Spectral versus interpolation norms in tracial nonassociative $\mathrm{L}^p$-spaces

Abstract: We investigate the metric structure of nonassociative $\mathrm{L}<sup>p$-spaces associated with tracial $\mathrm{JW}<sup>*$-algebras. While noncommutative $\mathrm{L}<sup>p$-spaces arising from von Neumann algebras enjoy a unique natural norm, the situation in the Jordan setting is more subtle. We compare two canonical definitions: the interpolation norm, arising from the complex method between the algebra and its predual, and the spectral norm, defined with the trace. We show that these two norms are equivalent but generally not isometric for $p \neq 2$, even in the associative case of nonabelian von Neumann algebras when viewed through the Jordan product, thereby answering an open question raised by the first author in a previous paper. We further analyze the geometry of these spaces in concrete examples as complex spin factors or the complexified Albert algebra. Finally, we discuss the relevance of these results to generalized probabilistic theories (GPTs), where Jordan structures arise naturally, and explain why $\mathrm{JBW}$-algebras and their preduals provide a natural framework for such models.

• The paper establishes that spectral and interpolation norms, while equivalent, are non-isometric for p ≠ 2 in tracial nonassociative L^p-spaces.
• It derives sharp equivalence constants and explicit formulas for key examples like complex spin factors and the complexified Albert algebra.
• The study links these norm results to applications in generalized probabilistic theories, advancing the mathematical framework for nonassociative integration.

Comparison of Spectral and Interpolation Norms in Tracial Nonassociative $\mathrm{L}^p$-Spaces

Introduction and Motivation

The paper "Spectral versus interpolation norms in tracial nonassociative $\mathrm{L}^p$-spaces" (2604.23232) addresses the metric structure of $\mathrm{L}^p$-spaces associated with tracial Jordan operator algebras, specifically focusing on situations where associativity is absent. While classical noncommutative $\mathrm{L}^p$-spaces linked to von Neumann algebras possess a canonical norm structure, the authors highlight subtle complexities when these spaces are constructed in the nonassociative Jordan framework. The study is motivated both by the mathematical foundations of quantum mechanics—where Jordan algebras naturally encode observables via the symmetrized product—and by emerging applications in generalized probabilistic theories (GPTs), which encompass quantum mechanics, real quantum systems, and exceptional cases such as octonionic models.

Mathematical Framework

Jordan Algebra Structures

The paper systematically develops the machinery of complex and real Jordan algebras and their operator analogs: $JBW^*$-algebras (weak* closed Jordan $*$-algebras) and $JW^*$-algebras (weak* closed Jordan subalgebras of a von Neumann algebra). The essential Jordan product $x \circ y = \frac{1}{2}(xy + yx)$, though commutative, is nonassociative and satisfies the Jordan identity, which is critical for modeling quantum observables.

Traces and Norms

A key concern is defining a meaningful norm for the $\mathrm{L}^p$-spaces attached to these algebras under a trace. Two principal constructions are investigated:

• Spectral Norm: Defined for $x \in M$ as $\mathrm{L}^p$0, where $\mathrm{L}^p$1 is a normal finite faithful trace. This generalizes the familiar Schatten $\mathrm{L}^p$2-norm in the associative case.
• Interpolation Norm: Derived via complex interpolation between $\mathrm{L}^p$3 and its predual $\mathrm{L}^p$4, leveraging embeddings defined by the trace and product structure.

The paper proves for $\mathrm{L}^p$5 that these norms, though equivalent, are not isometric—even in the associative framework when viewed through the Jordan lens. This result directly answers an open problem raised in prior work, showing a rigidity arising from the symmetrized product.

Main Results

Norm Equivalence and Non-Isometry

The authors compute sharp equivalence constants for the spectral and interpolation norms:

• For $\mathrm{L}^p$6, $\mathrm{L}^p$7.
• For $\mathrm{L}^p$8, $\mathrm{L}^p$9.
• For $\mathrm{L}^p$0, both norms coincide; for $\mathrm{L}^p$1, isometry does not hold except for normal elements and in the commutative case.

These constants are optimal, as demonstrated via explicit computation on $\mathrm{L}^p$2 matrices and embedding techniques.

Concrete Examples

The paper calculates the spectral norm explicitly for complex spin factors and the complexified Albert algebra. For spin factors (Jordan algebras constructed from Hilbert spaces via spin systems), explicit formulas are obtained for the $\mathrm{L}^p$3-norm of elements $\mathrm{L}^p$4, which depend on the Hilbertian norm of $\mathrm{L}^p$5 and the value of $\mathrm{L}^p$6. The same spectral machinery applies to the exceptional Jordan algebra $\mathrm{L}^p$7, showing that for selfadjoint elements, the norm reduces to the normalized $\mathrm{L}^p$8-norm of the spectral values.

Contractive Projections and Embeddings

Building on prior results, the authors show that contractively complemented subspaces arising from positive contractive projections onto noncommutative $\mathrm{L}^p$9-spaces can be isometrically realized as nonassociative $\mathrm{L}^p$0 spaces. This is significant for understanding structural decompositions in operator space theory, where projections correspond to distinct physical or statistical subsystems.

Application to Generalized Probabilistic Theories

A substantial theoretical implication is the compatibility of the $\mathrm{L}^p$1-algebra framework with GPTs. The predual spaces and their order structure provide a natural foundation for axiomatic probabilistic models extending quantum mechanics, including those with exceptional Jordan structure (e.g., real, complex, quaternionic, and octonionic models).

Spectral nonassociative $\mathrm{L}^p$2-norms establish the analytical groundwork for notions such as entropy and resource quantification in these GPTs, even where associativity is absent.

Implications and Further Directions

Practical and Theoretical Impact

The clarified relationship between spectral and interpolation norms provides robust tools for analytic and geometric investigations in nonassociative operator settings. It enables precise understanding of contractive projection ranges, spectral decompositions, and duality properties—key ingredients for quantum information theory and functional analytic approaches to quantum foundations. The rigorous treatment of Jordan structures advances the mathematical architecture for GPTs, supporting generalizations beyond standard Hilbert space quantum mechanics.

Open Questions

The paper identifies important open problems, such as:

• Whether the spectral norm formula defines a norm for $\mathrm{L}^p$3 for arbitrary measure spaces.
• Establishing the relation between the spectral norm and singular values in $\mathrm{L}^p$4, connecting analytic properties with spectral data.

Future research directions may include further exploration of entropy theory within the $\mathrm{L}^p$5 setting, investigation of duality structures, and characterization of nonassociative norm geometries relevant to quantum and GPT paradigms.

Conclusion

This paper gives a precise analysis of norm equivalence in tracial nonassociative $\mathrm{L}^p$6-spaces, establishing optimal constants, demonstrating non-isometricity in key cases, and providing explicit formulas for important Jordan algebra examples. Its results solidify the framework for nonassociative integration theory, projective decompositions, and GPT modeling, and open pathways for further mathematical and physical investigations into the structure of quantum and probabilistic theories.