Volume 1999, Issue 507

It is shown that the AH algebras satisfy a certain splitting property at the level of K -theory with torsion coefficients. The splitting property is used to prove the following: There are locally homogeneous C *-algebras which are not AH algebras. The class of AH algebras is not closed under countable inductive limits. There are real rank zero split quasidiagonal extensions of AH algebras which are not AH algebras.

Suppose that a simple C *-algebra A can be written as an inductive limit of , where X n,i are finite CW complexes with uniformly bounded dimensions. Such inductive limit C *-algebras are called AH algebras. In this paper, we will prove that for any n and ε > 0, there exists a homomorphism ψ n,m : A n → A m , which factors through an interval algebra , such that the spectra of ψ n,m and the spectra of φ n,m can be paired within ε provided that m is large enough. This result is essential for the classification of such simple C *-algebras.

Let A be the coordinate ring of an affine elliptic curve (over an infinite field k ) of the form X – { p }, where X is projective and p is a closed point on X . Denote by F the function field of X . We show that the image of H .(GL 2 ( A ), ℤ) in H .(GL 2 ( F ), ℤ) coincides with the image of H .(GL 2 ( k ), ℤ). As a consequence, we obtain numerous results about the K -theory of A and X . For example, if k is a number field, we show that r 2 ( K 2 ( A ) ⊗ ℚ) = 0, where r m denotes the m th level of the rank filtration.

J. Dadok has shown by classification that any polar representation has the same orbits as the isotropy representation of some symmetric space. A conceptual proof of this result is given subject to some restriction.

Let ℒ be an n -dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on ℒ and a scalar λ ∈ K , we introduce an associative algebra U ξ,λ (ℒ) of dimension p n over K . The algebra U ξ,1 (ℒ) is isomorphic to the reduced enveloping algebra U ξ (ℒ), while the algebra U ξ,0 (ℒ) is nothing but the reduced symmetric algebra S ξ (ℒ). Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple ℒ-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple ℒ-modules for all ℒ having a toral stabiliser of a linear function.