Universal Functions on Complex General Linear Groups

In 1929, Birkhoff proved the existence of an entire function F on with the property that for any entire function f there exists a sequence {a_k} of complex numbers such that {F(ζ+a_k)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoff's theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group *. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n,  ) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n,  ) be the general linear group of degree n over . We denote by (G) the set of all holomorphic functions on G. Similarly, we define ( ). Let K be the (G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n,  ) with f(0)(f(K)), where (f(K)) is the ( )-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any KB(G), for any function f holomorphic in some neighbourhood of K, and for any >0, there exists CG with max_ZK |F(CZ)−f(Z)|<.     

An Infinite Analogue of Rings with Stable Rank One

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings) and include examples like End (V), the ring of endomorphisms of a vector space V over some field , and ( ), the ring of all row- and column-finite matrices over . We show that the category of QB-rings is stable under the formation of corners, ideals, and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e., a quasi-invertible element. Finally we show that the C*-algebras that are QB-rings are exactly the extremally rich C*-algebras studied by L. G. Brown and the second author.     

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space (k) shares certain properties with the classical Bergman space L²_α of the unit disk. For example, the weighted Bergman kernels k^β_w(z)=(1−wz)^−β, 1β2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H² (k)L²_a, where the inclusion maps are contractive, and M_ζ, the operator of multiplication with the identity function ζ, defines a contraction operator on (k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace of (k) is a Bergman-type kernel as well. For the weighted Bergman kernels k^β this result even holds for all _ζ-invariant subspace of index 1, i.e., whenever the dimension of /ζ is one. Second, if is any multiplier invariant subspace of (k), and if we set *= z , then M_ζ is unitarily equivalent to M_ζ acting on a space of *-valued analytic functions with an operator-valued reproducing kernel of the type

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where V is a contractive analytic function V :  → ( ,  *), for some auxiliary Hilbert space . Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of (k^β), 1β2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.     

Asymptotic expression of the linear discrete best -approximation

Let h_p, 1<p<∞, be the best ℓ_p-approximation of the element from a proper affine subspace K of , hK, and let denote the strict uniform approximation of h from K. We prove that there are a vector and a real number a, 0a1, such that

for all p>1, where with γ_p=o(a^p/p).     

Division Algebras with PSL(2, q)-Galois Maximal Subfields

If G is a finite group and k is a field, then G is k-admissible if there exists a G-Galois extension L/k such that L is a maximal subfield of a k-division algebra. We prove that PSL(2, 7) is k-admissible for any number field which either fails to contain or which has two primes lying over the dyadic prime. In addition, PSL(2, 11) is shown to be admissible over or any number field k with at least two extensions of the dyadic prime. Indeed, there exist infinitely many linearly disjoint admissible extensions for these groups.     

Pervasive Algebras of Analytic Functions

We characterize those open U in the sphere such that A(U) is complex-pervasive, and those such that Re A(U) is real-pervasive. Pervasive means, roughly, that the uniform closure on each proper closed subset of bdy U is the space of all continuous functions (to or , as the case may be).     

On the Denseness of Rational Systems

This note characterizes the denseness of rational systems

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in C[−1, 1], where the nonreal poles in {a_k}^∞_k=1 \[−1, 1] are paired by complex conjugation. This extends an Achiezer's result.     

Identities on Units of Algebraic Algebras

Let be an algebraic algebra over an infinite field K and let ( ) be its group of units. We prove a stronger version of Hartley's conjecture for , namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in ( ), then satisfies a polynomial identity (PI). We also show that if is non-commutative, then is a PI-ring, provided f = 0 is satisfied by the non-central units of . In particular, is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in ( ) if and only if the same identity is satisfied in . The last fact remains true for generalized Laurent polynomial identities, provided that is locally finite.     

Constructing arithmetic subgroups of unipotent groups

Let G be a unipotent algebraic subgroup of some defined over . We describe an algorithm for finding a finite set of generators of the subgroup . This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.     

