We study the Daugavet property in tensor products of Banach spaces. We show that $L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$ has the Daugavet property when $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$ has the Daugavet property provided $X$ and $Y$ are $L_{1}$-preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.

中文翻译：

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张量积空间中的 DAUGAVET 属性

我们研究了 Banach 空间张量积中的 Daugavet 属性。我们表明$L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$有 Daugavet 属性时$\unicode[STIX]{x1D707}$和$\unicode[STIX]{x1D708}$是纯粹的非原子措施。此外，我们表明$X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$是否提供了 Daugavet 财产$X$和$Y$是$L_{1}$- 具有 Daugavet 属性的预对变量，特别是具有此属性的连续函数空间。使用相同的技术，我们还获得了关于射影张量积的粗糙度以及射影对称张量积的道加维性质的结果。