Norm equalities in pre-Hilbert \(C^{*}\)-modules
Let \((E,\langle\cdot,\cdot\rangle)\) be a pre-Hilbert module over a \(C^*\)-algebra \(A\) and \(\theta_{x,y}(z):=x\langle y,z\rangle\) for \(x, y, z \in E\). The author proves that the equality case in the triangle inequality \(\|\theta_{x,x}+\theta_{y,y}\| \leq\|\theta_{x,x}\|+\|\theta_{y,y}\|\) holds if and only if \(\|\langle x,y\rangle\| =\|x\|\,\|y\|\). \textit{Lj. Arambašić} and \textit{R. Rajić} [Linear Algebra Appl. 414, No.~1, 19--28 (2006; Zbl 1100.47006)] proved that, for two elements \(x, y \in E\) such that \(\langle x,y\rangle\) is a nonnull idempotent, \(\|x + y\| = \|x\|+\|y\|\) if and only if \(\|x\|\,\|y\|=1\). The author extends this result to the case when, for some \(\alpha \in\mathbb{C}\), \(\langle x,y\rangle -\alpha \notin \{0,1\}\) is an idempotent in the unital \(C^*\)-algebra \(A\) by showing that \(\|x + y\| = \|x\|+\|y\|\) if and only if \(\alpha=\|x\|\,\|y\|-1 \geq -1/2\). \textit{Z. Sebestyén} and \textit{D. Popovici} [J. Oper. Theory 56, No. 1, 3--15 (2006; Zbl 1122.47037)] proved that the equality case is attained in the triangle inequality for positive nonnull elements in a \(C^*\)-algebra if and only if the norm of their product equals the product of their norms. Using this fact, the author characterizes Pythagoras' equality in pre-Hilbert \(C^*\)-modules.
scientific article; zbMATH DE number 5982422