arXiv subjects
Publications and source records attributed to S. C. Power.
The natural lexicographic semigroupoids associated with Cantor product spaces indexed by countable linear orders are classified. Applications are given to the classification of triangular operator algebras which are direct limits of upper triangular matrix algebras.
It is shown that all the approximately finite dimensional C*-algebras which are not of Type I are isomorphic as Banach spaces. This generalises the matroid case given previously by Arazy. Analogous results are obtained for various families of triangular subalgebras of AF C*-algebras. In addition the classification of various continua of Type I AF C*-algebras is discussed.
A stable homology theory is defined for completely distributive CSL algebras in terms of the point-neighbourhood homology of the partially ordered set of meet-irreducible elements of the invariant projection lattice. This specialises to the simplicial homology of the underlying simplicial complex in the case of a digraph algebra. These groups are computable and useful. In particular it is shown that if the first spectral homology group is trivial then Schur automorphisms are automatically quasispatial. This motivates the introduction of essential Hochschild cohomology which we define by using the point weak star closure of coboundaries in place of the usual coboundaries.
A new homology is defined for a non-self-adjoint operator algebra and distinguished masa which is based upon cycles and boundaries associated with complexes of partial isometries in the stable algebra. Under natural hypotheses the zeroth order group coincides with the $\ K_0$\ group of the generated C*-algebra. Several identifications and applications are given, and in particular it is shown how stable homology is significant for the classification of regular subalgebras and regular limit algebras.
This is a revised and corrected version of a preprint circulated in 1990 in which various non-self-adjoint limit algebras are classified. The principal invariant is the scaled $K_0$ group together with the algebraic order on the scale induced by partial isometries in the algebra.
Let $A$ denote the alternation limit algebra, studied by Hopenwasser and Power, and by Poon, which is the closed direct limit of upper triangular matrix algebras determined by refinement embeddings of multiplicity $r_k$ and standard embeddings of multiplicity $s_k$. It is shown that the quotient of the isometric automorphism group by the approximately inner automorphisms is the abelian group $ \ZZ ^d$ where $d$ is the number of primes that are divisors of infinitely many terms of each of the sequences $(r_k)$ and $(s_k)$. This group is also the group of automorphisms of the fundamental relation of $A$.