I.
II.
III.
Hardy classes on non-commutative unit balls
Mihai Popa
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Non-commutative functions
- J. L. Taylor (Adv. in Math. ’72)
- D-V. Voiculescu (Asterisque ’95, also Jpn. J. of Math., ’08, Crelles
’09) (∼fully matricial functions)
- V. Vinnikov, D.S. Kaliuzhnyi-Verbovetskyi, M. P., S. Belinschi (2009 -’13)
- M. Aguiar (2011), M. Anshelevich (2011), B.Solel, P. Muhly (2013)
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
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IV.
V= vector space over C;
`∞
V n×n
X 0
• noncommutative sets: Ω ⊆ Vnc such that X ⊕ Y =
∈ Ωn+m
0 Y
n×n
for all X ∈ Ωn , Y ∈ Ωm , where Ωn = Ω ∩ V
• the non-commutative space over V: Vnc =
n=1
• upper admissible sets: Ω ⊆ Vnc such that for all X ∈ Ωn , Y ∈ Ωm
and all Z ∈ V n×m , there exists λ ∈ C, λ 6= 0, with
X λZ
∈ Ωn+m .
0
Y
Mihai Popa
Hardy classes on non-commutative unit balls
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III.
Examples of upper-admissible sets:
• Ω = Nilp V = the set of nilpotent matrices over V
• If V is a Banach space and Ω is a non-commutative set, open in
the sense that Ωn ⊆ V n×n is open for all n,
then Ω is upper admissible.
• Noncommutative upper/lower half-planes over a C∗ -algebra A:
H+ (Anc ) = {a ∈ Anc : =a > 0}
H− (Anc ) = {a ∈ Anc : =a < 0}
Mihai Popa
Hardy classes on non-commutative unit balls
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III.
Ω ⊆ Vnc = non-commutative (upper admissible) set
Noncommutative function:
f : Ω → Wnc such that
f (Ωn ) ⊆ Mn (W)
f respects direct sums: f (X ⊕ Y ) = f (X) ⊕ f (Y ) for all X ∈ Ωn ,
Y ∈ Ωm .
f respects similarities: f (T XT −1 ) = T f (X)T −1 for all X ∈ Ωn and
T ∈ GLn (C) such that T XT −1 ∈ Ωn .
Equivalently, f respects intertwinings:
if X ∈ Ωn , Y ∈ Ωm , S ∈ Cn×m such that XS = SY , then
f (X)S = Sf (Y )
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Ω ⊆ Vnc = non-commutative (upper admissible) set
Noncommutative function:
f : Ω → Wnc such that
f (Ωn ) ⊆ Mn (W)
f respects direct sums: f (X ⊕ Y ) = f (X) ⊕ f (Y ) for all X ∈ Ωn ,
Y ∈ Ωm .
f respects similarities: f (T XT −1 ) = T f (X)T −1 for all X ∈ Ωn and
T ∈ GLn (C) such that T XT −1 ∈ Ωn .
Equivalently, f respects intertwinings:
if X ∈ Ωn , Y ∈ Ωm , S ∈ Cn×m such that XS = SY , then
f (X)S = Sf (Y )
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Examples of nc-functions:
• non-commutative polynomials
V = Am , W = A
f (X1 , . . . , Xm ) = X1 X3 − X3 X1 + b1 X2 X4 b2 X5
N.B.: A nc polynomial is determined uniquely by this type of nc
function
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Examples of nc-functions:
• free holomorphic functions (G. Popescu)
f (X1 , . . . , Xn ) =
∞
X
X
Ai Xi1 · · · Xim
m=1 i=(i1 ,...,im )
where Xi1 , . . . , Xin are free elements in some operator algebra
• the generalized moment series of X ∈ A
φ : A −→ D cp B-bimodule map
e (1 − X b)−1 = MX = (Mn,X )n , where φ
e = (1n ⊗ φ) is the fully
φ
n
matricial extension of φ.
Mn,X (b) =
∞
X
(1n ⊗ φ)([X · b]k ),
k=0
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
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III.
MX indeed encodes all the moments of X, not only the symmetric
ones, since for


0
b1
0 ... 0
 0
0
b2 . . . 0 


.
.
.
.
.
.
.
.. ... ... 
b=
 ∈ Mn+1 (B)

 0
0
0 . . . bN 
0
0
0 ... 0
we have that

0
 0
n
(1N +1 ⊗ φ) [X · b] ) = 
 ...
0
Mihai Popa
...
...
...
...
0
0
...
0

φ(Xb1 · · · Xbn )

0
.

