Eduard Ortega

THE CUNTZ SEMIGROUP AND COMPARISON OF OPEN PROJECTIONS

Abstract We show that a number of naturally occurring comparison relations on positive elements in a C ⁎ -algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the C ⁎ -algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of—and is weaker than—a comparison notion defined by Peligrad and Zsido. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murray–von Neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the C ⁎ -algebra. We use these findings to give a new picture of the Cuntz semigroup.

Two-Sided Localization of Bimodules

We extend to bimodules Schelters localization of a ring with respect to Gabriel filters of left and right ideals. Our two-sided localization of bimodules provides an endofunctor on a convenient bicategory of rings with filters of ideals. This is used to study the Picard group of a ring relative to a filter of ideals.

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

Abstract We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C ⁎ -algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.

The Corona Factorization Property and refinement monoids

The Corona Factorization Property of a C � -algebra, originally defined to study extensions of C � -algebras, has turned out to say something important about intrinsic struc- tural properties of the C � -algebra. We show in this paper that a �-unital C � -algebra A of real rank zero has the Corona Factorization Property if and only if its monoid V(A) of Murray- von Neumann equivalence classes of projections in matrix algebras over A has a certain (rather weak) comparability property that we call the Corona Factorization Property (for monoids). We show that a projection in such a C � -algebra is properly infinite if (and only if) a multiple of it is properly infinite. The latter result is obtained from some more general result we establish about conical refinement monoids. We show that the set of order units (together with the zero-element) in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility; and if u is an element in a refinement monoid such that nu is properly infinite, then u can be written as a sum u = s + t such that ns and nt are properly infinite.

Algebraic Cuntz-Pimsner rings

From a system consisting of a ring R ,ap air ofR-bimodules Q and P and an R-bimodule homomorphism ψ : P ⊗ Q → R, we construct a Z-graded ring T(P,Q,ψ) called the Toeplitz ring and (for certain systems) a Z-graded quotient O(P,Q,ψ) of T(P,Q,ψ) called the Cuntz–Pimsner ring. These rings are the algebraic analogues of the Toeplitz C ∗ -algebra and the Cuntz–Pimsner C ∗ -algebra associated to a C ∗ -correspondence (also called a Hilbert bimodule). This new construction generalizes, for example, the algebraic crossed product by a single automorphism, fractional skew monoid rings by a single corner automorphism and Leavitt path algebras. We also describe the structure of the graded ideals of our graded rings in terms of pairs of ideals of the coefficient ring and show that our Cuntz–Pimsner rings satisfy the Graded Uniqueness Theorem.

Simple Cuntz–Pimsner rings

Abstract Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz–Pimsner ring contains a non-zero graded ideal, when a relative Cuntz–Pimsner ring is simple, and when every ideal in a relative Cuntz–Pimsner ring is graded, are given. A “Cuntz–Krieger uniqueness theorem” for relative Cuntz–Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz–Pimsner rings are introduced. As applications of these results, a uniqueness result for the Toeplitz algebra of a directed graph and characterizations of when crossed products of a ring by a single automorphism and fractional skew monoid rings of a single corner isomorphism are simple, are obtained.

The Structure of Stacey Crossed Products by Endomorphisms

We describe simplicity and purely infiniteness (in simple case) of the Stacey crossed product \(A \times _{\beta }\mathbb{N}\) in terms of conditions of the C ∗-dynamical system (A, β).