On the \(C^*\)-valued triangle equality and inequality in Hilbert \(C^*\)-modules. (English) Zbl 1174.47012
In the paper, it is proved that if \(\mathcal{A}\) is a \(C^{\ast }\)-algebra and \(a,b\in \mathcal{A}\), then \(\left| a+b\right| =\left| a\right| +\left| b\right| \) if and only if \(a^{\ast }b=\left| a\right| \left| b\right| \). From this result, it is deduced that if \(V\) is a Hilbert \(C^{\ast }\)-module over a \(C^{\ast }\)-algebra and \(x,y\in V\), then \(\left| x+y\right| =\left| x\right| +\left| y\right| \) if and only if \(\left\langle x,y\right\rangle =\left| x\right| \left| y\right| \). The proof uses the result proved by T. Ando and T. Hayashi [J. Oper. Theory 58, No. 1, 463–468 (2007; Zbl 1150.47001)], in the case \(\mathcal{A}=\mathbb {B}\left( H\right) \). Related questions are also investigated.
Reviewer: Dumitru Popa (Constanta)
Keywords:
\(C^*\)-algebra; Hilbert \(C^*\)-module; linking algebra; \(C^*\)-valued triangle equality; \(C^*\)-valued triangle inequalityCitations:
Zbl 1150.47001References:
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