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}
.