Functional Analysis

Suppose any solution of a linear impulsive delay differential equation x ˙ (t)+ ∑ i=1 m A i (t)x[ h i (t)]=0, t≥0,x(s)=0,s<0, x( τ j +0)= B j x( τ j −0)+ α j , j=1,2,..., is bounded for any bounded sequence { α i } . The conditions ensuring exponential stability of this equation are presented. The behavior of solutions of the non-homogeneous equation is analyzed.

Consider a linear impulsive equation in a Banach space x ˙ (t)+A(t)x(t)=f(t), t≥0, x( τ i +0)= B i x( τ i −0)+ α i , with lim i→∞ τ i =∞ . Suppose each solution of the corresponding semi-homogeneous equation x ˙ (t)+A(t)x(t)=0, (2) is bounded for any bounded sequence { α i } . The conditions are determined ensuring (a) the solution of the corresponding homogeneous equation has an exponential estimate; (b) each solution of (1),(2) is bounded on the half-line for any bounded f and bounded sequence { α i } ; (c) lim t→∞ x(t)=0 for any f, α i tending to zero; (d) exponential estimate of f implies a similar estimate for x .

The article contains a detailed description of the connection between finite depth inclusions of I I 1 -subfactors and finite C ∗ -tensor categories (i.e. C ∗ -tensor categories with dimension function for which the number of equivalence classes of irreducible objects is finite). The (N,N) -bimodules belonging to a I I 1 -subfactor N⊂M with finite Jones index form a C ∗ -tensor category with dimension function. Conversely, taking an object of a finite C ∗ -tensor category C we construct a subfactor A⊂R of the hyperfinite I I 1 -factor R with finite index and finite depth. For this subfactor we compute the standard invariant and show that the C ∗ -tensor category of the corresponding (A,A) -bimodules is equivalent to a subcategory of C. We illustrate the results for the C ∗ -tensor category of the unitary finite dimensional corepresentations of a finite dimensional Hopf-*-algebra.

We compute K -theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $\R^k.$ We discuss the relation between our results and the η -invariant.

We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C*-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K_1, and stable rank 1. In the unital case, the K_0 group separates the tracial states. The APFP passes to matrix algebras. and if I is an ideal in A such that I and A/I have the APFP, then so does A. We also give some new examples of C*-algebras with the APFP, including type II_1 factors and infinite-dimensional simple unital direct limits with slow dimension growth, real rank zero, and trivial K_1 group. An infinite- dimensional simple unital direct limit with slow dimension growth and with the APFP must have real rank zero, but we also give examples of unital algebras with the APFP which do not have real rank zero. Our analysis also leads to the introduction of a new concept of rank for a C*-algebra that may be of interest in the future.

The notions of Busby-Smith and Green type twisted actions are extended to discrete unital inverse semigroups. The connection between the two types, and the connection with twisted partial actions, are investigated. Decomposition theorems for the twisted crossed products are given.

We consider problems associated with the computation of spectra of self-adjoint operators in terms of the eigenvalue distributions of their n x n sections. Under rather general circumstances, we show how these eigenvalues accumulate near points of the essential spectrum of the given operator, and we prove that their averages converge to a measure concentrated precisely on the essential spectrum. In the primary cases of interest, namely the discretized Hamiltonians of one-dimensional quantum systems, this limiting measure is associated with a tracial state on a certain simple C*-algebra. These results have led us to conclude that one must view this kind of numerical analysis in the context of C*-algebras.

The recently developed theory of partial actions of discrete groups on C ∗ -algebras is extended. A related concept of actions of inverse semigroups on C ∗ -algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

If Au=−div(a(x,Du)) is a monotone operator defined on the Sobolev space W 1,p ( R n ) , 1<p<+∞ , with a(x,0)=0 for a.e. x∈ R n , the capacity C A (E,F) relative to A can be defined for every pair (E,F) of bounded sets in R n with E⊂F . We prove that C A (E,F) is increasing and countably subadditive with respect to E and decreasing with respect to F . Moreover we investigate the continuity properties of C A (E,F) with respect to E and F .

Partial actions of discrete groups on C ∗ -algebras and the associated crossed products have been studied by Exel and McClanahan. We characterise these crossed products in terms of the spectral subspaces of the dual coaction, generalising and simplifying a theorem of Exel for single partial automorphisms. We then use this characterisation to identify the Cuntz algebras and the Toeplitz algebras of Nica as crossed products by partial actions.