Embeddings of symmetric operator spaces into Lp-spaces, 1 ≤ p < 2, on finite von Neumann algebras

Abstract Let E (0, ∞) be a symmetric function space on (0, ∞) such that the set E (0, ∞) ∩ L ∞ (0, ∞) is distinct from the set L p (0, ∞)∩ L ∞ (0, ∞), 1 ≤ p &lt; 2, and let $$E(\cal M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:math> be the corresponding symmetric operator space associated with an atomless semifinite σ -finite von Neumann algebra $$\cal{M}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> equipped with a semifinite infinite faithful normal trace τ . We show that there exists a noncommutative probability space $$(\cal{N},\sigma)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:math> such that E (0, ∞) embeds into $$L_{p}(\cal{N})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:math> if and only if there exists a noncommutative probability space $$(\hat{\cal{N}},\hat{\sigma})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow> <mml:mover> <mml:mi>σ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>)</mml:mo> </mml:math> such that $$E(\cal{M})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:math> embeds into $$L_{p}(\hat{\cal{N}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo>(</mml:mo> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>)</mml:mo> </mml:math> . We also establish a discrete version of this result for symmetric sequence space ℓ E . These extend and complement earlier results in [37,40,41,55].