[Paper Review] Similarities and differences between real and complex Banach spaces: an overview and recent developments

Mohammad Sal Moslehian, Gustavo A. Muñoz-Fernández+2 more|arXiv (Cornell University)|Jul 8, 2021

Advanced Banach Space Theory4 citations

TL;DR

This paper provides a comprehensive comparative analysis of real and complex Banach spaces, highlighting fundamental differences in their structural and operator-theoretic properties. It systematically examines the complexification process, demonstrates how key results in complex Banach spaces fail in the real setting, and identifies conditions under which real counterparts hold, with particular focus on spectral theory, invariant subspaces, polynomial inequalities, and Jordan structures in real operator algebras.

ABSTRACT

There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.

Motivation & Objective

• To clarify the subtle yet critical differences between results in real and complex Banach space theory.
• To investigate the limitations and failures of extending complex results to the real setting, especially in spectral theory and operator algebras.
• To examine the role of complexification in translating results from complex to real Banach spaces and vice versa.
• To identify conditions under which real Banach spaces behave similarly to their complex counterparts, particularly in the context of polynomial inequalities and operator algebras.
• To analyze the behavior of geometric and algebraic structures—such as unitaries, symmetries, and projections—in real JB*-triples and real C*-algebras.

Proposed method

• Systematic comparison of real and complex Banach spaces using the complexification functor to relate real structures to their complex extensions.
• Analysis of various norms on the complexification of a real Banach space, including regular and strictly convex complexifications.
• Study of the extension of linear operators, polynomials, and multilinear mappings from real to complex spaces, with attention to injectivity and spectral properties.
• Application of advanced tools from functional analysis, including the spatial numerical range, numerical index, and the Russo–Dye theorem.
• Investigation of Jordan and C*-algebraic structures in the real setting, including real JB*-algebras and real non-commutative JB*-algebras.
• Use of counterexamples and structural theorems (e.g., Ferenczi’s example) to demonstrate non-isomorphism between real-isometric but complex-non-isomorphic Banach spaces.

Experimental results

Research questions

• RQ1 In what ways do spectral properties of operators differ between real and complex Banach spaces?
• RQ2 To what extent can results in complex Banach spaces—such as the existence of invariant subspaces or the validity of the Mazur–Ulam theorem—be extended to the real setting?
• RQ3 How do polynomial inequalities (e.g., Bohnenblust–Hille, polarization constants) behave differently in real versus complex Banach spaces?
• RQ4 What are the structural differences between real and complex Banach algebras, particularly in the context of real C*-algebras and real JB*-triples?
• RQ5 Under what conditions is a real Banach space with numerical index 1 necessarily finite-dimensional, and how does this compare to the complex case?

Key findings

• There exist real Banach spaces that are isometric as real spaces but non-isomorphic as complex spaces, demonstrating that real isometry does not imply complex isomorphism.
• The numerical index 1 problem in real Banach spaces implies finite-dimensionality for reflexive spaces, a result that does not hold in the complex case, where infinite-dimensional examples remain open.
• In real JB*-triples, a norm-one element is geometrically unitary if and only if it is a vertex of the unit ball and induces a JB-algebra structure via the product $ x \circ y = \{x,u,y\} $.
• The existence of a geometrically unitary element in a real JB*-triple is equivalent to the triple being triple-isomorphic to a unital JB-algebra.
• In real non-commutative JB*-algebras, every Jordan ∗-homomorphism is automatically contractive, and every Jordan ∗-monomorphism is an isometry, a property not universally shared in the complex case.
• The Russo–Dye theorem holds in the real setting for real JB*-algebras, but its extension to real C*-algebras requires careful analysis of surjective isometries and their complex affine extensions.