We investigate conditions for the extendibility of continuous algebra homomorphisms $ϕ$ from the Fourier algebra $A(F)$ of a locally compact group $F$ to the Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$ to maps between the corresponding $L^\infty$ algebras which are weak* continuous. When $ϕ$ is completely bounded and $F$ is amenable, it is induced by a piecewise affine map $α: Y\to F$ where $Y\subseteq G$. We show that extendibility of $ϕ$ is equivalent to $α$ being an open map. We also study the dual problem for contractive homomorphisms $ϕ: L^1(F)\to M(G)$. We show that $ϕ$ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map $θ$ (Greenleaf [1965], Stokke [2011]) is a proper map.
Every homomorphism $φ: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $α: Y \rightarrow Δ(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $Δ(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $α$ for which $φ=j_α: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractiv…
We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms in the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. We prove that the existence of idempotents of arbitrarily large norm in B(G) implies the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for every locally compact group H. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for some amenable locally compact group H implies the existence of idempotents of arb…
We prove that if $ρ: A(H) \to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $α: G \to H$, then $ρ$ extends to a w*-w* continuous map between the corresponding $L^\infty$ algebras if and only if $α$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $α$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haa…
We characterize completely bounded normal Jordan $*$-homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative $\mathrm{L}^p$-spaces.
Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal{A})$ and Jordan $*$-homomorphisms on $\operatorname{Lip}(X,\mathcal{A})$. More precisely, for any unital $C^{*}$-algebra $\mathcal{A}$, we prove that every Jordan $*$-homomorphism on $C(X, \mathcal{A})$ and every Jordan $*$-homomorphism on $\operatorname{Lip}(X,\mathcal{A})$ is represented as a weighted composition operator by using the irreducible representations of $\mathcal{A}$. In addition, when $\mathcal{A}_1$ and $\mathcal{A}_2$ are primitive $C^{*}$-algebras, we characterize the Jordan …
We show that a bounded, linear map between C*-algebras is a weighted $\ast$-homomorphism (the central compression of a $\ast$-homomorphism) if and only if it preserves zero-products, range-orthogonality, and domain-orthogonality. It follows that a self-adjoint, bounded, linear map is a weighted $\ast$-homomorphism if and only if it preserves zero-products. As an application we show that a linear map between C*-algebras is completely positive, order zero in the sense of Winter-Zacharias if and only if it is positive and preserves zero-products.
We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$ to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a manner reminiscent of several tracial approximation properties. We are particularly interested how regularity properties in the Elliott classification program pass from $B$ to $A$. Among them, we show that the strict comparison property and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting.
We introduce the Jordan-strict topology on the multipliers algebra of a JB$^*$-algebra, a notion which was missing despite the fourty years passed after the first studies on Jordan multipliers. In case that a C$^*$-algebra $A$ is regarded as a JB$^*$-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C$^*$-strict topology. We prove that every JB$^*$-algebra $\mathfrak{A}$ is J-strict dense in its multipliers algebra $M(\mathfrak{A})$, and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonalit…
We define a tracial analogue of the sequentially split $*$-homomorphism between $C^*$-algebras of Barlak and Szabó and show that several important approximation properties related to the classification theory of $C^*$-algebras pass from the target algebra to the domain algebra. Then we show that the tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$ gives rise to a tracial version of sequentially split $*$-homomorphism from $A\rtimes_αG$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \…
Let $A$ and $B$ be C*-algebras and $φ\colon A\to B$ be a $*$-homomorphism. We discuss the properties of the kernel and (co-)image of the induced map $\mathrm{K}_{0}(φ)\colon \mathrm{K}_{0}(A) \to \mathrm{K}_{0}(B)$ on the level of K-theory. In particular, we are interested in the case that the co-image is torsion free, and show that it holds when $A$ and $ B $ are commutative and unital, $B$ has real rank zero, and $φ$ is unital and injective. We also show that $ A$ is embeddable in $B$ if $ \mathrm{K}_{0}(φ)$ is injective and $A$ has stable rank one and real rank zero.
We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coa…
Texas Christian University（德克萨斯基督教大学）；University of Southern Denmark（南丹麦大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；University of Ottawa（渥太华大学）；University of Oxford（牛津大学）
Let $n\ge 2$, let $\mathcal{R}_n$ be the graph consisting of one vertex and $n$ loops and let $\mathcal{R}_{n^-}$ be its Cuntz splice. Let $L_n=L(\mathcal{R}_n)$ and $L_{n^-}=L(\mathcal{R}_{n^-})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^-}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied for example when $\ell$ is a field or a PID, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^-}$ nor in the opposite direction.
