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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 $C(G)$.
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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 $Tinmathcal{A}(X)$.
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 $Phicolon L^{p/m}(mathcal{M},tau)to X$.
Let $G$ be a compact group. For $1leq pleqinfty$ we introduce a class of Banach function algebras $mathrm{A}^p(G)$ on $G$ which are the Fourier algebras in the case $p=1$, and for $p=2$ are certain algebras discovered in cite{forrestss1}. In the case $p ot=2$ we find that $mathrm{A}^p(G)cong mathrm{A}^p(H)$ if and only if $G$ and $H$ are isomorphic compact groups. These algebras admit natural operator space structures, and also weighte
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a metaplectic representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the metaplectic representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cheredniks basic representation. This allows us to introduce a new family of metaplectic polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper.
Let $G$ be a compact connected Lie group. The question of when a weighted Fourier algebra on $G$ is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on $G$ with the order of growth strictly bigger than the half of the dimension of the group. The case of SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.
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