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.     

On idempotent <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated to a von Neumann algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L_p(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L_p(M, τ), then A² ∈ L_p(M, τ). Let n be a positive integer, n > 2, and A = Aⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ_t(A) belong to the set {0} ∪ [‖A^n?2‖^?1, ‖A‖] for all t > 0; 2) either μ_t(A) ≥ 1 for all t > 0 or there is a t₀ > 0 such that μ_t(A) = 0 for all t > t₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L_p(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L₁(M, τ) and if A = A³ and A ? A² ∈ M, then τ(A) ∈ R.     

Density of characters of bounded <Emphasis Type="Italic">p</Emphasis>-adic analytic functions in the topological dual

Let K be an ultrametric complete algebraically closed field, let D be a disk {x ∈ K ‖x| < R} (with R in the set of absolute values of K) and let A be the Banach algebra of bounded analytic functions in D. The vector space generated by the set of characters of A is dense in the topological dual of A if and only if K is not spherically complete. Let H(D) be the Banach algebra of analytic elements in D. The vector space generated by the set of characters of H(D) is never dense in the topological dual of H(D).     

Centralizers of generalized skew derivations on multilinear polynomials

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x ₁,..., x _n ) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r ₁,..., r _n ): r _i ∈ R} be the set of all evaluations of f(x ₁,..., x _n ) in R, while A = {[G (f(r ₁,..., r _n )), f(r ₁,..., r _n )]: r _i ∈ R}, and let C _R (A) be the centralizer of A in R; i.e., C _R (A) = {a ∈ R: [a, x] = 0, ? _x ∈ A }. We prove that if A ≠ (0), then C _R (A) = Z(R).     

Differences of Idempotents In <Emphasis Type="Italic">C</Emphasis>*-Algebras and the Quantum Hall Effect

Let ? be a trace on the unital C*-algebra A and M _? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P ? Q ∈ M _? , then ?((P ? Q)²ⁿ⁺¹) = ?(P ? Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ M _? with ?(U-A) ∈ R. Then I-A, I?U ∈ M _? and ?(I?U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.     

New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n ? m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) ? w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).     

Equations in finite fields with restricted solution sets. I (Character sums)

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z _p and for h(x) ∈ Z _p [x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with a ∈ A, b ∈ B, x ∈ Z _p , and for large subsets A, B, C, D of Z _p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results.     

Order,type and cotype of growth for <Emphasis Type="Italic">p</Emphasis>-adic entire functions: A survey with additional properties

Let IK be a complete ultrametric algebraically closed field and let A(IK) be the IK-algebra of entire functions on IK. For an f ∈ A(IK), similarly to complex analysis, one can define the order of growth as \(\rho \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {\log |f|\left( r \right)} \right)}}{{\log r}}\). When ρ(f) ≠ 0,+∞, one can define the type of growth as \(\sigma \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {|f|\left( r \right)} \right)}}{{{r^\rho }\left( f \right)}}\). But here, we can also define the cotype of growth as \(\psi \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{q\left( {f,r} \right)}}{{{r^\rho }\left( f \right)}}\) where q(f, r) is the number of zeros of f in the disk of center 0 and radius r. Many properties described here were first given in the Houston Journal, but new inequalities linking the order, type and cotype are given in this paper: we show that ρ(f)σ(f) ≤ ψ(f) ≤ eρ(f)σ(f). Moreover, if ψ or σ are veritable limits, then ρ(f)σ(f) = ψ(f) and this relation is conjectured in the general case. Several other properties are examined concerning ρ, σ, ψ for f and f’. Particularly,we show that if an entire function f has finite order, then \(\frac{{f'}}{{{f^2}}}\) takes every value infinitely many times.     

Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Traces of Higher Negative Orders for a String with a Singular Weight

We study the linear operator pencil A(λ) = L?λV, λ ∈ ?, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W ₂ ^?1 [0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums \(\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}} \), p ≥ 2, where λ_n, n ∈ ?, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L^?1 are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.     

