Projective tensor products where every element is norm-attaining

Abstract In this paper we analyse when every element of $$X{\widehat{\otimes }}_\pi Y$$ <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> attains its projective norm. We prove that this is the case if X is the dual of a subspace of a predual of an $$\ell _1(I)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> space and Y is 1-complemented in its bidual under approximation property assumptions. This result allows us to provide some new examples where X is a Lipschitz-free space. We also prove that the set of norm-attaining elements is dense in $$X{\widehat{\otimes }}_\pi Y$$ <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> if, for instance, $$X=L_1(\mu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and Y is any Banach space, or if X has the metric $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> -property and Y is a dual space with the RNP.