In the program to classify C-algebras, it is very important to find abstract conditions which are sufficient to imply that a given algebra has tracial rank zero, in the sense of Huaxin Lin. Even in the presence of a unique trace, we show that the union of the known necessary conditions is not enough.

We give a classification theorem for unital separable nuclear simple C∗-algebras with tracial rank no more than one. Let A and B be two unital separable simple nuclear C∗-algebras with TR(A), TR(B) ≤ 1 which satisfy the universal coefficient theorem. We show that A ∼= B if and only if there is an order and unit preserving isomorphism γ = (γ0, γ1, γ2) : (K0(A),K0(A)+, [1A],K1(A), T (A)) ∼= (K0(B...

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on L, 1 < p < ∞, and of a perfect dyadic singular integral operator. This allows to deduce a local T (b) theorem for singular integral operators from the one for perfect dyadic singular integral operators obtained by Hofmann, Muscalu, Tao, Thi...

Abstract Rotations $$f_\alpha $$ fα of the one-dimensional torus (equipped with normalized Lebesgue measure) by an irrational angle $$\alpha xmlns:mml="http://www.w3.org/1998/Math/MathML">α are known to be dynamical systems rank one. This is equi...

Assume M is a type II1 von Neumann algebra, the algebra of n × n matrices over another von Neumann algebra N , with the tracial state τM. In this article, we discuss the properties of semicircular elements that are free with, with respect to τM, the algebra Dn consisting of scalar diagonal matrices in M. Then we define a concept “matricial distance” of two elements in M and compute it when the ...