On the AF embeddability of crossed products of AF algebras by the integers

Nathanial P. Brown

Home/Archive/arXiv

arXiv · funct-an/9710004

On the AF embeddability of crossed products of AF algebras by the integers

Nathanial P. Brown

View original source ↗Read source document ↗Export metadata ↓

Abstract

It is shown that if A is an AF algebra then a crossed product of A by the integers can be embedded into an AF algebra if and only if the crossed product is stably finite. This equivalence follows from a simple K-theoretic characterization of AF embeddability. It is then shown that if a crossed product of an AF algebra by the integers is AF embeddable then the AF embedding can be chosen in such a way as to induce a rationally injective map on K_0 of the crossed product.

Explore related subjects

funct-an math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Nathanial P. Brown. 1997-10-28. On the AF embeddability of crossed products of AF algebras by the integers. https://arxiv.org/abs/funct-an/9710004

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2020

Isomorphism classes for quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.

funct-an↗

arXiv2007

Quantum Mechanics and Operator algebras on the Hilbert ball

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

funct-an↗

arXiv2004

No More Than Mechanics. I

One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different $\hbar$ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in $n$-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.

funct-an↗