[Paper Review] Nuclearity properties and $C^*$-envelopes of operator system inductive limits

Preeti Luthra, Ajay Kumar|arXiv (Cornell University)|Dec 5, 2016

Advanced Operator Algebra Research11 references3 citations

TL;DR

This paper investigates the preservation of $C^*$-envelopes and various nuclearity properties under inductive limits in the category of operator systems. It establishes that the $C^*$-envelope of an inductive limit of operator systems is isomorphic to the inductive limit of their $C^*$-envelopes under certain conditions, and proves that several nuclearity properties—such as $(\min,\mathrm{el})$-, $(\min,\mathrm{ess})$-, and $(\mathrm{el},\mathrm{c})$-nuclearity—are preserved under inductive limits when the systems satisfy appropriate technical conditions, including $C^*_e$-increasing inclusions and unitary density in their envelopes.

ABSTRACT

We study the relationship between $C^*$-envelopes and inductive limit of operator systems. Various operator system nuclearity properties of inductive limit for a sequence of operator systems are also discussed.

Motivation & Objective

• To investigate the relationship between $C^*$-envelopes and inductive limits of operator systems.
• To determine whether various operator system nuclearity properties are preserved under inductive limits.
• To establish conditions under which the $C^*$-envelope of an inductive limit of operator systems equals the inductive limit of their $C^*$-envelopes.
• To analyze the preservation of $(\min,\mathrm{el})$-, $(\min,\mathrm{ess})$-, and $(\mathrm{el},\mathrm{c})$-nuclearity in inductive limits.
• To clarify the role of $C^*_e$-increasing inclusions and unitary density in $C^*$-envelopes for nuclearity preservation.

Proposed method

• Utilizes the universal property of $C^*$-envelopes and the construction of inductive limits in the category of operator systems.
• Applies the fact that inductive limits commute with $C^*$-envelopes when the system is $C^*_e$-increasing and satisfies unitary density conditions.
• Employs known results on tensor product nuclearity: $(\min,\mathrm{el})$-nuclearity corresponds to exactness, and $(\mathrm{el},\mathrm{c})$-nuclearity corresponds to DCEP.
• Uses the characterization of $(\min,\mathrm{ess})$-nuclearity via $C^*$-nuclearity of the $C^*$-envelope and the preservation of $C^*$-nuclearity under inductive limits.
• Applies the fact that maximal commuting tensor product $\mathrm{c}$ coincides with the maximal tensor product when one factor is a $C^*$-algebra.
• Leverages the left-injectivity of the $\mathrm{el}$ tensor product to ensure compatibility of inductive limits with tensor products.

Experimental results

Research questions

• RQ1 Under what conditions does the $C^*$-envelope of an inductive limit of operator systems coincide with the inductive limit of their $C^*$-envelopes?
• RQ2 Is $(\min,\mathrm{el})$-nuclearity preserved under inductive limits of operator systems?
• RQ3 Does $C^*$-nuclearity of the $C^*$-envelope imply $(\min,\mathrm{ess})$-nuclearity of the operator system, and is this preserved under inductive limits?
• RQ4 Is $(\mathrm{el},\mathrm{c})$-nuclearity (i.e., DCEP) preserved under inductive limits of operator systems?
• RQ5 What role do $C^*_e$-increasing inclusions and unitary density in $C^*$-envelopes play in preserving nuclearity under inductive limits?

Key findings

• The $C^*$-envelope of the inductive limit of an increasing sequence of operator systems is isomorphic to the inductive limit of their $C^*$-envelopes, provided the system is $C^*_e$-increasing and each $\mathcal{S}_n$ contains enough unitaries in $C^*_e(\mathcal{S}_n)$.
• If each $\mathcal{S}_n$ is $(\min,\mathrm{el})$-nuclear, then the inductive limit $\displaystyle\lim_{\longrightarrow}\mathcal{S}_n$ is also $(\min,\mathrm{el})$-nuclear.
• If each $\mathcal{S}_n$ is $(\min,\mathrm{ess})$-nuclear and the system is $C^*_e$-increasing with unitary density, then $\displaystyle\lim_{\longrightarrow}\mathcal{S}_n$ is $(\min,\mathrm{ess})$-nuclear.
• If each $\mathcal{S}_n$ has DCEP (i.e., is $(\mathrm{el},\mathrm{c})$-nuclear), then the inductive limit $\displaystyle\lim_{\longrightarrow}\mathcal{S}_n$ also has DCEP.
• The preservation of $C^*$-nuclearity under inductive limits ensures that $C^*$-nuclearity of $C^*_e(\mathcal{S}_n)$ implies $(\min,\mathrm{ess})$-nuclearity of $\mathcal{S}_n$, and this property is inherited by the inductive limit.
• The inductive limit of operator systems preserves the maximal commuting tensor product $\mathrm{c}$, and when one factor is a $C^*$-algebra, $\mathrm{c}$ coincides with the maximal tensor product, enabling the use of $C^*$-nuclearity arguments.