About the ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk . Let Mult(A,‖⋅‖) be the set of continuous multiplicative semi-norms of A, let Multm(A,‖⋅‖) be the subset of the ?∈Mult(A,‖⋅‖) whose kernel is a maximal ideal and let Multa(A,‖⋅‖) be the subset of the ?∈Multm(A,‖⋅‖) whose kernel is of the form (if ?∈Multm(A,‖⋅‖)?Multa(A,‖⋅‖), the kernel of ? is then of infinite codimension). The main problem we examine is whether Multa(A,‖⋅‖) is dense inside Multm(A,‖⋅‖) with respect to the topology of simple convergence. This a first step to the conjecture of density of Multa(A,‖⋅‖) in the whole set Mult(A,‖⋅‖): this is the corresponding problem to the well-known complex corona problem. We notice that if ?∈Multm(A,‖⋅‖) is defined by an ultrafilter on D, ? lies in the closure of Multa(A,‖⋅‖). Particularly, we shaw that this is case when a maximal ideal is the kernel of a unique ?∈Multm(A,‖⋅‖). Thus, if every maximal ideal is the kernel of a unique ?∈Multm(A,‖⋅‖), Multa(A,‖⋅‖) is dense in Multm(A,‖⋅‖). And particularly, this is the case when K is strongly valued. In the general context, we find a subset of Multm(A,‖⋅‖)?Multa(A,‖⋅‖) which is included in the closure of Multa(A,‖⋅‖). More generally, we show that if ψ∈Mult(A,‖⋅‖) does not define the Gauss norm on polynomials (‖⋅‖), then it is characterized by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ does not lie in the closure of Multa(A,‖⋅‖), then its restriction to polynomials is the Gauss norm.     

Weak generators of the algebral ∞ and the commutant of a model operator

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H^∞. It is known that ϕ+BH^∞ is a weak^* generator of the algebra H^∞/BH^∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak^* generator of the algebra l^∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B₁·…·B_N, where N ∈ℕ, and B₁, …, B_N are Blaschke products satisfying condition (C), there exists a function ϕ∈H^∞ such that ϕ(Λ) is a weak^* generator of the algebra l^∞, and ϕ+BH^∞ is not a weak^* generator of the algebra H^∞/BH^∞. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 73–85. Translated by M. F. Gamal'.     

Uniqueness of the uniform norm and adjoining identity in Banach algebras

LetA _e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,A _e admits exactly one uniform norm (not necessarily complete) if so doesA. This is used to show that the spectral extension property carries over fromA to A _e . Norms onA _e that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥_op onA _e defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥_op restricted toA is equivalent to ∥ · ∥).     

Dual space,carleson measure and sequence interpolation onA p (ϕ)(O<p<1)

LetD={z∈Σ:|z|<1} and ϕ be a normal function on [0,1). Forp∈(0,1) such a function ϕ is used to define a Bergman spaceA ^p (ϕ) onD with weight ϕ ^p (|·|)/(1-|·|²). In this paper, the dual space ofA ^p (ϕ) is given, four characteristics of Carleson measure onA ^p (ϕ) are obtained. Moreover, as an application, three sequence interpolation theorems inA ^p (ϕ) are derived. Supported by the Doctoral Program Foundation of Institute of Higher Education, P.R. China.     

Large antipodal families

A family {A _i | i ∈ I} of sets in ℝ ^d is antipodal if for any distinct i, j ∈ I and any p ∈ A _i , q ∈ A _j , there is a linear functional ϕ:ℝ ^d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ _i∈I A _i . We study the existence of antipodal families of large finite or infinite sets in ℝ³. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.     

Commutators in model spaces

Let θ be an inner function, let K _θ = H ² ⊖ θH ², and let S_θ : K_θ → S_θ be defined by the formula S_θf = P_θzf, where f ∈ K_θ is the orthogonal projection of H² onto K_θ. Consider the set A of all trace class operators L : K_θ → K_θ, L = ∑(·,u_n)v_n, ∑∥u_n∥∥v_n∥ < ∞ (u_n, v_n ∈ K_θ), such that ∑ū_n v_n ∈ H ₀ ¹ . It is shown that trace class commutators of the form XS_θ − S_θX (where X is a bounded linear operator on K_θ) are dense in A in the trace class norm. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 54–61.     

