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Register a new userHomomorphisms from AH-algebras
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Huaxin Lin
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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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Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $phi, psi: Cto A$ are approximately unitarily equivalent if and only if $$ [phi]=[psi] {rm in} KL(C,A) and taucirc phi=taucirc psi tforal tauin T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $kappain KL(C,A)$ with $kappa(K_0(C)_+setminus {0})subset K_0(A)_+setminus {0}$ and with $kappa([1_C])=[1_A]$ and a continuous affine map $lambda: T(A)to T_{mathtt{f}}(C)$ which is compatible with $kappa,$ where $T_{mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $phi: Cto A$ such that $$ [phi]=kappaandeqn taucirc phi(c)=lambda(tau)(c) $$ for all $cin C_{s.a.}$ and $tauin T(A).$ 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 $T_{mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $kappain KL(C(X), A)$ with $kappa(K_0(C(X))_+setminus{0})subset K_0(A)_+setminus {0}$ for which there is no hm $h: C(X)to A$ so that $[h]=kappa.$
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.
An n-homomorphism between algebras is a linear map $phi : A to B$ such that $phi(a_1 ... a_n) = phi(a_1)... phi(a_n)$ for all elements $a_1, >..., a_n in A.$ Every homomorphism is an n-homomorphism, for all n >= 2, but the converse is false, in general. Hejazian et al. [7] ask: Is every *-preserving n-homomorphism between C*-algebras continuous? We answer their question in the affirmative, but the even and odd n arguments are surprisingly disjoint. We then use these results to prove stronger ones: If n >2 is even, then $phi$ is just an ordinary *-homomorphism. If n >= 3 is odd, then $phi$ is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C*-algebras.
We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation which gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of $H^{infty}(bb{T})otimescl B(cl H)$.
Starting with a vertex-weighted pointed graph $(Gamma,mu,v_0)$, we form the free loop algebra $mathcal{S}_0$ defined in Hartglass-Penneys article on canonical $rm C^*$-algebras associated to a planar algebra. Under mild conditions, $mathcal{S}_0$ is a non-nuclear simple $rm C^*$-algebra with unique tracial state. There is a canonical polynomial subalgebra $Asubset mathcal{S}_0$ together with a Dirac number operator $N$ such that $(A, L^2A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify $(mathcal{S}_0, A, N)$ yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini-Schramm convergence for vertex-weighted pointed graphs. As our $rm C^*$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov-Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet-Jones-Shyakhtenko (GJS) $rm C^*$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS $rm C^*$-algebras of many infinite families of planar algebras converge in quantum Gromov-Hausdorff distance.
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