Searcharxiv⌕ Search

arXiv subjects

George A. Elliott

Publications and source records attributed to George A. Elliott.

At least 37 records · Page 2Linked to original sources
arXiv2021

On the bundle of KMS state spaces for flows on a Z-absorbing C*-algebra

We obtain three results: 1) Every compact simplex bundle with exactly one point in the fiber over 0 is the KMS bundle of a periodic flow on the Jiang-Su algebra. 2) Let A be a separable unital C*-algebra with a unique trace state. Suppose that A tensorially absorbs the Jiang-Su algebra. The (weak) cocycle-conjugacy classes of flows that are not approximately inner are uncountable. 3) Let B be a separable, simple, unital, purely infinite and nuclear C*-algebra in the UCT class. Assume that the K1 group of B is torsion free. Every proper simplex bundle with empty fiber over 0 is the KMS bundle of a periodic flow on B.

math.OA↗
arXiv2021

The bundle of KMS state spaces for flows on a unital C*-algebra

It is shown that any bundle of KMS state spaces which can occur for a flow on a unital separable C*-algebra with a trace state can also be realized by a flow on any given unital infinite-dimensional simple AF algebra with a tracial state space affinely homeomorphic to the fiber in the bundle over 0.

math.OA↗
arXiv2020

The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $\mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.

math.OA↗
arXiv2020

Decomposition rank of approximately subhomogeneous C*-algebras

It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous C*-algebra are locally approximated by a certain class of more tractable subhomogeneous algebras, namely, a non-commutative generalization of the class of cell complexes. The result is applied to show that Jiang-Su stable minimal Z-crossed products have finite decomposition rank.

math.OA↗
arXiv2020

The category of ordered Bratteli diagrams

A category structure for ordered Bratteli diagrams is proposed in which isomorphism coincides with the notion of equivalence of Herman, Putnam, and Skau. It is shown that the natural one-to-one correspondence between the category of Cantor minimal systems and the category of simple properly ordered Bratteli diagrams is in fact an equivalence of categories. This gives a Bratteli-Vershik model for factor maps between Cantor minimal systems. We give a construction of factor maps between Cantor minimal systems in terms of suitable maps (called premorphisms) between the corresponding ordered Bratteli diagrams, and we show that every factor map between two Cantor minimal systems is obtained in this way. Moreover, solving a natural question, we are able to characterize Glasner and Weiss's notion of weak orbit equivalence of Cantor minimal systems in terms of the corresponding C*-algebra crossed products.

math.OA↗
arXiv2019

Stable homotopy, 1-dimensional NCCW complexes, and Property (H)

In this paper, we show that the homomorphisms between two unital one-dimensional NCCW complexes with the same KK-class are stably homotopic, i.e., with adding on a common homomorphism (with finite dimensional image), they are homotopic. As a consequence, any one-dimensional NCCW complex has the Property (H).

math.OA↗
arXiv2018

Simple stably projectionless C*-algebras with generalized tracial rank one

We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some structural question concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras.

math.OA↗
arXiv2018

Thomsen-Li's Theorem Revisited

In this paper, we prove a general version of Thomsen-Li's Theorem--a Krein-Milman type theorem for C*-algebras. To be precise, for a Markov operator on $C[0,1]$ which preserves certain subspace of $C[0,1]$, we approximate it by an average of homomorphisms on $C[0,1]$ in the strong operator topology, additionally we require that this average also preserves the same subspace. Such results could be useful for constructing homomorphisms for subhomogeneous C*-algebras.

math.OA↗
arXiv2016

The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $\mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.

math.OA↗
arXiv2016

The C*-algebra of a minimal homeomorphism of zero mean dimension

Let $X$ be an infinite compact metrizable space, and let $σ: X\to X$ be a minimal homeomorphism. Suppose that $(X, σ)$ has zero mean topological dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_σ\mathbb Z$ is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong A\otimes\mathcal Z$. This implies that $A$ is classifiable when $(X, σ)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang-Su algebra.

math.OA↗
arXiv2015

The classification of simple separable unital locally ASH algebras

Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $A\otimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $\textrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.

math.OA↗
arXiv2014

The category of Bratteli diagrams

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF algebras (and at the same time, of Glimm's classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.

math.OA↗
arXiv2013

The isomorphism relation for separable C*-algebras

We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.

math.OA↗
arXiv2009

The cone of lower semicontinuous traces on a C*-algebra

The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra.

math.OA↗
arXiv2007

Towards a theory of classification

The well-known difficulties arising in a classification which is not set-theoretically trivial---involving what is sometimes called a non-smooth quotient---have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor---the very existence of which is already a surprise. Here the notion of such a functor is developed abstractly, and a number of examples are considered (including those which have arisen for various classes of operator algebras).

math.OA↗