Unital AH-algebra Research Articles

Symmetries of simple $A\mathbb{T}$-algebras

Let A be a unital simple A\mathbb{T} -algebra of real rank zero. Given an order two automorphism h: K_1(A)\to K_1(A) , we show that there is an order two automorphism \alpha : A\to A such that \alpha_{*0}=\mathrm {id} , \alpha_{*1}=h and the action of \mathbb{Z}_2 generated by \alpha has the tracial Rokhlin property. Consequently, C^*(A,\mathbb{Z}_2,\alpha) is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus our main result can be considered as the \mathbb{Z}_2 -action analogue of the Lin-Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case.

The radius of comparison of the tensor product of a C∗-algebra with C(X)

Let X be a compact metric space, let A be a unital AH-algebra with large matrix sizes, and let B be a stably finite unital C∗-algebra. Then we give a lower bound for the radius of comparison of C(X)⊗B and prove that the dimension-rank ratio satisfies drr(A)=drr(C(X)⊗A). We also give a class of unital AH-algebras A with rc(C(X)⊗A)=rc(A). We further give a class of stably finite exact Z-stable unital C∗-algebras with nonzero radius of comparison.

The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

AbstractLetGbe a finite group, letAbe an infinite-dimensional stably finite simple unital C*-algebra, and let$\alpha \colon G \to {\operatorname {Aut}} (A)$be an action ofGonAwhich has the weak tracial Rokhlin property. Let$A^{\alpha }$be the fixed point algebra. Then the radius of comparison satisfies${\operatorname {rc}} (A^{\alpha }) \leq {\operatorname {rc}} (A)$and${\operatorname {rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{{{\operatorname{card}}} (G))} \cdot {\operatorname {rc}} (A)$. The inclusion of$A^{\alpha }$inAinduces an isomorphism from the purely positive part of the Cuntz semigroup${\operatorname {Cu}} (A^{\alpha })$to the fixed points of the purely positive part of${\operatorname {Cu}} (A)$, and the purely positive part of${\operatorname {Cu}} ( C^* (G, A, \alpha ) )$is isomorphic to this semigroup. We construct an example in which$G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$,Ais a simple unital AH algebra,$\alpha $has the Rokhlin property,${\operatorname {rc}} (A)> 0$,${\operatorname {rc}} (A^{\alpha }) = {\operatorname {rc}} (A)$, and${\operatorname {rc}} ({C^* (G, A, \alpha)} ) = ({1}/{2}) {\operatorname {rc}} (A)$.

On a lifting question of Blackadar

Let A be the AF algebra whose scaled ordered group K0(A) is (G⊕H,(G+∖{0})⊕H∪{(0,0)},g˜⊕0), where (G,G+,g˜) is the scaled ordered group K0(B) of a unital simple AF algebra B, and H is a countable torsion-free Abelian group. Let σ be an order 2 scaled ordered automorphism of K0(A), defined by σ(g,h)=(g,−h), where (g,h)∈G⊕H. We show that there is an order 2 automorphism α of A such that α∗=σ. This gives a partial answer to a lifting question posed by Blackadar. Incidentally, the lift α we construct has the tracial Rokhlin property. Consequently, the crossed product C∗(Z2,A,α) is a unital simple AH algebra with no dimension growth.

Homomorphisms from AH-algebras

Let [Formula: see text] be a general unital AH-algebra and let [Formula: see text] be a unital simple [Formula: see text]-algebra with tracial rank at most one. Suppose that [Formula: see text] are two unital monomorphisms. We show that [Formula: see text] and [Formula: see text] are approximately unitarily equivalent if and only if [Formula: see text] [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are continuous affine maps from tracial state space [Formula: see text] of [Formula: see text] to faithful tracial state space [Formula: see text] of [Formula: see text] induced by [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] and [Formula: see text] are induced homomorphisms s from [Formula: see text] into [Formula: see text], where [Formula: see text] is the space of all real affine continuous functions on [Formula: see text] and [Formula: see text] is the closure of the image of [Formula: see text] in the affine space [Formula: see text]. In particular, the above holds for [Formula: see text], the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements [Formula: see text], an affine map [Formula: see text] and a homomorphisms [Formula: see text], there exists a unital monomorphism [Formula: see text] such that [Formula: see text] and [Formula: see text].

The general linear group as a complete invariant for $C^*$-algebras

In 1955 Dye proved that two von Neumann factors not of type I2n are isomorphic (via a linear or a conjugate linear u -isomorphism) if and only if their unitary groups are isomorphic as abstract groups. We consider an analogue for C u -algebras. We show that the topological gen- eral linear group is a classifying invariant for simple, unital AH-algebras of slow dimension growth and of real rank zero, and the abstract gen- eral linear group is a classifying invariant for unital Kirchberg algebras in UCT.

