A note on the triangle inequality for the \(C^{\ast}\)-valued norm on a Hilbert \(C^{\ast}\)-module (Q2853308)

Let \({\mathcal X}\) be a Hilbert \(C^*\)-module over a \(C^*\)-algebra \(\mathcal{A}\). For every \(x \in {\mathcal X}\), the absolute value of \(x\) is defined as the unique positive square root of \(\langle x,x \rangle \in \mathcal{A}\). In general, the triangle inequality \(|x+y| \leq |x|+|y|\) need not hold. \textit{Lj. Arambašić} and \textit{R. Rajić} [Acta Math. Hung. 119, No. 4, 373--380 (2008; Zbl 1174.47012)] proved that, if \(\mathcal{A}\) is a \(C^*\)-algebra with a unit \(e\), then for every \(x, y \in \mathcal{X}\) and \(\varepsilon > 0\), there are \(a, b \) in the closed unit ball of \(\mathcal{A}\) such that \(|x+y| \leq a|x|a^*+b|y|b^*+\varepsilon e\). \textit{B. Kolarec} [Math. Inequal. Appl. 12, No. 4, 745--751 (2009; Zbl 1201.46051)] proved that, if \(x, y \in{\mathcal X}\) and \(|x| , |y|\) are in the center of \(\mathcal{A}\), then \(|x+y| \leq |x|+|y|\) is valid. In the paper under review, the author proves that \(|x+y| \leq |x|+|y|\) holds for all \(x, y \in \mathcal{X}\) if and only if the closed linear span of \(\{\langle x,y\rangle: x, y \in \mathcal{X}\}\) is a commutative \(C^*\)-subalgebra of \(\mathcal{A}\).