...
0
Hardy classes on non-commutative unit balls
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Difference-differential calculus
Nc functions admit a nice differential calculus. The
difference-differential operators can be calculated directly by
evaluation on block-triangular matrices.
X Z
f (X) ∆R f (X, Y )(Z)
f(
)=
0 Y
0
f (Y )
The operator Z 7→ ∆R f (X, Y )(Z) is linear and
f (Y ) = f (X) + ∆R (X, Y )(X − Y )
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
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II.
III.
IV.
The Taylor-Taylor expansion:
If f : Ω −→ Wnc is a non-commutative function, Ω=upper-admissible
set, X ∈ Ωn . Then for each N and X ∈ Ωn we have that
f (Y ) =
N
X
k=0
∆kR f (X, . . . , X )(X − Y, . . . , X − Y )
| {z } |
{z
}
k+1 times
k times
+1
+∆N
f (X, . . . , X , Y )(X − Y, . . . , X − Y )
R
|
{z
}
| {z }
N +1 times
N +1 times
XY
Moreover, if 0 ∈ Ω then for X ∈ Ω we have
f (X) =
∞
X
k=0
]
k
∆
R f (0, . . . , 0)(X, . . . , X )
| {z } | {z }
k+1
k
]
k
where ∆
R f (0, . . . , 0) are the fully matricial extension of the multilinear
| {z }
k+1
maps ∆kR f (0, . . . , 0) : V k −→ W.
| {z }
k+1
Mihai Popa
Hardy classes on non-commutative unit balls
I.
II.
III.
IV.
The Taylor-Taylor expansion:
If f : Ω −→ Wnc is a non-commutative function, Ω=upper-admissible
set, X ∈ Ωn . Then for each N and X ∈ Ωn we have that
f (Y ) =
N
X
k=0
∆kR f (X, . . . , X )(X − Y, . . . , X − Y )
| {z } |
{z
}
k+1 times
k times
+1
+∆N
f (X, . . . , X , Y )(X − Y, . . . , X − Y )
R
|
{z
}
| {z }
N +1 times
N +1 times
XY
Moreover, if 0 ∈ Ω then for X ∈ Ω we have
f (X) =
∞
X
k=0
]
k
∆
R f (0, . . . , 0)(X, . . . , X )
| {z } | {z }
k+1
k
]
k
where ∆
R f (0, . . . , 0) are the fully matricial extension of the multilinear
| {z }
k+1
maps ∆kR f (0, . . . , 0) : V k −→ W.
| {z }
k+1
Mihai Popa
Hardy classes on non-commutative unit balls
I.
II.

X

 0

 .

f
 ..

 ..
 .
0

f (X)

 0

 .
=
 ..

 ..
 .
0
Z1
X
..
.
···
0
..
.
..
.
..
.
···
···
..
.
..
.
X
0
III.

0

.. 

. 



0 


Z k 
Y
∆R f (X, X)(Z1 )
f (X)
..
.
···
IV.
···
..
.
..
.
..
.
···
∆kR f (X, . . . , X, Y )(Z1 , . . . , Zk )
···
..
.
f (X)
0
Mihai Popa



∆k−1
R f (X, . . . , X, Y )(Z2 , . . . , Zk )

..
.

.