Asymptotic unitary equivalence in C*-algebras

Let C = C(X) be the unital C*-algebra of all continuous functions on a finite CW complex X and let A be a unital simple C*-algebra with tracial rank at most one. We show that two unital monomorphisms φ,ψ: C → A are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries {ut: t ∈ [0, 1)} ⊂ A such that limt→1u*tφ(f)ut = ψ(f) for all f ∈ C(X) if and only if [φ] = [ψ] in KK(C,A), τ ◦φ = τ ◦ψ for all τ ∈ T(A), and φ† = ψ†, where T(A) is the simplex of tracial states of A and φ†, ψ†: U∞(C)/DU∞(C) → U∞(A)/DU∞(A) are the induced homomorphisms and where U∞(A) = ∪k=1∞U(Mk(A)) and U∞(C) = ∪k=1∞U(Mk(C)) are usual infinite unitary groups, respectively, and DU∞(A) and DU∞(C) are the commutator subgroups of U∞(A) and U∞(C), respectively. We actually prove a more general result for the case in which C is any general unital AH-algebra.

A Short Return to Simple Ah-Algebras with Real Rank Zero

Let $A$ be a unital simple AH-algebra with stable rank one and real rank zero such that $kx=0$ for all $x\in\operatorname{ker}\rho_A$, the subgroup of infinitesmal elements in $K_0(A)$, and for the same integer $k\ge 1$. We show that $A$ has tracial rank zero and is isomorphic to a unital simple AH-algebra with no dimension growth.

Mean dimension and AH-algebras with diagonal maps

Mean dimension for AH-algebras with diagonal maps is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of Toms' comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras with diagonal maps are shown to have mean dimension zero: the class of AH-algebras with at most countably many extremal traces, and the class of AH-algebras with numbers of extreme traces which induce same state on the K0-group being uniformly bounded (in particular, this class includes AH-algebras with real rank zero).

Pure Infiniteness of the Crossed Product of an AH-Algebra by an Endomorphism

Abstract It is shown that simplicity of the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

Unitary groups as a complete invariant

Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. We show that if the unitary groups of two simple unital AH-algebras of slow dimension growth and of real rank zero are isomorphic as abstract groups, then their K0-ordered groups are isomorphic. Also, using Gong and Dadarlatʼs classification theorem, we prove that such C⁎-algebras are isomorphic if and only if their unitary groups are isomorphic as topological groups. For simple, unital purely infinite C⁎-algebras, we show that two unital Kirchberg algebras are ⁎-isomorphic if and only if their unitary groups are isomorphic as abstract groups.

Approximate unitary equivalence in simple $C^{*}$-algebras of tracial rank one

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $\phi, \psi: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $\phi$ and $\psi$ are approximately unitarily equivalent if and only if [\phi]=[\psi] in KL(C,A) \tau\circ \phi=\tau\circ \psi for all tracial states of A and \phi^{\ddag}=\psi^{\ddag}, here \phi^{\ddag} and \psi^{\ddag} are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by \phi and \psi, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.

Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.

-actions on UHF algebras of infinite type

We prove that all strongly outer Z -actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable Z -actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other.

$${\mathbb{Z}}$$ -Actions on AH Algebras and $${\mathbb{Z}^2}$$ -Actions on AF Algebras

We consider Z-actions (single automorphisms) on a unital simple AH algebra with real rank zero and slow dimension growth and show that the uniform outerness implies the Rohlin property under some technical assumptions. Moreover, two Z-actions with the Rohlin property on such a C^*-algebra are shown to be cocycle conjugate if they are asymptotically unitarily equivalent. We also prove that locally approximately inner and uniformly outer Z^2-actions on a unital simple AF algebra with a unique trace have the Rohlin property and classify them up to cocycle conjugacy employing the OrderExt group as classification invariants.

Asymptotically unitary equivalence and asymptotically inner automorphisms

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple $C^*$-algebra with tracial rank zero. Suppose that $\phi_1, \phi_2\colon \ C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t\colon \ t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*\phi_1(a)u_t=\phi_2(a)\quad\hbox{\rm for all}\quad a\in C $$ if and only if $[\phi_1]=[\phi_2]$ in $KK(C,A),$ $\tau\circ \phi_1=\tau\circ \phi_2$ for all $\tau\in T(A)$ and the rotation map ${\tilde\eta}_{\phi_1,\phi_2}$ associated with $\phi_1$ and $\phi_2$ is zero. In particular, an automorphism $\alpha$ on a unital separable simple $C^*$-algebra $A$ in ${\cal N}$ with tracial rank zero is asymptotically inner if and only if $$ [\alpha]=[{\rm id}_A]\quad{\rm in}\quad KK(A,A) $$ and the rotation map ${\tilde\eta}_{\phi_1, \phi_2}$ is zero. Let $A$ be a unital AH-algebra (not necessarily simple) and let $\alpha\in Aut(A)$ be an automorphism. As an application, we show that the associated crossed product $A\rtimes_{\alpha}{\Bbb Z}$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a strictly positive $\alpha$-invariant tracial state.

