Rank functions on rings derived from group rings

The ring D_G derived from the group ring KG is shown to posses exactly one pseudo-rank function in case G is an infinite, but locally and residually finite, group and K is a field. The function is described naturally as the limit of the sequence of normalized ranks of all approximation matrices. This fact is applied to an investigation of the direct finiteness of the maximal quotient ring of D_G .     

The Projective Class Ring of Basic and Split Hopf Algebras

We describe the structure of the complexification of the projective class ring of a basic and split Hopf algebra using a positive integer determined by the composition series of the projective cover of the trivial module. If q is a root of unity of order n, the projective class algebra of u_q ⁺(sl₂) is the product of n–1 copies of the dual numbers over C and two copies of C.     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

A survey of recent results on congruence lattices of lattices

We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ring-theoretical, and topological results. Received March 13, 2002; accepted in final form September 24, 2002. RID="h1" ID="h1"     

K-Theory, Hermitian K-Theory and the Karoubi Tower

The goal of this paper is to present a new view on the link between the well known K-theory of finitely generated projective modules and much less understood K-theory of Hermitian forms on finitely generated projective modules.For a given Hermitian ring (R, , ), we obtain the K-theory space KR equipped with an involution and the Hermitian K-theory space KHerm(R, , ). The fixed subspace ² is homotopy equivalent to the Hermitian K-theory space KHerm(R,, , ). We construct the Karoubi Tower diagram, which is obtained by iterating Karoubi's construction of the homotopy fibers of the forgetful and hyperbolic maps. Using an interesting factorization of these maps, we prove various homotopy properties of the Karoubi Tower. The homotopy inverse limit of the Karoubi Tower is homotopy equivalent to the homotopy fibre of the inclusion of the fixed set ² into the homotopy fixed set ^{h 2}. Considering Karoubi's fundamental periodicity theorem, the Karoubi Tower generalizes the low dimensional connections between Hermitian K-theory and K-theory groups. Illustrative examples of the Karoubi Tower are given by the finite field case and the classical Hermitian rings over , and . Considering topological K-theory for these cases, the Karoubi Tower comprises the classical Bott periodicity.Another important application of the Karoubi Tower is an elegant and comprehensive generalization of the classical invariants of quadratic forms.     

Representations of distributive semilattices in ideal lattices of various algebraic structures

We study the relationships among existing results about representations of distributive semilattices by ideals in dimension groups, von Neumann regular rings, C*-algebras, and complemented modular lattices. We prove additional representation results which exhibit further connections with the scattered literature on these different topics. Received March 2, 1998; accepted in final form November 9, 2000.     

The Karoubi Tower and K-Theory Invariants of Hermitian Forms

By the use of the Karoubi Tower diagram we generalize the classical invariants of quadratic forms. Similar to Quillen's higher K-theory generalization of the classical K-theory groups, these invariants are an extension of the classical invariants by the use of homotopy theory. The iterated forgetful maps in the Karoubi Tower are KR valued and yield a generalization of the standard (rank, discriminant and total Hasse–Witt) invariants of quadratic forms in two directions. First, we get invariants of all degrees. Second, these invariants are defined for every Hermitian ring. They yield and generalize the Clifford invariant in the case of a field of characteristic different from 2, or in the case of an arithmetic Dedekind domain containing .     

étale Descent for Two-Primary Algebraic K -Theory of Totally Imaginary Number Fields

We show that the natural map from 2-adic algebraic K-theory to 2-adic étale K-theory induces an isomorphism in positive degrees for rings of 2-integers in totally imaginary number fields. In this sense, 2-adic algebraic K-theory of totally imaginary number fields satisfies étale descent in positive degrees.     

Homology of Linear Groups with Coefficients in the Adjoint Action and K-Theory

The main purpose of this paper is to compute some homology groups of linear groups with coefficients in the adjoint action which have appeared in different works. Our method uses various weight decompositions in group homology, algebraic K–theory, and Hochschild homology. This paper originated in a question posed by Cathelineau which is related to Hilbert's third problem on the scissors congruence of polyhedra.     

Anderson Duality in K-Theory and Im(J)-Theory

We study Anderson duality for connective K-theory l and Im(J)-theory A, the latter being the p-localization of Quillen's algebraic K-theory of a suitably chosen finite field. The interplay between Anderson duality and Postnikov decomposition results in a construction of two families of natural transformations from ordinary cohomology theory to l^* and A^*-theory, which are related to the Chern character. These operations are then used to study l^*(l) and A^*(A), the cohomology operations of connective K-theory and Im(J)-theory.     

Noncommutative Chern Characters of Compact Lie Group C*-Algebras

For compact Lie groups, the Chern characters K^*(G) Q H^* _DR(G;Q) have been already constructed. In this paper, we construct and study the corresponding noncommutative Chern characters. They are homomorphisms ch_C*: K_*(C^*(G)) from quantum K-groups into entire current periodic cyclic homology groups of group C^*-algebras. We also obtain the corresponding algebraic version ch_alg: K_*(C^*(G)) HP_*(C^*(G)), which can be identified with the classical Chern character K_* (C(T)) HP_* (C(T)), where T is the maximal torus of G.     

Local “superlinearity” and “sublinearity” for the p-Laplacian

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems −Δpu=fλ(x,u), , where Ω is a bounded domain in RN, N>p, and λ>0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti-Brezis-Cerami type in a more general form, namely λa(x)uq+b(x)ur, where 0?q<p−1<r?p^∗−1. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis-Nirenberg result on local minimization in and , a C1,α estimate for equations of the form −Δpu=h(x,u) with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper-lower solutions for the p-Laplacian.     

Generalized spherical and simplicial coordinates

Elementary trigonometric quantities are defined in l2,p analogously to that in l2,2, the sine and cosine functions are generalized for each p>0 as functions sinp and cosp such that they satisfy the basic equation p|cosp(φ)|+p|sinp(φ)|=1. The p-generalized radius coordinate of a point ξ∈Rn is defined for each p>0 as . On combining these quantities, ln,p-spherical coordinates are defined. It is shown that these coordinates are nearly related to ln,p-simplicial coordinates. The Jacobians of these generalized coordinate transformations are derived. Applications and interpretations from analysis deal especially with the definition of a generalized surface content on ln,p-spheres which is nearly related to a modified co-area formula and an extension of Cavalieri's and Torricelli's indivisibeln method, and with differential equations. Applications from probability theory deal especially with a geometric interpretation of the uniform probability distribution on the ln,p-sphere and with the derivation of certain generalized statistical distributions.