Frobenius Differential-Algebraic Universums on Complex Algebraic Curves

In terms of differential generators and differential relations for a finitely generated commutative- associative differential C-algebra A (with a unit element) we study and determine necessary and sufficient conditions for the fact that under any Taylor homomorphism \(\widetilde \psi \)M: A → C[[z]] the transcendence degree of the image \(\widetilde \psi \)M(A) over C does not exceed 1 \(\left( {\widetilde \psi M{{\left( a \right)}^{\underline{\underline {def}} }}\sum\limits_{m = 0}^\infty {\psi M\left( {{a^{\left( m \right)}}} \right)} } \right)\frac{{{z^m}}}{{m!}}\), where a ∈ A, M ∈ Spec_CA is a maximal ideal in A, a^(m) is the result of m-fold application of the signature derivation of the element a, and ψM is the canonic epimorphism A → A/M).     

On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ? T _g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A _k ) _k=1 ^∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V _q f(x) = (Σ _k≥1|π _k+1(f)(x) ? π _k (f)(x)|q)^1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L ¹(Ω, A, µ) we have µ({x ∈ Ω: V _q f(x) > λ}) ≤ C(∫_Ω | f | dµ/λ). Finally, some concrete examples are constructed.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

Generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ I ⁿ , then one of the following conditions holds:

1. (1)
   there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
    
2. (2)
   there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
    
3. (3)
   there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
    
4. (4)
   f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

   1. (a)
      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
       
   2. (b)
      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
       
    

     

Existence of linear Pfaffian systems whose lower characteristic set has positive measure in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaffian system ?x/?t i = A _i (t)x, x ∈ R ⁿ , t = (t ₁, t ₂, t ₃) ∈ R ₊ ³ , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure.     

Waiting times and stopping probabilities for patterns in Markov chains

Suppose that C is a finite collection of patterns. Observe a Markov chain until one of the patterns in C occurs as a run. This time is denoted by τ. In this paper, we aim to give an easy way to calculate the mean waiting time E(τ) and the stopping probabilities P(τ = τA)with A ∈ C, where τA is the waiting time until the pattern A appears as a run.     

Semigroup graded algebras and graded PI-exponent

Let S be a semigroup. We study the structure of graded-simple S-graded algebras A and the exponential rate PIexp ^S-gr(A):= lim_n→∞ \(\sqrt[n]{{c_n^{S - gr}\left( A \right)}}\) of growth of codimensions c _n ^S-gr (A) of their graded polynomial identities. This is of great interest since such algebras can have non-integer PIexp ^S-gr(A) despite being finite dimensional and associative. In addition, such algebras can have a non-trivial Jacobson radical J(A). All this is in strong contrast with the case when S is a group since in the group case J(A) is trivial, PIexp ^S-gr(A) is always integer and, if the base field is algebraically closed, then PIexp ^S-gr(A) equals dimA. Without any restrictions on the base field F, we classify graded-simple S-graded algebras A for a class of semigroups S which is complementary to the class of groups. We explicitly describe the structure of J(A) showing that J(A) is built up of pieces of a maximal S-graded semisimple subalgebra of A which turns out to be simple. When F is algebraically closed, we get an upper bound for \({\overline {\lim } _{n \to \infty }}\sqrt[n]{{c_n^{S - gr}\left( A \right)}}\). If A/J(A) ≈ M ₂(F) and S is a right zero band, we show that this upper bound is sharp and PIexp ^S-gr(A) indeed exists. In particular, we present an infinite family of graded-simple algebras A with arbitrarily large non-integer PIexp ^S-gr(A).     

On Generalized Derivations and Centralizers of Operator Algebras with Involution

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and A(H) ? B(H) be a standard operator algebra which is closed under the adjoint operation. Let F: A(H)→ B(H) be a linear mapping satisfying F(AA*A) = F(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H), where the associated linear mapping d: A(H) → B(H) satisfies the relation d(AA*A) = d(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H). Then F is of the form F(A) = SA ? AT for all A ∈ A(H) and some S, T ∈ B(H), that is, F is a generalized derivation. We also prove some results concerning centralizers on A(H) and semisimple H^*-algebras.