$L^p$空间上的Cuntz代数类
Analogs of Cuntz algebras on $L^p$ spaces
摘要：
对于$d = 2, 3, \ldots$和$p \in [1, \infty),$，我们定义了莱维特代数$L_d$在形式为$L^p (X, μ),$的空间上的一类表示$ρ$，我们称之为空间表示。我们证明了，对于固定的$d$和$p,$的Banach代数${\mathcal{O}_{d}^{p}}$作为$L_d$下的表示$ρ$的图像的封闭获得是相同的所有空间表示$ρ.$当$p = 2,$我们恢复通常的Cuntz代数${\mathcal{O}}_{d}.$我们给出了一些等价条件的表示是空间的。我们证明了对于$[1, \infty)$中的不同$p_1$和$p_2$以及$\{ 2, 3, \ldots \},$中的任意$d_1$和$d_2$，从${\mathcal{O}}_{d_1}^{p_1}$到${\mathcal{O}}_{d_2}^{p_2}.$不存在非零连续同态
英文摘要：
For $d = 2, 3, \ldots$ and $p \in [1, \infty),$ we define a class of representations $ρ$ of the Leavitt algebra $L_d$ on spaces of the form $L^p (X, μ),$ 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 $ρ$ is the same for all spatial representations $ρ.$ 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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