A simple separable C*-algebra which is not singly generated
It is shown that there is a simple unital separable AH algebra which is not singly generated.
math.OA↗
arXiv subjects
Chun Guang Li
Publications and source records attributed to Chun Guang Li.
Strict comparison holds in the uniform Roe algebra of a discrete amenable group
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.
math.OA↗
Villadsen algebras are singly generated
We show that Villadsen algebras, which are not Z-stable, are singly generated. More generally, we show that any simple unital AH algebra with diagonal maps is singly generated.
math.OA↗
Stable rank of $\mathrm{C}(X)\rtimesΓ$
It is shown that, for an arbitrary free and minimal $\mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $\mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, Γ)$, where $Γ$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $\mathrm{C}(X)\rtimesΓ$ has stable rank one. Moreover, in this case, the C*-algebra $\mathrm{C}(X)\rtimesΓ$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.
math.OA↗
Remarks on Villadsen algebras
It is shown that certain unital simple C*-algebras constructed by Villadsen are classified by the K0-group together with radius of comparison.
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$C^*$ exponential length of commutators unitaries in $AH$ algebras
For each unital $C^*$-algebra $A$, we denote $cel_{CU}(A)=\sup\{cel(u):u\in CU(A)\}$, where $cel(u)$ is the exponential length of $u$ and $CU(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that $cel_{CU}(A)=2π$ provided that $A$ is an $AH$ algebras with slow dimension growth whose real rank is not zero. On the other hand, we prove that $cel_{CU}(A)\leq 2π$ when $A$ is an $AH$ algebra with ideal property and of no dimension growth (if we further assume $A$ is not of real rank zero, we have $cel_{CU}(A)= 2π$).
math.OA↗