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 $L_1(\mathcal{M},\tau ){\hat{\otimes }}_{\pi }Y$ 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.