Nathanial P. Brown

C*-algebras and finite-dimensional approximations

Fundamental facts Basic theory: Nuclear and exact

THE CUNTZ SEMIGROUP, THE ELLIOTT CONJECTURE, AND DIMENSION FUNCTIONS ON C ∗ -ALGEBRAS

\textrm{C}^*

Invariant means and finite representation theory of *-algebras

-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and Kirchbergs factorization property Quasidiagonal C*-algebras AF embeddablity Local reflexivity and other tensor product conditions Summary and open problems Special topics: Simple

Free entropy dimension in amalgamated free products

\textrm{C}^*

Uniform embeddings of bounded geometry spaces into reflexive Banach space

-algebras Approximation properties for groups Weak expectation property and local lifting property Weakly exact von Neumann algebras Applications: Classification of group von Neumann algebras Herreros approximation problem Counterexamples in

Topological entropy of free product automorphisms

\textrm{K}

Quasi-diagonality and the finite section method

-homology and

Connes' embedding problem and Lance's WEP

\textrm{K}

AF Embeddings and the Numerical Computation of Spectra in Irrational Rotation Algebras

-theory Appendices: Ultrafilters and ultraproducts Operator spaces, completely bounded maps and duality Lifting theorems Positive definite functions, cocycles and Schoenbergs Theorem Groups and graphs Bimodules over von Neumann algebras Bibliography Notation index Subject index.

Dual entropy in discrete groups with amenable actions

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.