Isometric reflections in two dimensions and dual $\mathsf{L}^1$-structures

Francisco J. Garc\'{i}a-Pacheco

doi:10.4134/BKMS.2012.49.6.1275

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2012; 49(6): 1275-1289

Printed November 30, 2012

https://doi.org/10.4134/BKMS.2012.49.6.1275

Isometric reflections in two dimensions and dual $\mathsf{L}^1$-structures

Francisco J. Garc\'{i}a-Pacheco

Texas A\&M University

Abstract

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Abstract

In this manuscript we solve in the positive a question informally proposed by Enflo on the measure of the set of isometric reflection vectors in non-Hilbert $2$-dimensional real Banach spaces. We also reformulate equivalently the separable quotient problem in terms of isometric reflection vectors. Finally, we give a new and easy example of a real Banach space whose dual has a non-trivial $\mathsf{L}$-summand that does not come from an $M$-ideal in the predual.

Keywords: isometric reflection vector, $\mathsf{L}^1$-summand vector

MSC numbers: Primary 46B20, 46C05, 46B04