∆R f (X, Y )(Zk )
f (Y )
Hardy classes on non-commutative unit balls
I.
II.
III.
Examples of nc-functions:
• the generalized Cauchy transform of X: GX
φ : A −→ D cp B-bimodule map
(n)
GX = (GX )n , where
GX : H+ (Mn (B)) 3 b 7→ GX (b) = φ⊗1n [(b−X ⊗1n )−1 ] ∈ H− (Mn (D))
(n)
(n)
• the non-commutative R-transform of X: RX
MX (b) − 1 = Rν (bMν (b))
Applications in Free Probability Theory
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Examples of nc-functions:
• the generalized Cauchy transform of X: GX
φ : A −→ D cp B-bimodule map
(n)
GX = (GX )n , where
GX : H+ (Mn (B)) 3 b 7→ GX (b) = φ⊗1n [(b−X ⊗1n )−1 ] ∈ H− (Mn (D))
(n)
(n)
• the non-commutative R-transform of X: RX
MX (b) − 1 = Rν (bMν (b))
Applications in Free Probability Theory
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Examples of nc-functions:
• the generalized Cauchy transform of X: GX
φ : A −→ D cp B-bimodule map
(n)
GX = (GX )n , where
GX : H+ (Mn (B)) 3 b 7→ GX (b) = φ⊗1n [(b−X ⊗1n )−1 ] ∈ H− (Mn (D))
(n)
(n)
• the non-commutative R-transform of X: RX
MX (b) − 1 = Rν (bMν (b))
Applications in Free Probability Theory
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Framework: finite dimensional vector spaces: (Cm )nc
Operator space structure on Cm : - A collection of norms
k · k = {k · kn on Cn×n } such that
• kX ⊕ Y kn+m ≤ max{kXkn , kY km }
• kT XSkm ≤ kT kkXkn kSk
X ∈ Cn×n , Y ∈ Cm×m , T ∈ Cm×n , S ∈ Cn×m .
We shall be concerned with the following two operator space
structures on Cm :
• kXk(1) = max{kX1 k, . . . , kXm k}
P
1
∗
2
• kXk(2) = k m
i=1 Xi Xi k
for X = (X1 , X2 , . . . , Xm ) ∈ (Cn×n )m
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Framework: finite dimensional vector spaces: (Cm )nc
Operator space structure on Cm : - A collection of norms
k · k = {k · kn on Cn×n } such that
• kX ⊕ Y kn+m ≤ max{kXkn , kY km }
• kT XSkm ≤ kT kkXkn kSk
X ∈ Cn×n , Y ∈ Cm×m , T ∈ Cm×n , S ∈ Cn×m .
We shall be concerned with the following two operator space
structures on Cm :
• kXk(1) = max{kX1 k, . . . , kXm k}
P
1
∗
2
• kXk(2) = k m
i=1 Xi Xi k
for X = (X1 , X2 , . . . , Xm ) ∈ (Cn×n )m
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
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II.
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IV.
• For the norm k · k(1) , the non-commutative unit ball is
∞
a
(Dm )nc =
{(X1 , . . . , Xm ) ∈ (Cn×n )m : kXj k < 1}
n=1
with distinguished boundary
bd(Dm )nc =
∞
a
{(X1 , . . . , Xm ) ∈ (Cn×n )m : Xj∗ Xj = In } = U(n)m
n=1
• For the norm k · k(2) , the non-commutative unit ball is
(Bm )nc =
∞
a
{(X1 , . . . , Xm ) ∈ (Cn×n )m :
n=1
m
X
Xi∗ Xi < In },
i=1
with distinguished boundary
bd(Bm )nc =
∞
a
{(X1 , . . . , Xm ) ∈ (Cn×n )m :
n=1
m
X
Xi∗ Xi = In }
i=1
Mihai Popa
Hardy classes on non-commutative unit balls
I.
II.
III.
IV.
• For the norm k · k(1) , the non-commutative unit ball is
∞
a
(Dm )nc =
{(X1 , . . . , Xm ) ∈ (Cn×n )m : kXj k < 1}
n=1
with distinguished boundary
bd(Dm )nc =
∞
a
{(X1 , . . . , Xm ) ∈ (Cn×n )m : Xj∗ Xj = In } = U(n)m
n=1
• For the norm k · k(2) , the non-commutative unit ball is
(Bm )nc =
∞
a
{(X1 , . . . , Xm ) ∈ (Cn×n )m :
n=1
m
X
Xi∗ Xi < In },
i=1
with distinguished boundary
bd(Bm )nc =
∞
a
{(X1 , . . . , Xm ) ∈ (Cn×n )m :
n=1
m
X
