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Automatic continuity of $$M$$-norms on $$C^{*}$$-algebras. (English) Zbl 1225.46043
Summary: Elements $$a$$ and $$b$$ of a $$C^*$$-algebra are called orthogonal $$(a\perp b)$$ if $$a^*b=ab^*=0$$. We say that vectors $$x$$ and $$y$$ in a Banach space $$X$$ are semi-$$M$$-orthogonal $$(x\perp_{SM}y)$$ if $$\|x\pm y\|\geq \max\{\|x\|,\|y\|\}$$. We prove that every linear bijection $$T:A\to X$$, where $$X$$ is a Banach space, $$A$$ is either a von Neumann algebra or a compact $$C^*$$-algebra, and $$T(a)\perp_{SM}T(b)$$ whenever $$a\perp b$$, must be continuous. Consequently, every complete (semi-)$$M$$-norm on a von Neumann algebra or on a compact $$C^*$$-algebra is automatically continuous.

##### MSC:
 46L05 General theory of $$C^*$$-algebras 47B48 Linear operators on Banach algebras
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