P. Jones’ interpolation theorem for noncommutative martingale Hardy spaces

Source abstract

Let M \mathcal {M} be a semifinite von Nemann algebra equipped with an increasing filtration ( M n ) n ≥ 1 (\mathcal {M}_n)_{n\geq 1} of (semifinite) von Neumann subalgebras of M \mathcal {M} . For 0 > p ≤ ∞ 0>p \leq \infty , let h p c ( M ) \mathsf {h}_p^c(\mathcal {M}) denote the noncommutative column conditioned martingale Hardy space associated with the filtration ( M n ) n ≥ 1 (\mathcal {M}_n)_{n\geq 1} and the index p p . We prove that for 0 > p > ∞ 0>p>\infty , the compatible couple ( h p c ( M ) , h ∞ c ( M ) ) \big (\mathsf {h}_p^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big ) is K K -closed in the couple ( L p ( N ) , L ∞ ( N ) ) \big (L_p(\mathcal {N}), L_\infty (\mathcal {N}) \big ) for an appropriate amplified semifinite von Neumann algebra N ⊃ M \mathcal {N}\supset \mathcal {M} . This may be viewed as a noncommutative analogue of P. Jones interpolation of the couple ( H 1 , H ∞ ) (H_1, H_\infty ) . As an application, we prove a general automatic transfer of real interpolation results from couples of symmetric quasi-Banach function spaces to the corresponding couples of noncommutative conditioned martingale Hardy spaces. More precisely, assume that E E is a symmetric quasi-Banach function space on ( 0 , ∞ ) (0, \infty ) satisfying some natural conditions, 0 > θ > 1 0>\theta >1 , and 0 > r ≤ ∞ 0>r\leq \infty . If ( E , L ∞ ) θ , r = F (E,L_\infty )_{\theta ,r}=F , then $$( h E c ( M ) , h ∞ c ( M ) ) θ , r = h F c ( M ) . \big (\mathsf {h}_E^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )_{\theta , r}=\mathsf {h}_{F}^c(\mathcal {M}).$$ As an illustration, we obtain that if Φ \Phi is an Orlicz function that is p p -convex and q q -concave for some 0 > p ≤ q > ∞ 0>p\leq q>\infty , then the following interpolation on the noncommutative column Orlicz-Hardy space holds: for 0 > θ > 1 0>\theta >1 , 0 > r ≤ ∞ 0>r\leq \infty , and Φ 0 − 1 ( t ) = [ Φ − 1 ( t ) ] 1 − θ \Phi _0^{-1}(t)=[\Phi ^{-1}(t)]^{1-\theta } for t > 0 t>0 , $$( h Φ c ( M ) , h ∞ c ( M ) ) θ , r = h Φ 0 , r c ( M ) , \big (\mathsf {h}_\Phi ^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )_{\theta , r}=\mathsf {h}_{\Phi _0, r}^c(\mathcal {M}),$$ where h Φ 0 , r c ( M ) \mathsf {h}_{\Phi _0,r}^c(\mathcal {M}) is the noncommutative column Hardy space associated with the Orlicz-Lorentz space L Φ 0 , r L_{\Phi _0,r} .