arXiv subjects

Benjamín Itzá-Ortiz

Publications and source records attributed to Benjamín Itzá-Ortiz.

4 recordsLinked to original sources

arXiv2005

Realization of a simple higher dimensional noncommutative torus as a transformation group C*-algebra

Let $θ$ be a nondegenerate skew symmetric real $d$ by $d$ matrix, and let $A_θ$ be the corresponding simple higher dimensional noncommutative torus. Suppose that $d$ is odd, or that $d$ is greater or equal to 4 and the entries of $θ$ are not contained in a quadratic extension of $\mathbb{Q}$. Then $A_θ$ is isomorphic to the transformation group C*-algebra obtained from a minimal homeomorphism of a compact connected one dimensional space locally homeomorphic to the product of the interval and the Cantor set. The proof uses classification theory of C*-algebras.

math.OA↗

arXiv2005

Continuous and discrete flows on operator algebras

Let $(N,\R,θ)$ be a centrally ergodic W* dynamical system. When $N$ is not a factor, we show that, for each $t\not=0$, the crossed product induced by the time $t$ automorphism $θ_t$ is not a factor if and only if there exist a rational number $r$ and an eigenvalue $s$ of the restriction of $θ$ to the center of $N$, such that $rst=2π$. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if $(A,\R,α)$ is a minimal unital C* dynamical system and $A$ is either prime or commutative but not simple, then, for each $t\not=0$, the crossed product induced by the time $t$ automorphism $α_t$ is not simple if and only if there exist a rational number $r$ and an eigenvalue $s$ of the restriction of $α$ to the center of $A$, such that $rst=2π$.

math.OA↗

arXiv2005

The C*-algebras Associated to Time-$t$ Automorphisms of Mapping Tori

We find the range of a trace on the $K_0$ group of a crossed product by a time-$t$ automorphism of a mapping torus. We also find a formula to compute the Voiculescu-Brown entropy for such an automorphism. By specializing to the commutative setting, we prove that the crossed products by minimal time-t homeomorphisms of suspensions built over strongly orbit equivalent Cantor minimal systems have isomorphic Elliott invariants. As an application of our results we give examples of dynamical systems on (compact metric) connected 1-dimensional spaces which are not flip conjugate (because of different entropy) yet their associated crossed products have isomorphic Elliott invariants.

math.OA↗

arXiv2004

Eigenvalues, K-theory and Minimal Flows

Let $(Y,T)$ be a minimal suspension flow built over a dynamical system $(X,S)$ and with (strictly positive, continuous) ceiling function $f\colon X\to\R$. We show that the eigenvalues of $(Y,T)$ are contained in the range of a trace on the $K_0$-group of $(X,S)$. Moreover, a trace gives an order isomorphism of a subgroup of $K_0(C(X)\rtimes_S\mathbb{Z})$ with the group of eigenvalues of $(Y,S)$. Using this result, we relate the values of $t$ for which the time-$t$ map on minimal suspension flow is minimal, with the $K$-theory of the base of this suspension.

math.OA↗