Analogs of Cuntz algebras on $L^p$ spaces

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