[Paper Review] $L^p$-operator algebras associated with oriented graphs

Guillermo Cortiñas⋆, María Eugenia Rodríguez|arXiv (Cornell University)|Dec 23, 2017

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper introduces an $L^p$-operator algebra $\mathcal{O}^p(Q)$ associated with any countable oriented graph $Q$, generalizing Phillips' $L^p$-Cuntz algebras. It establishes that $\mathcal{O}^p(Q)$ is isometrically isomorphic to the closure of any nonzero spatial $L^p$-representation of the Leavitt path algebra $L_Q$, and proves that $\mathcal{O}^p(Q)$ is simple as an $L^p$-operator algebra if and only if $L_Q$ is simple, with no nonzero continuous homomorphism existing between $\mathcal{O}^p(Q)$ and $\mathcal{O}^{p'}(Q)$ when $p \neq p'$ and $L_Q$ is purely infinite simple.

ABSTRACT

For each $1\le p

Motivation & Objective

• To construct a universal $L^p$-operator algebra $\mathcal{O}^p(Q)$ associated with any countable oriented graph $Q$, generalizing $L^p$-Cuntz algebras.
• To characterize when $\mathcal{O}^p(Q)$ is simple as an $L^p$-operator algebra.
• To establish the non-existence of nonzero continuous homomorphisms between $\mathcal{O}^p(Q)$ and $\mathcal{O}^{p'}(Q)$ for $p \neq p'$ when $L_Q$ is purely infinite simple.
• To unify and generalize results from Phillips on $L^p$-Cuntz algebras to the broader class of Leavitt path algebras via $L^p$-operator algebras.

Proposed method

• Define $\mathcal{O}^p(Q)$ as the completion of the Leavitt path algebra $L_Q$ under the supremum norm induced by all spatial $L^p$-representations.
• Use spatial representations $\rho: L_Q \to \mathcal{L}(L^p(X))$ with $X$ a $\sigma$-finite measure space, where generators map to partial isometries satisfying spatiality conditions.
• Establish a spatiality criterion: a nondegenerate representation is spatial if and only if its restriction to the $0,1$-subalgebra is contractive and operator norms of generators are at most 1.
• Prove that $\mathcal{O}^p(Q)$ is isometrically isomorphic to the closure $\overline{\rho(L_Q)}$ for any nonzero spatial representation $\rho$, via a uniqueness theorem.
• Apply results from Andô and Gardella-Thiel to show that representations of $\mathcal{O}^p(Q)$ factor through spatial representations under finiteness or $p > 1$.
• Use $\ell^p(\mathbb{N})$-representations and subspace isomorphism obstructions (e.g., from Lindstrom-Tzafriri and Albin-Kalton) to rule out homomorphisms between $\mathcal{O}^p(Q)$ and $\mathcal{O}^{p'}(Q)$ for $p \neq p'$.

Experimental results

Research questions

• RQ1 When is the $L^p$-operator algebra $\mathcal{O}^p(Q)$ simple as an $L^p$-operator algebra?
• RQ2 Under what conditions is $\mathcal{O}^p(Q)$ isometrically isomorphic to the closure of the image of a nonzero spatial $L^p$-representation of $L_Q$?
• RQ3 Can there exist a nonzero continuous homomorphism between $\mathcal{O}^p(Q)$ and $\mathcal{O}^{p'}(Q)$ for $p \neq p'$ when $L_Q$ is purely infinite simple?
• RQ4 How does the simplicity of $L_Q$ relate to the simplicity of $\mathcal{O}^p(Q)$ as an $L^p$-operator algebra?
• RQ5 What is the relationship between $\mathcal{O}^p(Q)$ and the tight $L^p$-operator algebra $F_{\text{tight}}^p(\mathcal{S}(Q))$ for $p \notin \{1,2\}$?

Key findings

• For any countable oriented graph $Q$, the $L^p$-operator algebra $\mathcal{O}^p(Q)$ is isometrically isomorphic to the closure $\overline{\rho(L_Q)}$ of the image of any nonzero spatial $L^p$-representation $\rho: L_Q \to \mathcal{L}(L^p(X))$.
• The algebra $\mathcal{O}^p(Q)$ is simple as an $L^p$-operator algebra if and only if the Leavitt path algebra $L_Q$ is simple.
• If $L_Q$ is purely infinite simple and $p \neq p'$, then there is no nonzero continuous homomorphism $\mathcal{O}^p(Q) \to \mathcal{O}^{p'}(Q)$, generalizing Phillips' result for $L^p$-Cuntz algebras.
• For $p \notin \{1,2\}$, $\mathcal{O}^p(Q)$ coincides with the tight semigroup $L^p$-operator algebra $F_{\text{tight}}^p(\mathcal{S}(Q))$ associated with the inverse semigroup $\mathcal{S}(Q)$ generated by $Q$, establishing a bridge to earlier work by Gardella and Lupini.
• When $Q = \mathcal{R}_n$ (one vertex, $n$ loops), $\mathcal{O}^p(Q) = \mathcal{O}_n^p$, the $L^p$-Cuntz algebra, showing consistency with Phillips' construction.
• The existence of a nonzero continuous homomorphism $\mathcal{O}^p(Q) \to \mathcal{O}^{p'}(Q')$ implies that $L^{p'}(X)$ must contain a subspace isomorphic to $\ell^p(\mathbb{N})$, which is ruled out under the stated conditions via known results on $L^p$-space embeddings.