) of probability measures onX, and a transformationT _K on the spaceK (X) of closed subsets ofX. This note investigates which of the topological properties ofT∶X→X (like distality, transitivity, mixing property etc. ...) are “inherited” byT _M∶M (X)→M (X) andT _K∶K (X)→K (X).