Exchange Rings Generated by Their Units

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈U（R） such that 1R ± u ∈ U（R）, if and only if for any a ∈ R, there exists a u ∈ U（R） such that a ± u∈ U（R）. Phrthermore, we prove that, for any A ∈ Mn（R）（n ≥ 2）, there exists a U ∈ GLn（R） such that A ± U ∈ GLn（R）.     

The Corona problem on a complete ultrametric algebraically closed field

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult _m (A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult ₁(A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x ? a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult ₁(A, ‖. ‖) is dense in Mult _m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ? _p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult ₁(A, ‖. ‖), which suggest that Mult ₁(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).     

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains \frac12\frac{1}{2} and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism.     

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     

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

On the equivalence of measures on loop space

Let K be a simply-connected compact Lie Group equipped with an Ad _K -invariant inner product on the Lie Algebra ?, of K. Given this data, there is a well known left invariant “H ¹-Riemannian structure” on L(K) (the infinite dimensional group of continuous based loops in K), as well as a heat kernel ν_T(k ₀, ·) associated with the Laplace-Beltrami operator on L(K). Here T > 0, k ₀∈L(K), and ν _T (k ₀, ·) is a certain probability measure on L(K). In this paper we show that ν₁(e,·) is equivalent to Pinned Wiener Measure on K on ? _s0 ≡<x _t : t∈ [0, s ₀]> (the σ-algebra generated by truncated loops up to “time”s ₀). Recevied: 9 September 1999 / Revised version: 13 March 2000 / Published online: 22 November 2000     

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. The third author is Corresponding author.     

Strongly approximative similarity of operators

Two operators A, B ∈ B（H） are said to be strongly approximatively similar, denoted by A -sas B, if （i） given ε 〉 0, there exist Ki ∈ B（H） compact with ｜｜Ki｜｜ 〈ε（i = 1,2） such that A＋K1 and B ＋ K2 are similar; （ii） σ0（A） = σ0（B） and dim H（λ; A） = dim H（λ; B） for each λ ∈ σ0（A）. In this paper, we prove the following result. Let S,T ∈ B（H） be quasitriangular satisfying： （i） σ（T） = σ（S） = σw（S） is connected and σe（S） = σlre（S）; （ii） ρs-F（S） ∩ σ（S） consists of at most finite components and each component Ω satisfies that Ω = int Ω, where int Ω is the interior of Ω. Then, S -sas T if and only if S and T are essentially similar.     

Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V^{(m + 1)} \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V^(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

Generalizations of spectrally multiplicative surjections between uniform algebras

Let $ A $ A and ℬ be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: $ A $ A → ℬ with S(1)=T(1)= 1 and α ∈ ℂ \ {0} satisfy r(S(a)T(b) − α)= r(ab− α) for all a,b ∈ $ A $ A , then S=T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ _π (S(p)T(q)) ⊂ σ _π (ρ(p) τ(q)) for all p,q ∈ I, where σ _π (f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A.     

Algebras whose equivalence relations are congruences

It is proved that all the equivalence relations of a universal algebra A are its congruences if and only if either |A| ≤ 2 or every operation f of the signature is a constant (i.e., f(a ₁ , . . . , a _n ) = c for some c ∈ A and all the a ₁ , . . . , a _n ∈ A) or a projection (i.e., f(a ₁ , . . . , a _n ) = a _i for some i and all the a ₁ , . . . , a _n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x _a  = y _a for all x, y ∈ G). All the equivalence relations of a semigroup S are right congruences if and only if either |S| ≤ 2 or S can be represented as S = A∪B, where A is an inflation of a right zero semigroup, and B is the empty set or a left zero semigroup, and ab = a, ba = a ² for a ∈ A, b ∈ B. If G is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid G are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.     

Nonstandard arithmetic of iterated polynomials

LetK be an algebraic number field of finite degree andf(X,T) a polynomial overK. For eachφ(X)∈Z[X], we denote byE(φ) the set of all integersa with φ ^m (a) =φ ⁿ (a) for somem≠n. In this paper, we give a condition for a polynomialφ(X)∈Z[X] to satisfy the following; If forn∈N, there existr∈K anda∈Z−E(φ) such thatf r, φ ^m (a)=0, then there exists a rational functiong(X) overK andk∈N such thatf(g(T)), φ ^k (T))=0 .