Andrew S. Toms

Strongly self-absorbing

Say that a separable, unital C*-algebra V ? C is strongly self absorbing if there exists an isomorphism V such that lx> ai>e approximately unitarily equivalent *-homomorphisms. We study this class of algebras, which includes the Cuntz algebras ?2, Ooo, the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of ?00 with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra V we characterise when a separable C*-algebra absorbs V tensorially (i.e., is P-stable), and prove closure properties for the class of separable Testable C* algebras. Finally, we compute the possible If-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C*-algebras.

C^{*}

Four signal-to-noise ratio (SNR) estimators for quaternary phase-shift keying (QPSK)-like signaling are proposed and examined. Two are based on receiver statistics directly related to the SNR while two others are based on receiver statistics inversely related to the SNR. The results show that the estimators based on the inverse of the SNR perform better than the estimators based on the SNR.

-algebras

We report on recent progress in the program to classify separable amenable C∗-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program’s successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

Comparison of four SNR estimators for QPSK modulations

Abstract We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliotts classification program, proving that the usual form of the Elliott conjecture is equivalent, among -stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C*-algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of simple unital C*-algebras.

Regularity properties in the classification program for separable amenable C*-algebras

Let X be a compact infinite metric space of finite covering dimension and α : X → X a minimal homeomorphism. We prove that the crossed product

Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture

{\mathcal{C}(X) \rtimes_\alpha \mathbb{Z}}

An algebraic approach to the radius of comparison

absorbs the Jiang–Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

Comparison Theory and Smooth Minimal C*-Dynamics

Abstract Jiang and Su and (independently) Elliott discovered a simple, nuclear, infinite-dimensional C*-algebra ℒ̵ having the same Elliott invariant as the complex numbers. For a nuclear C*-algebra A with weakly unperforated K*-group the Elliott invariant of A ⊗ ℒ̵ is isomorphic to that of A. Thus, any simple nuclear C*-algebra A having a weakly unperforated K*-group which does not absorb ℒ̵ provides a counterexample to Elliotts conjecture that the simple nuclear C*-algebras will be classified by the Elliott invariant. In the sequel we exhibit a separable, infinite-dimensional, stably finite instance of such a non-ℒ̵-absorbing algebra A, and so provide a counterexample to the Elliott conjecture for the class of simple, nuclear, infinite-dimensional, stably finite, separable C*-algebras.