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms $${\phi, \psi: C\to A}$$ are approximately unitarily equivalent if and only if $$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$$ where T(A) is the tracial state space of A. In this paper we prove the following: Given $${\kappa\in KL(C,A)}$$ with $${\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$ and with κ([1 C ]) = [1 A ] and a continuous affine map $${\lambda: T(A)\to T_{\mathfrak f}(C)}$$ which is compatible with κ, where $${T_{\mathfrak f}(C)}$$ is the convex set of all faithful tracial states, there exists a unital monomorphism $${\phi: C\to A}$$ such that $$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$$ for all $${c\in C_{s.a.}}$$ and $${\tau\in T(A).}$$ Denote by $${{\rm Mon}_{au}^e(C,A)}$$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$$ where KLT(C, A)++ is the set of compatible pairs of elements in KL(C, A)++ and continuous affine maps from T(A) to $${T_{\mathfrak f}(C).}$$ Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and $${\kappa\in KL(C(X), A)}$$ with $${\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}$$ for which there is no homomorphism h: C(X) → A so that [h] = κ.

Lifting KK-elements, asymptotic unitary equivalence and classification of simple [formula omitted]-algebras

Let A and C be two unital simple C ∗ -algebras with tracial rank zero. Suppose that C is amenable and satisfies the Universal Coefficient Theorem. Denote by K K e ( C , A ) + + the set of those κ in K K ( C , A ) for which κ ( K 0 ( C ) + ∖ { 0 } ) ⊂ K 0 ( A ) + ∖ { 0 } and κ ( [ 1 C ] ) = [ 1 A ] . Suppose that κ ∈ K K e ( C , A ) + + . We show that there is a unital monomorphism ϕ : C → A such that [ ϕ ] = κ . Suppose that C is a unital AH-algebra and λ : T ( A ) → T f ( C ) is a continuous affine map for which τ ( κ ( [ p ] ) ) = λ ( τ ) ( p ) for all projections p in all matrix algebras of C and any τ ∈ T ( A ) , where T ( A ) is the simplex of tracial states of A and T f ( C ) is the convex set of faithful tracial states of C. We prove that there is a unital monomorphism ϕ : C → A such that ϕ induces both κ and λ. Suppose that h : C → A is a unital monomorphism and γ ∈ Hom ( K 1 ( C ) , Aff ( A ) ) . We show that there exists a unital monomorphism ϕ : C → A such that [ ϕ ] = [ h ] in K K ( C , A ) , τ ○ ϕ = τ ○ h for all tracial states τ and the associated rotation map can be given by γ. Denote by K K T ( C , A ) + + the set of compatible pairs ( κ , λ ) , where κ ∈ K L e ( C , A ) + + and λ is a continuous affine map from T ( A ) to T f ( C ) . Together with a result on asymptotic unitary equivalence in [H. Lin, Asymptotic unitary equivalence and asymptotically inner automorphisms, arXiv:math/0703610, 2007], this provides a bijection from the asymptotic unitary equivalence classes of unital monomorphisms from C to A to ( K K T ( C , A ) + + , Hom ( K 1 ( C ) , Aff ( T ( A ) ) ) / R 0 ) , where R 0 is a subgroup related to vanishing rotation maps. As an application, combining these results with a result of W. Winter [W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, arXiv:0708.0283v3, 2007], we show that two unital amenable simple Z -stable C ∗ -algebras are isomorphic if they have the same Elliott invariant and the tensor products of these C ∗ -algebras with any UHF-algebra have tracial rank zero. In particular, if A and B are two unital separable simple Z -stable C ∗ -algebras with unique tracial states which are inductive limits of C ∗ -algebras of type I, then they are isomorphic if and only if they have isomorphic Elliott invariants.

Tracial Rokhlin property for automorphisms on simple -algebras

AbstractLet A be a unital simple $A\mathbb {T}$-algebra of real rank zero. Given an isomorphismγ1:K1(A)→K1(A), we show that there is an automorphism α:A→A such that α*1=γ1 and α has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes _{\alpha }\mathbb {Z}$ is a simple unital AH-algebra with real rank zero. We also show that automorphisms with the Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.

On the classification problem for nuclear C∗-algebras

We exhibit a counterexample to Elliott’s classification conjecture for simple, separable, and nuclear C ∗ -algebras whose construction is elementary, and demonstrate the necessity of extremely fine invariants in distinguishing both approximate unitary equivalence classes of automorphisms of such algebras and isomorphism classes of the algebras themselves. The consequences for the program to classify nuclear C ∗ -algebras are far-reaching: one has, among other things, that existing results on the classification of simple, unital AH algebras via the Elliott invariant of K-theoretic data are the best possible, and that these cannot be improved by the addition of continuous homotopy invariant functors to the Elliott invariant.