Terry A. Loring

A universal multicoefficient theorem for the Kasparov groups

that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. There are advantages, in some circumstances, to using HomΛ(K(A),K(B)) in place of KK(A,B). These advantages derive from the fact that K(A) can be equipped with order and scale structures similar to those on K0(A). With this additional structure, the “Λ−module” K(A) becomes a powerful invariant of C*algebras. We show that it is a complete invariant for the class of real-rank-zero AD algebras. The AD algebras are a certain kind of approximately subhomogeneous C∗-algebras which may have torsion in K1 [Ell]. In addition to classifying these algebras, we calculate their automorphism groups up to approximately innerautomorphisms.

Invariants of almost commuting unitaries

Abstract Two invariants of pairs of almost commuting unitary matrices have been defined and used to identify those pairs which are bounded away from commuting pairs. The first, denoted k(U, V) for unitaries U,VϵMn( C ), involved K-theory, while the second was defined via winding numbers. We show that these invariants coincide. We also establish some upper bounds on ¦k(U, V)¦.

FINITE-DIMENSIONAL REPRESENTATIONS OF FREE PRODUCT C*-ALGEBRAS

Our main theorem is a characterization of C*-algebras that have a separating family of finite-dimensional representations. This characterization makes possible a solution to a problem posed by Goodearl and Menaul. Specifically, we prove that the free product of such C*-algebras again has this property.

Disordered topological insulators via C*-algebras

The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.

COMPUTING CONTINGENCIES FOR STABLE RELATIONS

Close ties between approximation properties for relations on C*-algebra elements and lifting results for the universal C*-algebras the relations generate is a very widespread and useful phenomenon in C*-algebra theory. In this paper, we explore how to achieve results of this kind when the approximation properties and the lifting results are true only in special cases determined by K-theoretical contingencies. To interpolate between properties of these two basic types, we must investigate C*-algebras given by softened relations, in particular with emphasis on their K-theory. A surprisingly weak correlation between the K-theory of the C*-algebras given by exact and softened relations leads to delicate problems which must be treated with care. As an example of a set of relations which are prone to an analysis of this kind we study the pairs of unitaries commuting up to rational rotation.

K-theory and pseudospectra for topological insulators

We derive formulas and algorithms for Kitaevs invariants in the periodic table for topological insulators and superconductors for finite disordered systems on lattices with boundaries. We find that K-theory arises as an obstruction to perturbing approximately compatible observables into compatible observables. We derive formulas in all symmetry classes up to dimension two, and in one symmetry class in dimension three, that can be computed with sparse matrix algorithms. We present algorithms in two symmetry classes in 2D and one in 3D and provide illustrative studies regarding how these algorithms can detect the scaling properties of phase transitions.

Projectivity, transitivity and AF-telescopes

Continuing our study of projective C∗-algebras, we establish a projective transitivity theorem generalizing the classical Glimm-Kadison result. This leads to a short proof of Glimm’s theorem that every C∗-algebra not of type I contains a C∗-subalgebra which has the Fermion algebra as a quotient. Moreover, we are able to identify this subalgebra as a generalized mapping telescope over the Fermion algebra. We next prove what we call the multiplier realization theorem. This is a technical result, relating projective subalgebras of a multiplier algebra M(A) to subalgebras of M(E), whenever A is a C∗-subalgebra of the corona algebra C(E) = M(E)/E. We developed this to obtain a closure theorem for projective C∗-algebras, but it has other consequences, one of which is that if A is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective C∗-algebra, then A is MF. The last part of the paper contains a proof of the projectivity of the mapping telescope over any AF (inductive limit of finite-dimensional) C∗-algebra. Translated to generators, this says that in some cases it is possible to lift an infinite sequence of elements, satisfying infinitely many relations, from a quotient of any C∗-algebra.

ClassifyingC*-algebras via ordered, mod-p K-theory

We introduce an order structure on K 0 ( ) 9 K0(; Z/p ) . This group may also be thought of as Ko(; 7z @ Z/p ) . We exhibit new examples of real-rank zero C*-algebras that are inductive limits of finite dimensional and dimension-drop algebras, have the same ordered, graded K-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure on K0(; Z ~3 Z / p ) naturally distinguishes these algebras. The same invariant is used to give an isomorphism theorem for such realrank zero inductive limits. As a corollary we obtain an isomorphism theorem for all real-rank zero approximately homogeneous C*-algebras that arise from systems of bounded dimension growth and torsion-free K0 group. At the 1980 Kingston conference, Effros posed the problem of finding suitable invariants for use in studying C*-algebras that are limits of sequences of homogeneous C*-algebras. These are now called almost homogeneous (AH) C*-algebras. The classification of AH algebras is a rapidly developing field and we will not attempt to summarize all this activity. Instead, we will focus on the growth of the invariants used. Specifically, we consider an AH algebra A that is the direct limit o f a system of the form

Almost commuting matrices, localized Wannier functions, and the quantum Hall effect

For models of noninteracting fermions moving within sites arranged on a surface in three-dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are K-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward and does not require translational invariance or introduce a flux torus. We further show that there is a Z2 index obstruction to approximating almost commuting self-dual matrices by exactly commuting self-dual matrices and present additional conjectures regarding the approximation of almost commuting real and self-dual matrices by exactly commuting real and self-dual matrices. The motivation for considering this problem is the case of physical systems with additional antiunitary symmetries such as t...

AF embeddings of C(T2) with a prescribed K-theory

Let A be a unital AF algebra without finite-dimensional quotients. Our main result is the classification of the group homomorphisms γ:K0(C(T2))→K0(A) which are induced by unital star-homomorphisms θ:C(T2)→A. Specifically, γ = θ∗ for some θ exactly when γ preserves the order unit and maps the reduced K-theory of the torus into the kernel of τ∗ for all bounded traces τ:A→C.