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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 $Tinmathcal{A}(X)$.
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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 $C(G)$.
We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelsons space.
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 $Phicolon L^{p/m}(mathcal{M},tau)to X$.
Given a complex Banach space $X$ and a joint spectrum for complex solvable finite dimensional Lie algebras of operators defined on $X$, we extend this joint spectrum to quasi-solvable Lie algebras of operators, and we prove the main spectral properties of the extended joint spectrum. We also show that this construction is uniquely determined by the original joint spectrum.
Starting with an additive process $(Y_t)_{tgeq0}$, it is in certain cases possible to construct an adjoint process $(X_t)_{tgeq0}$ which is itself additive. Moreover, assuming that the transition densities of $(Y_t)_{tgeq0}$ are controlled by a natural pair of metrics $mathrm{d}_{psi,t}$ and $delta_{psi,t}$, we can prove that the transition densities of $(X_t)_{tgeq0}$ are controlled by the metrics $delta_{psi,1/t}$ replacing $mathrm{d}_{psi,t}$ and $mathrm{d}_{psi,1/t}$ replacing $delta_{psi,t}$.
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