[Paper Review] Metric characterisation of unitaries in JB$^*$-algebras

María Cueto-Avellaneda, Antonio M. Peralta|arXiv (Cornell University)|Jul 10, 2019

Advanced Operator Algebra Research23 references4 citations

TL;DR

This paper establishes a novel geometric characterization of unitary elements in unital JB$^*$-algebras by showing that an extreme point $ u $ of the closed unit ball is unitary if and only if the set $ \mathcal{M}_u = \{ e \in \partial_e(\mathcal{B}_M) : \|u \pm e\| \leq \sqrt{2} \} $ contains an isolated point. The result provides a metric-topological criterion for unitarity based solely on the structure of extreme points and their norm relations.

ABSTRACT

Let $M$ be a unital JB$^*$-algebra whose closed unit ball is denoted by $\mathcal{B}_M$. Let $\partial_e(\mathcal{B}_M)$ denote the set of all extreme points of $\mathcal{B}_M$. We prove that an element $u\in \partial_e(\mathcal{B}_M)$ is a unitary if and only if the set $$\mathcal{M}_{u} = \{e\in \partial_e(\mathcal{B}_M) : \|u\pm e\|\leq \sqrt{2} \}$$ contains an isolated point. This is a new geometric characterisation of unitaries in $M$ in terms of the set of extreme points of $\mathcal{B}_M$.

Motivation & Objective

• To provide a geometric, metric-based characterization of unitary elements in unital JB$^*$-algebras using only the Banach space structure of the algebra.
• To extend the geometric characterizations of unitaries from C$^*$-algebras and JB$^*$-triples to the broader class of JB$^*$-algebras.
• To clarify the distinction between unitaries and other extreme points in JB$^*$-algebras by identifying a topological invariant—isolated points in a specific norm-bounded set of extreme points.
• To investigate the role of the set $ \mathcal{M}_u $ in determining unitarity, particularly in non-C$^*$-algebraic settings like JB-algebras and rectangular Cartan factors.

Proposed method

• The authors analyze the set $ \mathcal{M}_u = \{ e \in \partial_e(\mathcal{B}_M) : \|u \pm e\| \leq \sqrt{2} \} $ for $ u \in \partial_e(\mathcal{B}_M) $, where $ \partial_e(\mathcal{B}_M) $ is the set of extreme points of the closed unit ball of a unital JB$^*$-algebra $ M $.
• They use the theory of tripotents and the Peirce decomposition in JB$^*$-triples to analyze the structure of elements $ y \in \mathcal{M}_u $, particularly focusing on their spectral and orthogonality properties.
• The proof relies on constructing a one-parameter family $ y_\theta $ of extreme points in $ \mathcal{M}_u $ that converge to a given $ y \in \mathcal{M}_u $, demonstrating non-isolation when $ u $ is not unitary.
• They apply results from the theory of Brown-Pedersen quasi-invertibility and the structure of rectangular Cartan factors to analyze cases where $ u $ is not unitary.
• The argument uses the fact that in a JB$^*$-triple with a unitary element, the set $ \mathcal{E}_u $ of extreme points satisfying $ \|u \pm e\| \leq \sqrt{2} $ contains an isolated point if and only if $ u $ is unitary.
• The authors use a contradiction argument: assuming $ u $ is not unitary but $ \mathcal{M}_u $ contains an isolated point leads to a contradiction via the existence of a net $ y_\theta \to y $ with $ y_\theta \in \mathcal{M}_u $, $ y_\theta \neq y $.

Experimental results

Research questions

• RQ1 Can unitary elements in a unital JB$^*$-algebra be characterized purely by the topological structure of the set of extreme points of the unit ball?
• RQ2 Is the existence of an isolated point in the set $ \mathcal{M}_u = \{ e \in \partial_e(\mathcal{B}_M) : \|u \pm e\| \leq \sqrt{2} \} $ sufficient to guarantee that $ u $ is unitary?
• RQ3 Does the absence of isolated points in $ \mathcal{M}_u $ imply that $ u $ is not unitary, even when $ u $ is an extreme point of the unit ball?
• RQ4 How does the geometric structure of $ \mathcal{M}_u $ differ in JB$^*$-algebras that are not C$^*$-algebras, such as rectangular Cartan factors?
• RQ5 Can the characterization of unitaries via isolated points in $ \mathcal{M}_u $ be extended to general JB$^*$-triples?

Key findings

• An element $ u \in \partial_e(\mathcal{B}_M) $ is unitary in a unital JB$^*$-algebra $ M $ if and only if the set $ \mathcal{M}_u = \{ e \in \partial_e(\mathcal{B}_M) : \|u \pm e\| \leq \sqrt{2} \} $ contains an isolated point.
• The characterization is purely metric and topological, relying only on the norm and the set of extreme points of the unit ball, without reference to the algebraic involution or Jordan product.
• For non-unitary extreme points $ u $, the set $ \mathcal{M}_u $ contains no isolated points; in particular, every $ y \in \mathcal{M}_u $ is a limit point of other elements in $ \mathcal{M}_u $, as shown by constructing a net $ y_\theta \to y $ with $ y_\theta \in \mathcal{M}_u $.
• In rectangular Cartan factors of type 1 with $ \dim(H) > \dim(K) $, the set $ \mathcal{C}_u $ is pathwise-connected and contains no isolated points, consistent with the non-unitarity of $ u $.
• The result extends to JB$^*$-triples: a complete tripotent $ u $ is unitary if and only if $ \mathcal{E}_u = \{ e \in \partial_e(\mathcal{B}_E) : \|u \pm e\| \leq \sqrt{2} \} $ contains an isolated point.
• The characterization distinguishes unitaries from other extreme points in JB$^*$-algebras, particularly in cases where the standard geometric characterizations via states or dual spaces fail or differ, as in real JB-algebras.