Isomorphisms of non-commutative domain algebras
A. Arias and F. Latremoliere
University of Denver
October 15, 2011
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
1 / 19
Isomorphisms of Domain algebras
1
[AL1] Isomorphisms of non-commutative domain algebras, JOT (to
appear)
2
[AL2] Isomorphisms of non-commutative domain algebras II
(submitted)
Combines ideas from operator algebras and several complex variables
Non-commutative Domain Algebras: Multivariate, non commutative
operator algebras (Popescu, 2007)
Several complex variables
Cartan’s Lemma, 1932
Thullen’s characterization of bounded Reinhardt domains in C2 with
non-compact automorphism group, 1931
Sunada’s theorem, 1978
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
2 / 19
Setting: the Full Fock space
H is a Hilbert space
F 2 (H ) = C
H
H
2
H
3
.
If H is n - dimensional, then F 2 (H ) = `2 (Fn+ )
Fn+ is the free semigroup with n generators: g1 , g2 , . . . , gn .
`2 (Fn+ ) has orthonormal basis fδα : α 2 Fn+ g .
Left Creation Operators: isometries with orthogonal ranges
S1 , S2 , . . . , Sn : `2 Fn+ ! `2 Fn+ ,
Alvaro Arias (Institute)
Non-commutative domains
Si ( δ α ) = δ g i α .
October 15, 2011
3 / 19
Setting: the Full Fock space
H is a Hilbert space
F 2 (H ) = C
H
H
2
H
3
.
If H is n - dimensional, then F 2 (H ) = `2 (Fn+ )
Fn+ is the free semigroup with n generators: g1 , g2 , . . . , gn .
`2 (Fn+ ) has orthonormal basis fδα : α 2 Fn+ g .
Left Creation Operators: isometries with orthogonal ranges
S1 , S2 , . . . , Sn : `2 Fn+ ! `2 Fn+ ,
Si ( δ α ) = δ g i α .
We use Fn+ to represent arbitrary products: X1 , X2 , . . . , Xn are
variables and α = gi1 gi2
gik 2 Fn+ then
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
3 / 19
Setting: the Full Fock space
H is a Hilbert space
F 2 (H ) = C
H
H
2
H
3
.
If H is n - dimensional, then F 2 (H ) = `2 (Fn+ )
Fn+ is the free semigroup with n generators: g1 , g2 , . . . , gn .
`2 (Fn+ ) has orthonormal basis fδα : α 2 Fn+ g .
Left Creation Operators: isometries with orthogonal ranges
S1 , S2 , . . . , Sn : `2 Fn+ ! `2 Fn+ ,
Si ( δ α ) = δ g i α .
We use Fn+ to represent arbitrary products: X1 , X2 , . . . , Xn are
variables and α = gi1 gi2
gik 2 Fn+ then
Xα = Xi1 Xi2 . . . Xik .
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
3 / 19
Motivating example: non-commutative disc algebras
An is the norm closed algebra generated by S1 , . . . , Sn and the identity.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
4 / 19
Motivating example: non-commutative disc algebras
An is the norm closed algebra generated by S1 , . . . , Sn and the identity.
Model for row contractions (Popescu, 1991, VN Inequality)
If T1 , . . . , Tn 2 B (H ) and ∑i n Ti Ti
I , then there exists a unital
completely contractive homomorphism Φ : An ! B (H ) satisfying
Φ (Si ) = Ti for i
Alvaro Arias (Institute)
Non-commutative domains
n.
October 15, 2011
4 / 19
Motivating example: non-commutative disc algebras
An is the norm closed algebra generated by S1 , . . . , Sn and the identity.
Model for row contractions (Popescu, 1991, VN Inequality)
If T1 , . . . , Tn 2 B (H ) and ∑i n Ti Ti
I , then there exists a unital
completely contractive homomorphism Φ : An ! B (H ) satisfying
Φ (Si ) = Ti for i
n.
A simple proof for this uses Poisson Transforms, an explicit dilation.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
4 / 19
Completely contractive representations
The unital completely contractive representations of An on B (H ) are
given by
(
)
(T1 , . . . , Tn ) : Ti 2 B (H ) and
∑ Ti Ti
I.
i n
When dim (H ) = 1, we obtain the characters
(
r
Bn = (λ1 , . . . , λn ) 2 Cn : kλk2 = ∑ jλi j2
i n
Alvaro Arias (Institute)
Non-commutative domains
)
1 .