Strong bias of group generators: an obstacle to the “product replacement algorithm”

Let G be a finite group. Efficient generation of nearly uniformly distributed random elements in G, starting from a given set of generators of G, is a central problem in computational group theory. In this paper we demonstrate a weakness in the popular “product replacement algorithm,” widely used for this purpose. The main results are the following. Let be the set of generating k-tuples of elements of G. Consider the distribution of the first components of the k-tuples in induced by the uniform distribution over  . We show that there exist infinite sequences of groups G such that this distribution is very far from uniform in two different senses: (1) its variation distance from uniform is >1−ε; and (2) there exists a short word (of length (loglog|G|)^O(k)) which separates the two distributions with probability 1−ε. The class of groups we analyze is direct powers of alternating groups. The methods used include statistical analysis of permutation groups, the theory of random walks, the AKS sorting network, and a randomized simulation of monotone Boolean operations by group operations, inspired by Barrington's work on bounded-width branching programs. The problem is motivated by the product replacement algorithm which was introduced in [Comm. Algebra 23 (1995) 4931–4948] and is widely used. Our results show that for certain groups the probability distribution obtained by the product replacement algorithm has a bias which can be detected by a short straight line program.     

-Modules on Smooth Toric Varieties

Let X be a smooth toric variety. Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal . Let A denote the ring of differential operators on Spec(S). We show that the category of -modules on X is equivalent to a subcategory of graded A-modules modulo -torsion. Additionally, we prove that the characteristic variety of a -module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic -modules correspond to holonomic A-modules.     

Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F:  →  alg_f be a functor, where alg_f is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra A_aut(F) such that F factorizes through a functor :  →  coalg_f(A_aut(F)), where coalg_f(A_aut(F)) is the category of finite-dimensional measured A_aut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category coalg_f(A) and the forgetful functor ω: coalg_f(A) →  alg_f: we have A  A_aut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz.     

On Identity Bases of Epigroup Varieties

Let C _n and C _n be the varieties of all completely regular and of all completely simple semigroups, respectively, whose idempotent generated subsemigroups are periodic with period n. We use Ol'shanski 's theory of geometric group presentations to show that for large odd n these varieties (and similarly defined varieties of epigroups) do not have finitely axiomatizable equational theories.     

Markov inequality for polynomials of degree n with m distinct zeros

Let be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove thatfor every . This is far away from what we expect. We conjecture that the Markov factor 32·8^mn above may be replaced by cmn with an absolute constant c>0. We are not able to prove this conjecture at the moment. However, we think that our result above gives the best-known Markov-type inequality for on a finite interval when mc log n.     

Generalized Green Classes

Generalized Green classes are introduced; some basic properties of members in a generalized Green class are studied. Finally, we apply the results to (Λ), the Ringel–Hall algebra of a finite-dimensional hereditary algebra Λ over a finite field. In particular, it is proved that (Λ) belongs to a suitable generalized Green class, and that there is direct decomposition of spaces (Λ) =  (Λ)  J, where (Λ) is the composition algebra of Λ and J is a twisted Hopf ideal of (Λ), which is exactly the orthogonal complement of (Λ).     

Commuting Varieties of Lie Algebras over Fields of Prime Characteristic

Let K be an algebraically closed field of positive characteristic and let G be a reductive group over K with Lie algebra . This paper will show that under certain mild assumptions on G, the commuting variety ( ) is an irreducible algebraic variety.     

On Homogeneous Exact Categories

In the present paper we prove that a certain subcategory of the module category over some infinite-dimensional algebra R has almost split sequences and strongly homogeneous property; i.e., for each indecomposable module M in , there is an almost split sequence starting and also ending at M. It is also proved that except for a trivial case, is of wild representation type.     

Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σ_k has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δ_k∩Δ_k+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions r_k, k=1,…,m, whose zeros and poles lie in .     

Piecewise Smooth Spaces in Duality: Application to Blossoming

Given an (n+1)-dimensional space of piecewise smooth functions in which each basis has a non-vanishing Wronskian, and its dual space *, a canonical bilinear form is defined on × *, which provides a simple characterization of a contact of order rn. An intrinsic reproducing function is introduced, leading to Marsden-type identities. In the case of Chebyshev spaces connected with totally positive matrices, the bilinear form yields a general notion of blossom which can be extended to Chebyshev splines.     

A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

Let , with

-1=x_0n<x_1n<<x_nn<x_n+1,n=1