Orthogonality in Banach Spaces via projective tensor product

TL;DR

This paper characterizes Birkhoff-James orthogonality in Banach spaces using semi-inner products and projective tensor products, providing new insights into the structure of orthogonality in functional analysis.

Contribution

It establishes a novel characterization of orthogonality in Banach spaces through semi-inner products and tensor product representations, extending to $C^*$-algebras.

Findings

Orthogonality characterized by semi-inner products.

Representation of bilinear maps via projective tensor products.

Extension of results to $C^*$-algebras.

Abstract

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+\lambda y\|_{X} \geq \|x\|_{X}$ for all $\lambda \in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $\varphi$ on $X$ such that $\|\varphi\| = 1$, $\varphi(x,x)=\|x\|^2$ and $\varphi(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.