The Operator Daugavet Property in Semifinite Noncommutative $L_1$-Spaces

Abstract

Let $\mathcal M$ be a diffuse semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$. We prove that, for every nonzero Banach space $Y$, the projective tensor product $L_1(\mathcal M,\tau)\widehat{\otimes}_{\pi}Y$ has the operator Daugavet property. Moreover, the witnessing operators may always be chosen contractive. This extends the operator Daugavet phenomenon from atomless vector-valued $L_1$-spaces to the semifinite noncommutative setting and yields further Daugavet-type consequences for projective symmetric tensor products, all without approximation assumptions.