arXiv subjects
Publications and source records attributed to George A. Elliott.
We introduce Property (C) for a unital commutative sub-C*-algebra $D$ of a unital C*-algebra $A$, which is a version of the relative comparison property using almost normalizers. In the case that $D = \mathrm{C}(X)$ and $A = \mathrm{C}(X) \rtimesΓ$, where $(X, Γ)$ is a free and minimal dynamical system with uniform Rokhlin property, it turns out that this Property (C) is equivalent to the small boundary property of $(X, Γ)$.
It is shown that there is a simple unital separable AH algebra which is not singly generated.
Let $Γ$ be a countable discrete amenable group, and let $A=l^\infty(Γ) \rtimes Γ$. It is shown that if $a, b \in A \otimes \mathcal K$ are positive elements such that $$\mathrm{d}_τ(a) < \mathrm{d}_τ(b),\quad τ\in \mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$. Moreover, consider the universal minimal set $(M, Γ)$. The simple C*-algebra $\mathrm{C}(M)\rtimesΓ$ is shown to be AH in the strong sense that there is an increasing net of unital sub-C*-algebras $A_λ\subseteq A$, $λ\in Λ$, such that each $A_λ$ is a simple (separable) $\mathcal Z$-absorbing approximately homogeneous C*-algebra with real rank zero and $A = \bigcup_{λ\in Λ} A_λ$. In particular, $\mathrm{C}(M)\rtimesΓ$ is approximately divisible.
We construct a ``noncommutative'' Villadsen algebra $B$ and show that, given an extreme tracial state $ν$ on its canonical AF subalgebra, the subset of $T(B)$ consisting of those tracial states that equal $ν$ when restricted to the canonical AF subalgebra is the Poulsen simplex. In particular, if the canonical AF subalgebra has a unique trace, then $T(B)$ is the Poulsen simplex. We go on to show that in certain instances, the tracial cone of a ``classical'' AF-Villadsen algebra $D$ is isomorphic to the tracial cone of the algebra obtained from $D$ by deleting all point evaluations.
The authors' recent classification of Jesper Villadsen's remarkable generalization (based on a self-reproducing seed space) of Glimm's infinite tensor product (UHF) C*-algebras, by means of the Cuntz semigroup (in the case of a fixed, well-behaved, seed space), is extended to the analogous generalization of Bratteli's approximately finite-dimensional (AF) C*-algebras. Some progress is made in the direction of distinguishing between algebras based on different seed spaces.
In this paper, we prove results on the relative radius of comparison of C*-algebras and their crossed products, focusing on the non-unital setting. More precisely, let $A$ be a stably finite simple non-type-I (not necessarily unital) C*-algebra, let $G$ be a finite group, and let $α\colon G \to {\operatorname{Aut}} (A)$ be an action which has the weak tracial Rokhlin property. Let $a$ be a non-zero positive element in $A^α\otimes \mathcal{K}$. Then we show that the radius of comparison of $\operatorname{Cu} (A^α)$ relative to $[a]$ is bounded above by the radius of comparison of $\operatorname{Cu} (A)$ relative to $[a]$. If further $A$ is exact and $a$ is in the Pedersen ideal of $A^α\otimes \mathcal{K}$, then the radius of comparison of $\operatorname{Cu} (A\rtimes_α G)$ relative to $[a]$ is equal to its radius of comparison relative to $[p\cdot a]$, scaled by $1/|G|$, where $p$ is the averaging projection in the multiplier algebra of $(A \otimes \mathcal{K}) \rtimes_{α\otimes \operatorname{id}} G$. Moreover, the radius of comparison of $\operatorname{Cu} (A\rtimes_α G)$ relative to $[a]$ is bounded above by $1/|G|$ times the radius of comparison of $\operatorname{Cu} (A)$ relative to $[a]$. We also prove that the inclusion of $A^α$ in $A$ induces an isomorphism from the purely positive part of the Cuntz semigroup ${\operatorname{Cu}} (A^α)$ to the fixed point of the purely positive part of ${\operatorname{Cu}} (A)$. An important consequence of our results is that they apply to non-unital C*-algebras and give new insights into comparison theory of C*-algebras and their crossed products.
Let $X$ be a compact metrizable space, and let $Δ$ be a closed set of Borel probability measures on $X$. We study the small boundary property of the pair $(X, Δ)$. In particular, it is shown that $(X, Δ)$ has the small boundary property if it has a restricted version of property Gamma. As an application, it is shown that, if $A$ is the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^d$, where $(X, \mathbb Z^d)$ is a free minimal topological dynamical system, or if $A$ is an AH algebra with diagonal maps, then, $A$ is $\mathcal Z$-stable if the set of extreme tracial states is compact, regardless of its dimension.
A new proof following a suggestion of Kaplansky to use a result of Dixmier, and in this way avoid unbounded nets of operators, is given of the Kaplansky density theorem.
Let $Ω$ be a class of ${\rm C^*}$-algebras. In this paper, we study a class of not necessarily unital generalized tracial approximation ${\rm C^*}$-algebras, and the class of simple ${\rm C^*}$-algebras which can be generally tracially approximated by ${\rm C^*}$-algebras in $Ω$, denoted by ${\rm gTA}Ω$. Let $Ω$ be a class of unital ${\rm C^*}$-algebras and let $A$ be a simple unital ${\rm C^*}$-algebra. Then $A\in {\rm gTA}Ω$, if, and only if, $A\in {\rm WTA}Ω$ (where ${\rm TA}Ω$ is the class of weakly tracially approximable unital ${\rm C^*}$-algebras introduced by Elliott, Fan, and Fang).Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or are of tracial nuclear dimension at most $n$, or are $m$-almost divisible, or have the property $\rm SP$). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, is weakly ($n, m$)-almost divisible, has the property $\rm SP$) for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of generalized tracial approximation ${\rm C^*}$-algebras.
It is shown that certain unital simple C*-algebras constructed by Villadsen are classified by the K0-group together with radius of comparison.
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.
In order to realize all possible KMS-bundles on the Jiang-Su algebra, we introduce a class of C*-algebras which we call rationally approximately finite dimensional (RAF). Using these, we show that for a given proper simplex bundle $(S, π)$ with a singleton $π^{-1}(\{0\})$ and a unital separable monotracial C*-algebra $A$ absorbing the Jiang-Su algebra tensorially (for instance, the irrational rotation algebra), there exists a flow on $A$ whose KMS-bundle is isomorphic to $(S, π)$.
In this paper, we introduce some classes of generalized tracial approximation ${\rm C^*}$-algebras. Consider the class of unital ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or have tracial nuclear dimension at most $n$, or have the property $\rm SP$, or are $m$-almost divisible). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, has the property $\rm SP$, is weakly ($n, m$)-almost divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the corresponding class of generalized tracial approximation ${\rm C^*}$-algebras. As an application, let $A$ be an infinite-dimensional unital simple ${\rm C^*}$-algebra, and let $B$ be a centrally large subalgebra of $A$. If $B$ is tracially $\mathcal{Z}$-absorbing, then $A$ is tracially $\mathcal{Z}$-absorbing. This result was obtained by Archey, Buck, and Phillips in \cite{AJN}.
It is shown that the coloured isomorphism class of a unital, simple, $\mathcal{Z}$-stable, separable amenable C$^*$-algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex.
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.
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.
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.
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.