Orthogonality in Banach Spaces via projective tensor product

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.