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On complex structures and uniqueness of algebra norms in Banach spaces

W. Cuellar Carrera, V. Ferenczi

TL;DR

The paper investigates the algebra $L(X)/S(X)$ for infinite-dimensional Banach spaces, focusing on quantitative singularity notions and the interplay with complex structures, twisted sums, and symplectic forms. It extends Ferenczi–Galego's results, proves a universal lower bound $\,\|I_{|H}-J\|_S \ge 2$ for complex structures on a space and its hyperplane, and analyzes how near-Id perturbations of complex structures force equivalences. A central feat is the Kalton–Peck space $Z_2$: $L(Z_2)/S(Z_2)$ is not complete for $\|\cdot\|_S$, not *-isomorphic to a $C^*$-algebra under the quotient norm, and admits two inequivalent *-algebra norms; the work also builds a *-isometric representation $\\Lambda$ of $L(Z_2)/S(Z_2)$ into a larger operator algebra, and shows this image is not closed. Collectively, these results produce new inequivalent algebra norms on natural operator-quotients and illuminate the structure of twisted Hilbert spaces via interpolation and symplectic geometry, with implications for the existence and behavior of complex structures on spaces and their hyperplanes.

Abstract

For $X$ an infinite dimensional Banach space, we contribute to the study of the Banach algebra $L(X)/S(X)$, where $S(X)$ is the ideal of strictly singular operators. We extend results of Ferenczi-Galego (2007) by proving that $\|I-J\|_S \geq 2$, whenever $I$ is a complex structure on a real space $X$ and $J$ extends a complex structure on a hyperplane of $X$, and where $\|.\|_S$ denotes a certain algebra norm on $L(X)/S(X)$ dominated by the usual quotient norm $\|.\|$. We solve two questions of Kalton-Swanson (1982) by proving that if $X=Z_2$ the Kalton-Peck space, then $L(Z_2)/S(Z_2)$ a) is not complete for $\|.\|_S$ and b) that it is not *-isomorphic to a $C^*$-algebra for $\|.\|$. In particular $L(Z_2)/S(Z_2)$ admits two inequivalent *-algebra norms.

On complex structures and uniqueness of algebra norms in Banach spaces

TL;DR

The paper investigates the algebra for infinite-dimensional Banach spaces, focusing on quantitative singularity notions and the interplay with complex structures, twisted sums, and symplectic forms. It extends Ferenczi–Galego's results, proves a universal lower bound for complex structures on a space and its hyperplane, and analyzes how near-Id perturbations of complex structures force equivalences. A central feat is the Kalton–Peck space : is not complete for , not *-isomorphic to a -algebra under the quotient norm, and admits two inequivalent *-algebra norms; the work also builds a *-isometric representation of into a larger operator algebra, and shows this image is not closed. Collectively, these results produce new inequivalent algebra norms on natural operator-quotients and illuminate the structure of twisted Hilbert spaces via interpolation and symplectic geometry, with implications for the existence and behavior of complex structures on spaces and their hyperplanes.

Abstract

For an infinite dimensional Banach space, we contribute to the study of the Banach algebra , where is the ideal of strictly singular operators. We extend results of Ferenczi-Galego (2007) by proving that , whenever is a complex structure on a real space and extends a complex structure on a hyperplane of , and where denotes a certain algebra norm on dominated by the usual quotient norm . We solve two questions of Kalton-Swanson (1982) by proving that if the Kalton-Peck space, then a) is not complete for and b) that it is not *-isomorphic to a -algebra for . In particular admits two inequivalent *-algebra norms.
Paper Structure (23 sections, 58 theorems, 79 equations)

This paper contains 23 sections, 58 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Operator Ideals
  3. Twisted sums
  4. Complex structures
  5. Uniqueness of algebra norms
  6. Organization of the paper
  7. Quantitative notions of singularity for operators
  8. Notions of singularity
  9. Fredholm and inessential operators
  10. Complex structures and ideals
  11. Perturbations of complex structures
  12. Operators whose square are close to $-\operatorname{Id}$
  13. Complex structures and hyperplanes
  14. Quantified versions of singularity for quasi-linear maps
  15. Symplectic structures and twisted Hilbert spaces induced by complex interpolation
  16. ...and 8 more sections

Key Result

Theorem 1.6

If $X$ is a real Banach space, $I$ a complex structure on $X$ and $J$ a complex structure on its hyperplane $H$, then $\|I_{|H}-J\|_S \geq 2$.

Theorems & Definitions (123)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 113 more