Takesaki duality generalizes to weak* closed L^2-operator crossed products but fails to generalize to L^p-operator crossed products for p ≠ 2.
Analogs of Cuntz algebras on $L^p$ spaces
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
For $d = 2, 3, \ldots$ and $p \in [1, \infty),$ we define a class of representations $\rho$ of the Leavitt algebra $L_d$ on spaces of the form $L^p (X, \mu),$ which we call the spatial representations. We prove that for fixed $d$ and $p,$ the Banach algebra ${{\mathcal{O}}_{d}^{p}}$ obtained as the closure of the image of $L_d$ under the representation $\rho$ is the same for all spatial representations $\rho.$ When $p = 2,$ we recover the usual Cuntz algebra ${\mathcal{O}}_{d}.$ We give a number of equivalent conditions for a representation to be spatial. We show that for distinct $p_1$ and $p_2$ in $[1, \infty)$ and arbitrary $d_1$ and $d_2$ in $\{ 2, 3, \ldots \},$ there is no nonzero continuous homomorphism from ${\mathcal{O}}_{d_1}^{p_1}$ to ${\mathcal{O}}_{d_2}^{p_2}.$
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UNVERDICTED 2representative citing papers
Defines twisted crossed products of Banach algebras via families of representations and proves they form Banach algebras with universal properties; generalizes Packer-Raeburn trick to show L^p-twisted crossed products are stably isometrically isomorphic to untwisted ones.
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Takesaki duality for weak* closed $L^p$-operator crossed products
Takesaki duality generalizes to weak* closed L^2-operator crossed products but fails to generalize to L^p-operator crossed products for p ≠ 2.
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Twisted crossed products of Banach algebras
Defines twisted crossed products of Banach algebras via families of representations and proves they form Banach algebras with universal properties; generalizes Packer-Raeburn trick to show L^p-twisted crossed products are stably isometrically isomorphic to untwisted ones.