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Register a new userAF-embedding of the crossed products of AH-algebras by finitely generated abelian groups
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Authors
Huaxin Lin
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Let $X$ be a compact metric space and let $Lambda$ be a $Z^k$ ($kge 1$) action on $X.$ We give a solution to a version of Voiculescus problem of AF-embedding: The crossed product $C(X)rtimes_{Lambda}Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $Lambda: Gto Aut(C)$ be a monomorphism. We show that $Crtimes_{Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $Lambda$-invariant tracial state.
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We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.
Let $A$ be a unital AF-algebra whose Murray-von Neumann order of projections is a lattice. For any two equivalence classes $[p]$ and $[q]$ of projections we write $[p]sqsubseteq [q]$ iff for every primitive ideal $mathfrak p$ of $A$ either $p/mathfrak ppreceq q/mathfrak ppreceq (1- q)/mathfrak p$ or $p/mathfrak psucceq q/mathfrak p succeq (1-q)/mathfrak p.$ We prove that $p$ is central iff $[p]$ is $sqsubseteq$-minimal iff $[p]$ is a characteristic element in $K_0(A)$. If, in addition, $A$ is liminary, then each extremal state of $K_0(A)$ is discrete, $K_0(A)$ has general comparability, and $A$ comes equipped with a centripetal transformation $[p]mapsto [p]^Game$ that moves $p$ towards the center. The number $n(p) $ of $Game$-steps needed by $[p]$ to reach the center has the monotonicity property $[p]sqsubseteq [q]Rightarrow n(p)leq n(q).$ Our proofs combine the $K_0$-theoretic version of Elliotts classification, the categorical equivalence $Gamma$ between MV-algebras and unital $ell$-groups, and L os ultraproduct theorem for first-order logic.
Let $(mathcal G, Sigma)$ be an ordered abelian group with Haar measure $mu$, let $(mathcal A, mathcal G, alpha)$ be a dynamical system and let $mathcal Artimes_{alpha} Sigma $ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism [ mathcal Artimes_{alpha} Sigma sim_{s} big(mathcal A otimes mathcal K(mathcal G, Sigma, mu)big)rtimes_{alphaotimes {rm Ad}: rho} mathcal G, ] where $mathcal K(mathcal G, Sigma, mu)$ denotes the compact operators in the CSL algebra ${rm Alg}:mathcal L(mathcal G, Sigma, mu)$ and $rho$ denotes the right regular representation of $mathcal G$. We also show that there exists a complete lattice isomorphism between the $hat{alpha}$-invariant ideals of $mathcal Artimes_{alpha} Sigma$ and the $(alphaotimes {rm Ad}: rho)$-invariant ideals of $mathcal A otimes mathcal K(mathcal G, Sigma, mu)$. Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by $mathbb R$. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems $(mathcal A, mathcal G, alpha)$ for which the identity ${rm Rad}(mathcal A rtimes_alpha mathcal G)=({rm Rad}:mathcal A) rtimes_alpha mathcal G$ persists. A broad class of such dynamical systems is identified.
We consider a twisted noncommutative join procedure for unital $C^*$-algebras which admit actions by a compact abelian group $G$ and its discrete abelian dual $Gamma$, so that we may investigate an analogue of Baum-Dabrowski-Hajac noncommutative Borsuk-Ulam theory in the twisted setting. Namely, under what conditions is it guaranteed that an equivariant map $phi$ from a unital $C^*$-algebra $A$ to the twisted join of $A$ and $C^*(Gamma)$ cannot exist? This pursuit is motivated by the twisted analogues of even spheres, which admit the same $K_0$ groups as even spheres and have an analogous Borsuk-Ulam theorem that is detected by $K_0$, despite the fact that the objects are not themselves deformations of a sphere. We find multiple sufficient conditions for twisted Borsuk-Ulam theorems to hold, one of which is the addition of another equivariance condition on $phi$ that corresponds to the choice of twist. However, we also find multiple examples of equivariant maps $phi$ that exist even under fairly restrictive assumptions. Finally, we consider an extension of unital contractibility (in the sense of Dabrowski-Hajac-Neshveyev) modulo $k$.
Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. Suppose that $phi, psi: Cto A$ are two unital monomorphisms. We show that $phi$ and $psi$ are approximately unitarily equivalent if and only if beq[phi]&=&[psi] {rm in} KL(C,A), phi_{sharp}&=&psi_{sharp}tand phi^{dag}&=&psi^{dag}, eneq where $phi_{sharp}$ and $psi_{sharp}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{rm f}(C)$ of $C$ induced by $phi$ and $psi,$ respectively, and $phi^{ddag}$ and $psi^{ddag}$ are induced homomorphisms from $K_1(C)$ into $Aff(T(A))/bar{rho_A(K_0(A))},$ where $Aff(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $bar{rho_A(K_0(A))}$ is the closure of the image of $K_0(A)$ in the affine space $Aff(T(A)).$ In particular, the above holds for $C=C(X),$ 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 $kappain KL_e(C,A)^{++},$ an affine map $gamma: T(C)to T_{rm f}(C)$ and a hm $af: K_1(C)to Aff(T(A))/bar{rho_A(K_0(A))},$ there exists a unital monomorphism $phi: Cto A$ such that $[h]=kappa,$ $h_{sharp}=gamma$ and $phi^{dag}=af.$
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