AF-embeddings into C*-algebras of real rank zero

It is proved that every separable $C^*$-algebra of real rank zero contains an AF-sub-$C^*$-algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two $C^*$-algebras and such that every projection in a matrix algebra over the large $C^*$-algebra is equivalent to a projection in a matrix algebra over the AF-sub-$C^*$-algebra. This result is proved at the level of monoids, using that the monoid of Murray-von Neumann equivalence classes of projections in a $C^*$-algebra of real rank zero has the refinement property. As an application of our result, we show that given a unital $C^*$-algebra $A$ of real rank zero and a natural number $n$, then there is a unital $^*$-homomorphism $M_{n_1} \oplus ... \oplus M_{n_r} \to A$ for some natural numbers $r,n_1, ...,n_r$ with $n_j \ge n$ for all $j$ if and only if $A$ has no representation of dimension less than $n$.Comment: 28 page

Recasting the Elliott conjecture

Let A be a simple, unital, exact, and finite C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases -- Z-stable algebras all -- we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of Z-stable algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott's classification conjecture -- that K-theoretic invariants will classify separable and nuclear C*-algebras -- with the recent appearance of counterexamples to its strongest concrete form.Comment: 28 pages; several typos corrected, Lemma 3.4 added; to appear in Math. An

Simple Riesz groups having wild intervals

We prove that every partially ordered simple group of rank one which is not Riesz embeds into a simple Riesz group of rank one if and only if it is not isomorphic to the additive group of the rationals. Using this result, we construct examples of simple Riesz groups of rank one $G$, containing unbounded intervals $(D_n)_{n\geq 1}$ and $D$, that satisfy: (a) For each $n\geq 1$, $tD_n\ne G^+$ for every $t< q_n$, but $q_nD_n=G^+$ (where $(q_n)$ is a sequence of relatively prime integers); (b) For every $n\geq 1$, $nD\ne G^+$. We sketch some potential applications of these results in the context of K-Theory.Comment: 27 page