[Paper Review] A Note on Dominant Contractions of Jordan Algebras

Farrukh Mukhamedov, Seyi̇t Temir+1 more|Jun 18, 2008

Advanced Operator Algebra Research8 references4 citations

TL;DR

This paper establishes that for positive contractions $T$ and $S$ on the $L_1$-space of a semi-finite $JBW$-algebra with $T \leq S$, if $\|S^{n_0} - T^{n_0}\| < 1$ for some $n_0 \in \mathbb{N}$, then $\|S^n - T^n\| < 1$ for all $n \geq n_0$. The result extends Zaharopol's zero-two law to non-associative and non-commutative settings, using spectral and order-theoretic properties of $JBW$-algebras and the additivity of the norm on positive elements.

ABSTRACT

In the paper we consider two positive contractions $T,S:L^{1}(A,τ)\longrightarrow L^{1}(A,τ)$ such that $T\leq S$, here $(A, )$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in\mathbb{N}$ such that $\|S^{n_{0}}-T^{n_{0}}\|&lt;1$. Then we prove that $\|S^{n}-T^{n}\|&lt;1$ holds for every $n\geq n_{0}.$

Motivation & Objective

• To resolve Problem 1.2: whether $\|S^n - T^n\| < 1$ holds for all $n \geq n_0$ when $\|S^{n_0} - T^{n_0}\| < 1$ for positive contractions $T, S$ on $L_1(A,\tau)$ with $T \leq S$.
• To extend Zaharopol's zero-two law, which requires $\|S - T\| < 1$, to cases where $\|S - T\| = 1$ but $\|S^{n_0} - T^{n_0}\| < 1$ for some $n_0 > 1$.
• To establish the result in the non-associative setting of $JBW$-algebras, where von Neumann algebra techniques do not apply due to the existence of exceptional algebras.
• To demonstrate that the result holds in any partially ordered Banach space where the norm is additive on the positive cone, broadening its applicability beyond $L_1$-spaces.

Proposed method

• The authors work within the framework of semi-finite $JBW$-algebras equipped with a faithful, normal, semi-finite trace $\tau$, defining $L_1(A,\tau)$ as the completion of $\{x \in A : \tau(|x|) < \infty\}$ under the $L_1$-norm.
• They use the order structure induced by the positive cone $A^+$ and the spectral properties of positive operators on $L_1(A,\tau)$, particularly focusing on the behavior of iterated powers $T^n$ and $S^n$.
• The key technical tool is the additivity of the norm on positive elements: $\|x_1 - x_2\| = \|x_1\| + \|x_2\|$ for $x_1, x_2 \in A^+$, which allows control over the difference $\|S^n - T^n\|$.
• They analyze the operator norms via suprema over normalized positive vectors in $\mathbb{R}^2$, constructing explicit examples with matrices to verify the norm conditions.
• The proof relies on the fact that $S^{n_0}$ and $T^{n_0}$ are positive contractions with $\|S^{n_0} - T^{n_0}\| < 1$, and then applies Theorem 1.1 iteratively to $S^{n_0}$ and $T^{n_0}$, showing that $\|S^{nn_0} - T^{nn_0}\| < 1$ for all $n \in \mathbb{N}$, which is extended to all $n \geq n_0$ via continuity and order structure.
• A concrete counterexample is constructed using $2 \times 2$ matrices with $A = C = 1/2$, $B = D = 1/3$, $\lambda = 1/4$, showing $\|S - T\| = 1$ but $\|S^2 - T^2\| < 1$, proving the result is strictly stronger than Zaharopol's.

Experimental results

Research questions

• RQ1 Does $\|S^n - T^n\| < 1$ hold for all $n \geq n_0$ if $\|S^{n_0} - T^{n_0}\| < 1$ for positive contractions $T, S$ on $L_1(A,\tau)$ with $T \leq S$?
• RQ2 Can the zero-two law for $L_1$-contractions be extended to cases where $\|S - T\| = 1$, which invalidates the original condition of Theorem 1.1?
• RQ3 Is the result valid in non-associative algebras such as $JBW$-algebras, where standard von Neumann algebra methods fail due to exceptional algebras?
• RQ4 Does the norm additivity on the positive cone of a Banach space allow a generalization of the result beyond $L_1$-spaces?

Key findings

• The paper proves that if $\|S^{n_0} - T^{n_0}\| < 1$ for some $n_0 \in \mathbb{N}$, then $\|S^n - T^n\| < 1$ for all $n \geq n_0$, even when $\|S - T\| = 1$, thus resolving Problem 1.2 affirmatively.
• The result is established for positive contractions on $L_1(A,\tau)$ where $(A, \tau)$ is a semi-finite $JBW$-algebra, extending Zaharopol's zero-two law to non-associative and non-commutative settings.
• A concrete example is constructed with $A = C = 1/2$, $B = D = 1/3$, $\lambda = 1/4$, showing $\|S - T\| = 1$ but $\|S^2 - T^2\| = 5/12 < 1$, confirming the condition of the theorem is satisfied despite the failure of the original $\|S - T\| < 1$ condition.
• The key technical insight is that the norm additivity on positive elements — $\|x_1 - x_2\| = \|x_1\| + \|x_2\|$ for $x_1, x_2 \geq 0$ — allows the propagation of the strict norm inequality through iterated powers.
• The result generalizes to any partially ordered Banach space where the norm is additive on the positive cone, including $A(K)^*$, the dual of the space of continuous affine functions on a compact convex set $K$, as shown in Remark 3.3.
• The result applies to absolute contractions on von Neumann algebras, since their self-adjoint parts form $JBW$-algebras and such contractions extend to $L_1$-contractions, as shown in Remark 3.2.