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.     

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).     

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.     

-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.     

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.     

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.     

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.     

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

Full-size image

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.     

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.     

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.     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

The Linear Part of a Discontinuously Acting Euclidean Semigroup

Let ⁿ be a Euclidean space and let S be a Euclidean semigroup, i.e., a subsemigroup of the group of isometries of ⁿ. We say that a semigroup S acts discontinuously on ⁿ if the subset {s  S:sK ∩ K ≠ } is finite for any compact set K of ⁿ. The main results of this work areTheorem.If S is a Euclidean semigroup which acts discontinuously on ⁿ, then the connected component of the closure of the linear part ℓ(S) of S is a reducible group.Corollary.Let S be a Euclidean semigroup acting discontinuously on ⁿ; then the linear part ℓ(S) of S is not dense in the orthogonal group O(n).These results are the first step in the proof of the followingMargulis' Conjecture.If S is a crystallographic Euclidean semigroup, then S is a group.     

Analytic regularity for an operator with Treves curves

We study a partial differential operator with analytic coefficients, which is of the form “sum of squares”. is hypoelliptic on any open subset of , yet possesses the following properties: (1) is not analytic hypoelliptic on any open subset of that contains 0. (2) If u is any distribution defined near with the property that is analytic near 0, then u must be analytic near 0. (3) The point 0 lies on the projection of an infinite number of Treves curves (bicharacteristics).These results are consistent with the Treves conjectures. However, it follows that the analog of Treves conjecture, in the sense of germs, is false.As far as we know, is the first example of a “sum of squares” operator which is not analytic hypoelliptic in the usual sense, yet is analytic hypoelliptic in the sense of germs.     

An upper estimate in Turán's pure power sum problem

Let z₁, z₂, …, z_n be complex numbers, and write for their power sums. Let where the minimum is taken under the condition that . In this paper we prove that .     

Dilations of C*-Correspondences and the Simplicity of Cuntz–Pimsner Algebras

We develop a dilation theory for C*-correspondences, showing that every C*-correspondence E over a C*-algebra A can be universally embedded into a Hilbert C*-bimodule X_E over a C*-algebra A_E such that the crossed product A_E  is naturally isomorphic to A_EXE  . The Cuntz–Pimsner algebra _E is isomorphic to _{E E}  where _E and _E are quotients of A_E, resp. X_E.  If E is full and the left action is by generalized compact operators, then _E is an equivalence bimodule or, equivalently, an invertible C*-correspondence. In general, _E is merely an essential Hilbert C*-bimodule. Slightly extending previous results on crossed products by equivalence bimodules, we apply our dilation theory to show that for full C*-correspondences over unital C*-algebras, _E is simple if and only if E is minimal and nonperiodic, extending and simplifying results of Muhly and Solel and Kajiwara, Pinzari, and Watatani.     

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.     

Young measures generated by a class of integrands: a narrow epicontinuity and applications to Homogenization

Given a subset E of convex functions from into which satisfy growth conditions of order p>1 and an open bounded subset of , we establish the continuity of a map μΦ_μ from the set of all Young measures on equipped with the narrow topology into a set of suitable functionals defined in and equipped with the topology of Γ-convergence. Some applications are given in the setting of periodic and stochastic homogenization.     

A characterization and equations for minimal shape-preserving projections

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by the space of all bounded linear operators from X into V; let be the set of all projections in . For a given , we denote by the set of operators such that PSS. When , we characterize those for which P is minimal. This characterization is then utilized in several applications and examples.     

Invariants of 2×2-Matrices over Finite Fields

Let _q be the finite field with q elements, q=p^ν, p a prime, and Mat_2.2( _q) the vector space of 2×2-matrices over . The group GL(2, ) acts on Mat_2,2( _q) by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where the trace and determinant generate the ring of invariants, several new invariants appear in the case of finite fields.     

Local lagrange interpolation with cubic splines on tetrahedral partitions

We describe an algorithm for constructing a Lagrange interpolation pair based on C¹ cubic splines defined on tetrahedral partitions. In particular, given a set of points , we construct a set P containing and a spline space based on a tetrahedral partition whose set of vertices include such that interpolation at the points of P is well-defined and unique. Earlier results are extended in two ways: (1) here we allow arbitrary sets , and (2) the method provides optimal approximation order of smooth functions.