Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Q-curves of degree 5 and jacobian surfaces of GL2-type

We construct a parametric family {E ^(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q , and the family {C(s,t,u)} of genus two curves over Q covering E ^{(+)(s,t,u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E ^{(+)(s,t,u) and its twist E ⁽⁻⁾(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509. Received: 23 September 1997 / Revised version: 26 May 1998     

A banach space containing non-trivial limited sets but no non-trivial bounding sets

An example of a Banach spaceE is given with the following properties: Every bounding setA⊂E (i.e.f(A) is bounded for each holomorphic functionf:E →C) is relatively compact but there are relatively non-compact limited setsA (i.e.T(A) is relatively compact for each bounded linear mapT:E →c ₀).     

Estimation of the number of fixed points of map extensions

Letf:(X,A)→(X,A) be an extension of a given map ψ:A→A, where (X,A) is a pair of compact polyhedra. We shall introduce a special Nielsen number,SN(f|ψ), which is a lower bound for the number of fixed points onX-A for all extensions in the homotopy class off. It is shown that for many space pairs this lower bound is the best possible one, and that it can be realized without the by-passing condition.     

Oscillation of Solutions of Linear Differential Equations

This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential equations f″＋e^-zf′＋Q（z）f=F（z）,whereQ（z）≡h（z）e^cz and c∈R.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

Standard commutator formula,revisited

Let R be an associative ring with 1; A, B ⊴ R be its ideals; C(n, R, A) be the full congruence subgroup of level A in GL(n, R); and E(n, R, A) be the relative elementary subgroup of level A. We present a very easy proof of the following commutator formula: [E(n, R, A),C(n, R, B)] = [E(n, R, A), E(n, R, B)] for all commutative rings based exclusively on the absolute standard commutator formula and its non-commutative counterparts. This generalizes and strengethens Mason and Stothers’ results, our results, and those of Hazrat and Zhang. For comaximal ideals A + B = R, we show that this commutator is E(n, R, AB + BA).     

Group Chromatic Number of Graphs without K5-Minors

 Let G be a graph with a fixed orientation and let A be a group. Let F(G,A) denote the set of all functions f: E(G) ↦A. The graph G is A -colorable if for any function f∈F(G,A), there is a function c: V(G) ↦A such that for every directed e=u v∈E(G), c(u)−c(v)≠f(e). The group chromatic numberχ₁(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m under a given orientation D. In [J. Combin. Theory Ser. B, 56 (1992), 165–182], Jaeger et al. proved that if G is a simple planar graph, then χ₁(G)≤6. We prove in this paper that if G is a simple graph without a K ₅-minor, then χ₁(G)≤5. Received: August 18, 1999 Final version received: December 12, 2000     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χ_g(G). In this note we will prove the following results. (1) Let H₁ and H₂ be two subgraphs of G such that V(H₁)∩V(H₂)=∅ and V(H₁)∪V(H₂)=V(G). Then χ_g(G)≤min{max{χ_g(H₁), max_v∈V(H2)deg(v,G)+1},max{χ_g(H₂), max_u∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K_3,3-minor, then χ_g(G)≤5.     

Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C ^*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E ^*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E ^*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].      

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

Realisierbarkeit von darstellungen endlicher gruppen in einheitswurzelkörpern

Zusammenfassung LetD:G→GL(n,C) be an irreducible linear representation of a finite groupG with the characterX. IfD is realizible in Q(ξ _m ) and Q(ξ _m′ ) we give a condition for then realizability ofD in Q(ξ_(m′)). If the degreen is a prime ≠ 2, we show thatD realizible in Q(ξ _f ), wheref is the conductor of the abelian extensionQ(X)/Q.     

A counterexample in copositive approximation

The present paper gives a converse result by showing that there exists a functionf ∈C _[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE _n ⁽⁰⁾ (f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E _n(f) is the ordinary best polynomial approximation off of degreen.     

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.