The stable rank of full corners in C*-algebras

We give a treatment of Rieffel's theory of stable rank for C*-algebras in terms of left invertibility of generalized nonsquare matrices, and prove that if is a full projection in a unital C*-algebra , then the stable rank of the corner is at least as large as the stable rank of .

     

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.     

Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K‐ and L‐Theory

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraicKandLtheory of integral group rings and to the Baum–Connes Conjecture on the topologicalKtheory of reducedC*algebras of groups. The approach is through spectra over the orbit category of a discrete groupG.We give several points of view on the assembly map for a family of subgroups and characterize such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and their associated generalized homology and cohomology theories, and homotopy limits.     

On Generalized Hamming Weights for Galois Ring Linear Codes

The definition of generalized Hamming weights (GHW) for linear codes over Galois rings is discussed. The properties of GHW for Galois ring linear codes are stated. Upper and existence bounds for GHW of – linear codes and a lower bound for GHW of the Kerdock code over – are derived. GHW of some – linear codes are determined.     

Stable rank of corner rings

B. Blackadar recently proved that any full corner in a unital C*-algebra has K-theoretic stable rank greater than or equal to the stable rank of . (Here is a projection in , and fullness means that .) This result is extended to arbitrary (unital) rings in the present paper: If is a full idempotent in , then . The proofs rely partly on algebraic analogs of Blackadar's methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if where is a finitely generated projective generator, and can be generated by elements, then .

     

On the Stability of Diagonal Actions

In the note it is proved that, for an arbitrary action of a semisimple group G on an affine variety X, there is a positive integer n such that the diagonal action (m copies) is stable for any .     

On the Additive D-Stability of Matrices on the Basis of the Kharitonov Criterion

On the basis of the Kharitonov theorem, sufficient conditions on an matrix A are presented for the matrix to be stable for arbitrary , .     

Moduli Spaces of Stable Polygons and Symplectic Structures on\overline {\mathcal{M}} _{0,n}

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to , brings up a new tool to study the Kähler topology of . A wild but precise conjecture on the shape of the Kähler cone of is given in the end.     

On Generalized Stable Ideals

An ideal I of a ring R is generalized stable in case aR + bR = R with a ∈ I, b ∈ R implies that there exist s, t ∈ 1 + I such that s(a + by)t = 1 for a y ∈ R. We establish, in this article, necessary and sufficient conditions for an ideal of a regular ring to be generalized stable. It is shown that every regular square matrix over such ideals admits a diagonal reduction. These extend the corresponding results of generalized stable regular rings.     

An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup

Conjecturally, for an odd prime and a certain ring of -integers, the stable general linear group and the étale model for its classifying space have isomorphic mod cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if is regular and certain homology classes for vanish. We check that this criterion is satisfied for as evidence for the conjecture.

     

Janet’s approach to presentations and resolutions for polynomials and linear pdes

Janets algorithm to create normal forms for systems of linear pdes is outlined and used as a tool to construct resolutions for finitely generated modules over polynomial rings over fields as well as over rings of linear differential operators with coefficients in a differential field. The main result is that a Janet basis for a module allows to read off a Janet basis for the syzygy module. Two concepts are introduced: The generalized Hilbert series allowing to read off a basis (over the ground field) of the modules, once the Janet basis is constructed, and the Janet graph, containing all the relevant information connected to the Janet basis. In the context of pdes, the generalized Hilbert series enumerates the free Taylor coefficients for power series solutions. Rather than presenting Janets algorithm as a powerful computational tool competing successfully with more commonly known Gröbner basis techniques, it is used here to prove theoretical results.Received: 6 September 2004     

Supports of semi-stable probability measures on locally convex spaces

LetE be a locally convex space. Let be an absolutely convexly tight Radom semi-stable probability measure onE with index 1<2 and Lévy measureM. The main result of this paper shows that the closed semigroup generated by the support ofM and the negative of the barycenter ofM restricted to a suitable compact subset ofE is a (closed) linear space ofE, and that the support of is a suitable translate of this linear space. This result complements a few known results concerning the supports of stable and semi-stable probability measures. In particular, it extends an analogous result proved recently for the support of -stable probability measures 1<2 (Ref. 4). Related results concerning the support of Radon semi-stable probability measures onE of index 0<<1 are also discussed.The research of this author was partially supported by AFSOR Grant No. 90-0168The research of this author was supported by KBN Grant, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

A gaussian central limit theorem for trimmed products on simply connected step 2-Nilpotent Lie groups

A limit theorem due to J. Kuelbs and M. Ledoux, valid for dilatation-stable laws on type 2-Banach spaces, is carried over to stable laws on simply connected step 2-nilpotent Lie groupsG. We show that for products of i.i.d. random variables in the of attraction of a nondegenerate semigroup onG, where is a one-parameter automorphism group acting contracting onG, a certain intermediate trimming procedure, together with a suitable norming, always yields a nondegenerate centered Gaussian limit.     

Gromov translation algebras over discrete trees are exchange rings

It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras of matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers in the unit interval , the growth algebras (introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension in .

     

On Mittag-Leffler functions and related distributions

The distribution F () = 1 – E (–), 0 < 1; 0 , where E (x) is the Mittag-Leffler function is studied here with respect to its Laplace transform. Its infinite divisibility and geometric infinite divisibility are proved, along with many other properties. Its relation with stable distribution is established. The Mittag-Leffler process is defined and some of its properties are deduced.     

Stable Ranges For Morita Contexts

In this paper, we show that if rings A and B are (s, 2)-rings, then so is the ring of a Morita context (A, B, M, N, , ). Also we get analogous results for unit 1-stable ranges and GM-rings. These give new classes of rings satisfying such stable range conditions.2000 Mathematics Subject Classification: 16U99 16E50     

Tilting objects in abelian categories and quasitilted rings

D. Happel, I. Reiten and S. Smalø initiated an investigation of quasitilted artin -algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring . Here, employing a notion of -objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

     

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

Homotopy stability of orthogonal spaces in commutative banach algebras

We study the concept of real stable rank for a complex commutative Banach algebra A (rsr A). It is shown that this invariant has a behaviour completely analogous to the classical Bass stable rank; in particular, we establish a precise relation between both invariants.Further, we use this machinery to show that the connected components of the orthogonal spaces of A, O _k ⁿ (A)={ A ^k×n:^t=id_k×k}, stabilize when k and n increase in a way that depends on rsr A Finally, we prove an analogous stabilization result for the homotopy classes of maps from the j-sphere into O _k ⁿ (A), where j is an arbitrary positive integer.Research supported by a grant of the CONICET, Argentina.