We obtain new progress about the diameter two property and the Daugavet property in tensor product spaces. Namely, the main results of the paper are:<UnorderedList Mark="Bullet"> <ItemContent> If <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> has the WODP, then <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X \widehat{\otimes }_\varepsilon Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <msub> <mover accent="true"> <mo>⊗</mo> <mo stretchy="true">^</mo> </mover> <mi>ε</mi> </msub> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> has the DD2P for any Banach space Y. </ItemContent> <ItemContent> If X has the WODP, then <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X \widehat{\otimes }_\pi Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <msub> <mover accent="true"> <mo>⊗</mo> <mo stretchy="true">^</mo> </mover> <mi>π</mi> </msub> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> has the DD2P for any Banach space Y. </ItemContent> <ItemContent> If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Y^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>Y</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> have the WODP then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X \widehat{\otimes }_\varepsilon Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <msub> <mover accent="true"> <mo>⊗</mo> <mo stretchy="true">^</mo> </mover> <mi>ε</mi> </msub> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> has the Daugavet property. </ItemContent> </UnorderedList> The above improve many results in the literature and establish progress on some open questions.

Weak operator Daugavet property and weakly open sets in tensor product spaces

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We obtain new progress about the diameter two property and the Daugavet property in tensor product spaces. Namely, the main results of the paper are:

If $X^*$ $X^{*}$ has the WODP, then $X \widehat{\otimes }_\varepsilon Y$ $X {\hat{\otimes}}_{ε} Y$ has the DD2P for any Banach space Y.

If X has the WODP, then $X \widehat{\otimes }_\pi Y$ $X {\hat{\otimes}}_{π} Y$ has the DD2P for any Banach space Y.

If $X^*$ $X^{*}$ and $Y^*$ $Y^{*}$ have the WODP then $X \widehat{\otimes }_\varepsilon Y$ $X {\hat{\otimes}}_{ε} Y$ has the Daugavet property.

The above improve many results in the literature and establish progress on some open questions.