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The stability of non-commutative \(L^p\)-spaces. (La stabilité des espaces \(L^p\) non-commutatifs.) (French) Zbl 0914.46050
Summary: Let \(M\) be a von Neumann algebra and \(R\) the hyperfinite factor of type \(\text{II}_1\). We show that if \(M\) is not of type I then \(L^p(\mathbb{R})\) is a 1-complemented subspace of \(L^p(M)\) for all \(1\leq p<\infty\). We also show that \(M\) is of type I if and only if \(L^p(M)\) is stable for all \(1\leq p<\infty\).

MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L35 Classifications of \(C^*\)-algebras
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