Xi∗ Xi = In }
i=1
Mihai Popa
Hardy classes on non-commutative unit balls
I.
II.
III.
On (bd(Dm )nc )n = U(n)m there is the canonical Haar product
measure.
On (bd(Bm )nc )n ' U (mn)/U((m − 1)n) there exists also a canonical
U(mn)-invariant Radon measure νn of mass 1.
For f ∈ Alg{ui,j , ui,j : 1 ≤ i ≤ n, 1 ≤ j ≤ mn}, the measure νn is
actually easy to describe:
Z
Z
f (X)dνn (X) =
f (U )dUmn (U )
(bd(Bm )nc )n
U (mn)
for dUN the Haar measure on UN .
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
The Hardy H 2 spaces:
H 2 (Ω) ={f : Ω −→ Cnc , nc-function, locally bounded on slices
Z
sup sup
tr(f (rX)∗ f (rX))dωn < ∞}.
n
m
r<1
(bdΩ)n
m
for Ω ∈ {(D )nc , (B )nc }.
The Taylor-Taylor expansion around 0 for functions as above gives
f (X) =
∞
X
X
X w · fw ).
(
l=0 w∈Fm
|w|=l
for Fm the free semigroup in m generators and
w
X w = (X1 , . . . , Xm )(w1 ,...,wl ) = Xiw11 · · · Xil l
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
The Hardy H 2 spaces:
H 2 (Ω) ={f : Ω −→ Cnc , nc-function, locally bounded on slices
Z
sup sup
tr(f (rX)∗ f (rX))dωn < ∞}.
n
m
r<1
(bdΩ)n
m
for Ω ∈ {(D )nc , (B )nc }.
The Taylor-Taylor expansion around 0 for functions as above gives
f (X) =
∞
X
X
(
X w · fw ).
l=0 w∈Fm
|w|=l
for Fm the free semigroup in m generators and
w
X w = (X1 , . . . , Xm )(w1 ,...,wl ) = Xiw11 · · · Xil l
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• B.Collins 2003, B.Collins, P. Sniady 2006:
Integration theory on U(n) for functions generated by
{ui,j , ui,j : 1 ≤ i, j ≤ n};
dependent on the difficult to handle “ Weingarten function”
• Free Probabilities results:
Haar unitaries with independent entries and constant matrices
with limit distribution form an asympotically free family wrt
Z
tr(·)dUn .
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• B.Collins 2003, B.Collins, P. Sniady 2006:
Integration theory on U(n) for functions generated by
{ui,j , ui,j : 1 ≤ i, j ≤ n};
dependent on the difficult to handle “ Weingarten function”
• Free Probabilities results:
Haar unitaries with independent entries and constant matrices
with limit distribution form an asympotically free family wrt
Z
tr(·)dUn .
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• H 2 (Ω) are inner-product spaces with the inner product
Z
hf, gi = lim
lim
tr (g(rX)∗ f (rX)) dωN
N −→∞ r−→1−
(bdΩ)n
N.B.: the limit is not the supremum. For m = 2 and
f (X) = X1 X2 + X2 X1 ,
Z
1
(f (rX)∗ f (rX)) dωn = 2r2 (1 + 2 )
n
m
(bdDnc )n
• For each n, the boundary values f (X) = lim exist a. e.
r−→1−
The limit over r in the formula above con be replaced either by
the sup or the integral of the boundary value.
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• {X w }w∈Fm is a complete orthonormal system in H 2 ((Dm )nc );
X
moreover fw = hf, X w i and f =
fw X w in H 2 ((Dm )nc ).
w∈Fm
|w|
2
X w }w∈Fm is a complete orthonormal system in H 2 ((Bm )nc ):
X
moreover, fw = hf, m|w| X w i and f =
fw X w in H 2 ((Bm )nc ).
• {m
w∈Fm
The spaces H 2 ((Dm )nc ) and H 2 ((Bm )nc ) are not isomorphic to some
weighted l2 spaces on Fm . In fact, they are not complete.
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• {X w }w∈Fm is a complete orthonormal system in H 2 ((Dm )nc );
X
moreover fw = hf, X w i and f =
fw X w in H 2 ((Dm )nc ).
