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Register a new userOrthogonally additive polynomials on non-commutative $L^p$-spaces
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Let $mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $tau$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(mathcal{M},tau)$, with $0<p<infty$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(mathcal{M},tau)$ can be represented in the form $P(x)=Phi(x^m)$ $(xin L^p(mathcal{M},tau))$ for some continuous linear map $Phicolon L^{p/m}(mathcal{M},tau)to X$.
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Let $mathcal{M}$ be a von Neumann algebra, and let $0<p,qleinfty$. Then the space $Hom_mathcal{M}(L^p(mathcal{M}),L^q(mathcal{M}))$ of all right $mathcal{M}$-module homomorphisms from $L^p(mathcal{M})$ to $L^q(mathcal{M})$ is a reflexive subspace of the space of all continuous linear maps from $L^p(mathcal{M})$ to $L^q(mathcal{M})$. Further, the space $Hom_mathcal{M}(L^p(mathcal{M}),L^q(mathcal{M}))$ is hyperreflexive in each of the following cases: (i) $1le q<pleinfty$; (ii) $1le p,qleinfty$ and $mathcal{M}$ is injective, in which case the hyperreflexivity constant is at most $8$.
Based on the characterization of surjective $L^p$-isometries of unitary groups in finite factors, we describe all surjective $L^p$-isometries between Grassmann spaces of projections with the same trace value in semifinite factors.
Let $X$ and $Y$ be Banach spaces, let $mathcal{A}(X)$ stands for the algebra of approximable operators on $X$, and let $Pcolonmathcal{A}(X)to Y$ be an orthogonally additive, continuous $n$-homogeneous polynomial. If $X^*$ has the bounded approximation property, then we show that there exists a unique continuous linear map $Phicolonmathcal{A}(X)to Y$ such that $P(T)=Phi(T^n)$ for each $Tinmathcal{A}(X)$.
We apply Arvesons non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We do this by describing all representations of the Toeplitz algebra $mathcal{T}(G)$ that have unique extension when restricted to the tensor algebra $mathcal{T}_+(G)$. This yields an alternative proof to a result of Katsoulis and Kribs that the $C^*$-envelope of $mathcal T_+(G)$ is the Cuntz-Krieger algebra $mathcal O(G)$. We then generalize our dilation results further, to the context of colored directed graphs, by investigating free products of operator algebras. These generalizations rely on results of independent interest on complete injectivity and a characterization of representations with the unique extension property for free products of operator algebras.
Let $G$ be a compact group, let $X$ be a Banach space, and let $Pcolon L^1(G)to X$ be an orthogonally additive, continuous $n$-homogeneous polynomial. Then we show that there exists a unique continuous linear map $Phicolon L^1(G)to X$ such that $P(f)=Phi bigl(faststackrel{n}{cdots}ast f bigr)$ for each $fin L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1< pleinfty$, and $C(G)$.
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