[Paper Review] Arveson's extension theorem in *-algebras

Hossein Esslamzadeh, Lyudmila Turowska|arXiv (Cornell University)|Nov 20, 2013

Advanced Operator Algebra Research16 references3 citations

TL;DR

This paper establishes a purely algebraic generalization of Arveson's extension theorem by showing that the algebra of bounded operators on a Hilbert space, $\mathcal{B}(\mathcal{H})$, remains injective in the category of quasi operator systems—self-adjoint unital subspaces of unital $*$-algebras—using Archimedeanization as a key tool. The main result proves that any unital completely positive map from such a quasi operator system extends to the entire $*$-algebra.

ABSTRACT

Arveson's extension theorem asserts that B(H) is an injective object in the category of operator systems. Calling every self adjoint unital subspace of a unital *-algebra, a quasi operator system, we show that Arveson's theorem remains valid in the much larger category of quasi operator systems. This shows that Arveson's theorem as a non commutative extension of Hahn-Banach theorem, is of purely algebraic nature.

Motivation & Objective

• To investigate whether Arveson's extension theorem, known in functional analysis, has an underlying algebraic nature independent of $C^*$-algebraic structure.
• To extend the theorem beyond $C^*$-algebras by replacing the domain with bounded unital $*$-algebras and quasi operator systems.
• To demonstrate that $\mathcal{B}(\mathcal{H})$ remains injective in this broader category of quasi operator systems.
• To establish the universal property of Archimedeanization in the context of matrix-ordered $*$-vector spaces with matrix order units.
• To provide a new proof technique for Arveson-type extension theorems that avoids classical analytic methods and instead relies on algebraic and order-theoretic tools.

Proposed method

• Introduce the concept of a quasi operator system as a self-adjoint unital subspace of a unital $*$-algebra.
• Define the bounded subalgebra $\mathcal{A}_0$ of a unital $*$-algebra $\mathcal{A}$, consisting of elements $a$ with $a^*a \leq k1$ for some $k \in \mathbb{R}^+$.
• Use Archimedeanization to construct a universal object $\mathcal{X}_{\text{Arch}}$ from a quasi operator system $\mathcal{X}$, ensuring the positive cone becomes Archimedean.
• Apply the quotient map $P: \mathcal{X} \to \mathcal{X}_{\text{Arch}}$ and show it is unital and completely positive.
• Prove that any unital completely positive map $\phi: \mathcal{X} \to \mathcal{Y}$, where $\mathcal{Y}$ is a quasi operator system in a $*$-algebra with a proper Archimedean cone, factors uniquely through $\mathcal{X}_{\text{Arch}}$.
• Leverage the reducing ideal $\mathcal{A}_R$ and norm properties to ensure well-definedness and positivity of the induced map $\widetilde{\phi}$.

Experimental results

Research questions

• RQ1 Can Arveson's extension theorem be generalized to a purely algebraic setting beyond $C^*$-algebras?
• RQ2 Is the injectivity of $\mathcal{B}(\mathcal{H})$ as a codomain for completely positive maps intrinsic to algebraic and order-theoretic structure rather than analytic properties?
• RQ3 Does Archimedeanization provide a universal construction for extending completely positive maps from quasi operator systems?
• RQ4 Can the extension of a unital completely positive map from a quasi operator system be uniquely factored through its Archimedeanization?
• RQ5 What role does the bounded subalgebra $\mathcal{A}_0$ play in enabling such algebraic extensions?

Key findings

• The algebra $\mathcal{B}(\mathcal{H})$ is injective in the category of quasi operator systems, meaning any unital completely positive map from such a system to $\mathcal{B}(\mathcal{H})$ extends to the entire $*$-algebra.
• The Archimedeanization $\mathcal{X}_{\text{Arch}}$ of a bounded quasi operator system $\mathcal{X}$ is a universal object for unital completely positive maps into quasi operator systems with proper Archimedean cones.
• The quotient map $P: \mathcal{X} \to \mathcal{X}_{\text{Arch}}$ is unital and completely positive, and any unital completely positive map $\phi: \mathcal{X} \to \mathcal{Y}$ factors uniquely through $\mathcal{X}_{\text{Arch}}$.
• The kernel $N$ of the quotient map $P$ satisfies $\|x\|_{\mathcal{A}} = 0$ for all $x \in N$, ensuring that $\phi(N) = 0$ and thus $\widetilde{\phi}$ is well-defined.
• The induced map $\widetilde{\phi}: \mathcal{X}_{\text{Arch}} \to \mathcal{Y}$ is completely positive, as shown by the Archimedean property of the matrix order unit in $\mathcal{B}$.
• The construction relies on the reducing ideal $\mathcal{A}_R$ and the fact that $\|\cdot\|_{\mathcal{B}}$ is a norm, which ensures that $\phi(N) = 0$ implies well-definedness of $\widetilde{\phi}$.