Quasi‐convexly dense and suitable sets in the arc component of a compact group

Let G be an abelian topological group. The symbol $\widehat{G}Let G be an abelian topological group. The symbol $\widehat{G}$ denotes the group of all continuous characters $\chi :G\rightarrow {\mathbb T}$ endowed with the compact open topology. A subset E of G is said to be qc‐dense in G provided that χ(E)?φ([? 1/4, 1/4]) holds only for the trivial character $\chi \in \widehat{G}$, where $\varphi : {\mathbb R}\rightarrow {\mathbb T}={\mathbb R}/{\mathbb Z}$ is the canonical homomorphism. A super‐sequence is a non‐empty compact Hausdorff space S with at most one non‐isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super‐sequence converging to 0 that is qc‐dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism $r:\widehat{G}\rightarrow \widehat{G}_a$ defined by $r(\chi )=\chi \upharpoonright _{G_a}$ for $\chi \in \widehat{G}$, is a topological isomorphism. We show that an infinite compact group G is connected if and only if its arc component G_a contains a super‐sequence converging to the identity that is qc‐dense in G and generates a dense subgroup of G. We also offer a short alternative proof of the result of Hofmann and Morris on the existence of suitable sets of minimal size in the arc component of a compact connected group.     

Locally finite Lie algebras with root decomposition

Let K be an algebraically closed field of characteristic zero, $\frak {g}$ be a countably dimensional locally finite Lie algebra over K, and $\frak {h} \subset \frak {g}$ be a (a priori non-abelian) locally nilpotent subalgebra of $\frak {g}$ which coincides with its zero Fitting component. We classify all such pairs $(\frak {g}, \frak {h})$ under the assumptions that the locally solvable radical of $\frak {g}$ equals zero and that $\frak {g}$ admits a root decomposition with respect to $\frak {h}$. More precisely, we prove that $\frak {g}$ is the union of reductive subalgebras $\frak {g}_n$ such that the intersections $\frak {g}_n \cap \frak {h}$ are nested Cartan subalgebras of $\frak {g}_n$ with compatible root decompositions. This implies that $\frak {g}$ is root-reductive and that $\frak {h}$ is abelian. Root-reductive locally finite Lie algebras are classified in [6]. The result of the present note is a more general version of the main classification theorem in [9] and is at the same time a new criterion for a locally finite Lie algebra to be root-reductive. Finally we give an explicit example of an abelian selfnormalizing subalgebra $\frak {h}$ of $\frak {g} = \frak {sl}(\infty)$ with respect to which $\frak {g}$ does not admit a root decomposition.Work Supported in Part by the University of Hamburg and the Max Planck Institute for Mathematics, Bonn     

The influence of $\pi$-quasinormality of some subgroups of a finite group

A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following:Theorem 3.4. Let ${\cal F}$ be a saturated formation containing the supersolvable groups. Suppose that G is a group with a normal subgroup H such that $G/H \in {\cal F}$, and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in G, then $G \in {\cal F}$. Received: 10 May 2002     

On spherical ideals of Borel subalgebras

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of These are ideals of such that their G-saturation is a spherical G-variety. We classify all maximal spherical ideals of for all simple G.Received: 25 March 2004     

On the arc component of a locally compact Abelian group

For a topological group G, we denote by G _a the arc component of the neutral element and by the character group of G, i.e. the group of all continuous homomorphisms from G into T. We prove the following theorem: Let G be a connected locally compact abelian group and let be the embedding. Then is a topological isomorphism. In particular, the character group of the arc component of a compact abelian group is discrete. Some conclusions will be drawn.     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

2-极大子群的$F_s$-拟正规性

$F$是一个群系. $G$的子群$H$在$G$中称为$F_s$-拟正规的,如果存在$G$的正规子群$T$,使得$HT$在$G$中是$s$-置换的并且$(H\cap T)H_G/H_G$包含在$G/H_G$的$F$超中心$Z^F_\infty(G/H_G)$中.本文利用$F_s$-拟正规子群研究了有限群的结构.获得了某些新的结果.     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

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}}} $.     

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed(resp．bounded closed)boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X．Let K(X)denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance．Moreover,let KG(X)denote the closure of the set {A∈K(x)：A∩G=0}．We prove that the set of all A∈KG(X)(resp．A∈K(X)),such that the minimization (resp．maximization)problem min(A,G)(resp．max(A,G))is well posed,contains a dense Gδ-subset of KG(X)(resp．K(X))．thus extending the recent results due to Blasi,Myjak and Papini and Li．     

The dimension of the kernel in an intersection of starshaped sets

We establish the following Helly-type theorem: Let ${\cal K}$ be a family of compact sets in $\mathbb{R}^d$. If every d + 1 (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel contains a translate of set A, then $\cap \{ K : K\; \hbox{in}\; {\cal K} \}$ also is a starshaped set whose kernel contains a translate of A. An analogous result holds when ${\cal K}$ is a finite family of closed sets in $\mathbb{R}^d$. Moreover, we have the following planar result: Define function f on $\{0, 1, 2\}$ by f(0) = f(2) = 3, f(1) = 4. Let ${\cal K}$ be a finite family of closed sets in the plane. For k = 0, 1, 2, if every f(k) (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel has dimension at least k, then $\cap \{K : K\; \hbox{in}\; {\cal K}\}$ also is a starshaped set whose kernel has dimension at least k. The number f(k) is best in each case.Received: 4 June 2002     

A cohomological property of <Emphasis Type="Italic">p</Emphasis>-groups

Let p be a prime, a finite p-group, any finite group with order divisible by p, and any action of on . We show that the cardinality of the set of all derivations with respect to this action is a multiple of p. This generalises theorems of Frobenius and Hall. Received: 16 June 2003     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

Finite Gap Jacobi Matrices, II.?The Szeg? Class

Let \frake ì \mathbbR\frak{e}\subset\mathbb{R} be a finite union of disjoint closed intervals. We study measures whose essential support is \frake{\frak{e}} and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

关于图的符号边全控制数

Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.     

Positive-definite functions on infinite-dimensional groups

This paper concerns positive-definite functions on infinite-dimensional groups G. Our main results are as follows: first, we claim that if G has a σ-finite measure μ on the Borel field whose right admissible shifts form a dense subgroup G ₀, a unique (up to equivalence) unitary representation (H, T) with a cyclic vector corresponds to through a method similar to that used for the G–N–S construction. Second, we show that the result remains true, even if we go to the inductive limits of such groups, and we derive two kinds of theorems, those taking either G or G ₀ as a central object. Finally, we proceed to an important example of infinite-dimensional groups, the group of diffeomorphisms on smooth manifolds M, and see that the correspondence between positive-definite functions and unitary representations holds for under a fairy mild condition. For a technical reason, we impose condition (c) in Sect. 2 on the measure space throughout this paper. It is also a weak condition, and it is satified, if G is separable, or if μ is Radon. This research was partially supported by a Grant-in-Aid for Scientific Research (No.18540184), Japan Socieity of the Promotion of Science.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Zeros of Monomial Brauer Characters

Let G be a finite group and p be a fixed prime. A p-Brauer character of G is said to be monomial if it is induced from a linear p-Brauer character of some subgroup(not necessarily proper) of G. Denote by IBr_m(G) the set of irreducible monomial p-Brauer′characters of G. Let H = G′O~p′(G) be the smallest normal subgroup such that G/H is an abelian p′-group. Suppose that g ∈ G is a p-regular element and the order of gH in the factor group G/H does not divide |IBr_m(G)|. Then there exists ? ∈ IBr_m(G) such that ?(g) = 0.