Analogs of Cuntz algebras on Lp spaces

By N. Christopher Phillips. 2012

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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 ${\mathcal{O}}_{d_2}^{p_2}.$

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d0feb4902a3a9e539a216c49b7aa24498ac812f

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces (2013, methodology): “16 of [19], and remove the hypothesis of σ-finiteness there by using, in [10], Theorem 1.”