w∈Fm
|w|
2
X w }w∈Fm is a complete orthonormal system in H 2 ((Bm )nc ):
X
moreover, fw = hf, m|w| X w i and f =
fw X w in H 2 ((Bm )nc ).
• {m
w∈Fm
The spaces H 2 ((Dm )nc ) and H 2 ((Bm )nc ) are not isomorphic to some
weighted l2 spaces on Fm . In fact, they are not complete.
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• {X w }w∈Fm is a complete orthonormal system in H 2 ((Dm )nc );
X
moreover fw = hf, X w i and f =
fw X w in H 2 ((Dm )nc ).
w∈Fm
|w|
2
X w }w∈Fm is a complete orthonormal system in H 2 ((Bm )nc ):
X
moreover, fw = hf, m|w| X w i and f =
fw X w in H 2 ((Bm )nc ).
• {m
w∈Fm
The spaces H 2 ((Dm )nc ) and H 2 ((Bm )nc ) are not isomorphic to some
weighted l2 spaces on Fm . In fact, they are not complete.
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Consider the weighted l2 spaces:
X
l(2Dm )nc = {(αw )w∈Fm :
|α|2 < ∞}
w∈Fm
l(2Bm )nc = {(αw )w∈Fm :
X
w∈Fm
m
1
|α|2 < ∞}
m|w|
m
For Ω ∈ {(D )nc , (B )nc }, define
Ωbd =
∞
a
{X ∈ (Cm )n×n : {X w }w∈Fw ∈ l2 (Ω) ⊗ Cn×n }
n=1
•
1 m
(D )nc ( (Dm )nc, bd 6= (Dm )nc
m
• m(Dm )nc ( (Bm )nc, bd 6= (Bm )nc
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Consider the weighted l2 spaces:
X
l(2Dm )nc = {(αw )w∈Fm :
|α|2 < ∞}
w∈Fm
l(2Bm )nc = {(αw )w∈Fm :
X
w∈Fm
m
1
|α|2 < ∞}
m|w|
m
For Ω ∈ {(D )nc , (B )nc }, define
Ωbd =
∞
a
{X ∈ (Cm )n×n : {X w }w∈Fw ∈ l2 (Ω) ⊗ Cn×n }
n=1
•
1 m
(D )nc ( (Dm )nc, bd 6= (Dm )nc
m
• m(Dm )nc ( (Bm )nc, bd 6= (Bm )nc
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• The completions H 2 (Ω) of H 2 (Ω) can be identified with spaces of
nc functions on Ωbd :
H 2 (Ω) ={f : Ωbd −→ Cnc : f = nc function with T-T expansion at 0
X
f (X) =
fw X w
w∈Fm
2
for some {fw }w∈Fm ∈ lΩ
}
• Ωn ∩ (Ωbd )n consists of all X ∈ Ωn such that the evaluation
mapping
H 2 (Ω) 3 f 7→ f (X) ∈ Cn×n
is bounded
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• The completions H 2 (Ω) of H 2 (Ω) can be identified with spaces of
nc functions on Ωbd :
H 2 (Ω) ={f : Ωbd −→ Cnc : f = nc function with T-T expansion at 0
X
f (X) =
fw X w
w∈Fm
2
for some {fw }w∈Fm ∈ lΩ
}
• Ωn ∩ (Ωbd )n consists of all X ∈ Ωn such that the evaluation
mapping
H 2 (Ω) 3 f 7→ f (X) ∈ Cn×n
is bounded
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
• H 2 (Ω) are reproducing kernel Hilbert spaces, with the completely
positive non-commutative kernels
K(Dm )nc (X, Y ) =
∞ X
X
[
X w ⊗ (Y w )∗ ]
l=0 |w|=l
K(Bm )nc (X, Y ) =
∞
X
1 X w
[
X ⊗ (Y w )∗ ]
ml
l=0
Mihai Popa
|w|=l
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
H ∞ (Ω) = {f : Ω −→ Cnc : f nc function, sup kf (X)k ≤ ∞}
X∈Ω
∞
e = {f : Ω −→ Cnc : f nc function,
H (Ω)
sup
kf (X)k ≤ ∞}
X∈Ω∩Ωbd
M(Ω) = {f : Ω −→ Cnc : f nc function and a bdd multiplier for H 2 (Ω)}
e
H ∞ (Ω) ( M(Ω) ( H ∞ (Ω)
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
H ∞ (Ω) = {f : Ω −→ Cnc : f nc function, sup kf (X)k ≤ ∞}
X∈Ω
∞
e = {f : Ω −→ Cnc : f nc function,
H (Ω)
sup
kf (X)k ≤ ∞}
X∈Ω∩Ωbd
M(Ω) = {f : Ω −→ Cnc : f nc function and a bdd multiplier for H 2 (Ω)}
e
H ∞ (Ω) ( M(Ω) ( H ∞ (Ω)
Mihai Popa
Hardy classes on non-commutative unit balls
IV.
I.
II.
III.
Wg :Z+ ×
∞
[
Sn −→ C
n=1
Z
Wg(N, π) =
u1,1 · · · un,n u1,π(1) · · · un,π(n) dUN (U )
U (N )
Mihai Popa
Hardy classes on non-commutative unit balls
IV.