Analogs of Cuntz algebras on $L^p$ spaces

Abstract

For $d=2,3,\dots$ 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 $\{2,3,\dots \},$ there is no nonzero continuous homomorphism from ${\mathcal{O}}_{d_1}^{p_1}$ to ${\mathcal{O}}_{d_2}^{p_2}.$

arXiv comment60 pages; AMSLaTeX

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