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

Abstract 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 $$X^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> has the WODP, then $$X \widehat{\otimes }_\varepsilon Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>ε</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> has the DD2P for any Banach space Y . If X has the WODP, then $$X \widehat{\otimes }_\pi Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>π</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> has the DD2P for any Banach space Y . If $$X^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> and $$Y^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> have the WODP then $$X \widehat{\otimes }_\varepsilon Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:msub> <mml:mover> <mml:mo>⊗</mml:mo> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>ε</mml:mi> </mml:msub> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> has the Daugavet property. The above improve many results in the literature and establish progress on some open questions.