October 15, 2011
5 / 19
Completely contractive representations
The unital completely contractive representations of An on B (H ) are
given by
(
)
(T1 , . . . , Tn ) : Ti 2 B (H ) and
∑ Ti Ti
I.
i n
When dim (H ) = 1, we obtain the characters
(
r
Bn = (λ1 , . . . , λn ) 2 Cn : kλk2 = ∑ jλi j2
i n
)
1 .
The techniques and methods used to study An are associated with
multivariable interpolation results (Caratheodory, Nevanlinna-Pick)
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Non-commutative domains
October 15, 2011
5 / 19
A not pretty generalization: Arias and Popescu, 1999
Weighted shifts
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Non-commutative domains
October 15, 2011
6 / 19
A not pretty generalization: Arias and Popescu, 1999
Weighted shifts
For i n, de…ne Wi : `2 (Fn+ ) ! `2 (Fn+ ) by Wi δα =
where
ω α > 0 for all α 2 Fn+ and
jαj = jγj = 1.
ω αβγ
ω βγ
ω αβ
ωβ
q
ω βα
ω α δg i α ,
for any α, β, γ 2 Fn+ with
The weights are motivated by a paper of Quiggin (interpolation).
Then there exist aα ’s satisfying
agi > 0 for i n,
aα 0 for jαj 1 such that
I , and (W1 , . . . , Wn ) is the model for operators
∑ j α j 1 aα Wα Wα
satisfying ∑jαj 1 aα Tα Tα
I
The Poisson transform works!
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
6 / 19
A much better generalization: Popescu, 2007
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
7 / 19
A much better generalization: Popescu, 2007
If the aα ’s satisfy
then
Alvaro Arias (Institute)
8
a > 0 for i n,
>
< gi
aα 0 for jαj 1, and
1
>
n
: sup
2
a
= M < ∞,
∑jαj=k α
k 2N
Non-commutative domains
October 15, 2011
7 / 19
A much better generalization: Popescu, 2007
If the aα ’s satisfy
then
8
a > 0 for i n,
>
< gi
aα 0 for jαj 1, and
1
>
n
: sup
2
a
= M < ∞,
∑jαj=k α
k 2N
There exist weighted shifts W1f , W2f , . . . , Wnf on the Full Fock space
`2 (Fn+ ) that are the model for operator satisfying
∑
aα Tα Tα
I.
jαj 1
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
7 / 19
A much better generalization: Popescu, 2007
If the aα ’s satisfy
then
8
a > 0 for i n,
>
< gi
aα 0 for jαj 1, and
1
>
n
: sup
2
a
= M < ∞,
∑jαj=k α
k 2N
There exist weighted shifts W1f , W2f , . . . , Wnf on the Full Fock space
`2 (Fn+ ) that are the model for operator satisfying
∑
aα Tα Tα
I.
jαj 1
The symbol f = ∑jαj 1 aα Xα is used to identify the operators.
X1 , . . . , Xn are free variables and f is called positive regular n-free
formal power series.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
7 / 19
Completely contractive representations
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
8 / 19
Completely contractive representations
A (Df ) is the norm closure of the algebra generated by
W1f , W2f , . . . , Wnf and the identity.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
8 / 19
Completely contractive representations
A (Df ) is the norm closure of the algebra generated by
W1f , W2f , . . . , Wnf and the identity.
The unital completely contractive representations of A (Df ) on the
Hilbert space H are given by the non-commutative domains
(
)
Df (H ) =
Alvaro Arias (Institute)
(T1 , . . . , Tn ) : Ti 2 B (H ) and
Non-commutative domains
∑
aα Tα Tα
I.
jαj 1
October 15, 2011
8 / 19
Completely contractive representations
A (Df ) is the norm closure of the algebra generated by
W1f , W2f , . . . , Wnf and the identity.
The unital completely contractive representations of A (Df ) on the
Hilbert space H are given by the non-commutative domains
(
)
Df (H ) =
(T1 , . . . , Tn ) : Ti 2 B (H ) and
∑
aα Tα Tα
I.
jαj 1
When dim (H ) = 1, we obtain the characters, which are given by the
bounded domain:
(
)
Df (C) =
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(λ1 , . . . , λn ) 2 Cn :
Non-commutative domains
∑
jαj 1
aα j λ α j2
1 .
October 15, 2011
8 / 19
Example
f = X1 + X2 . Then W1f = S1 and W2f = S2
I
fS1 , S2 g is the model for row contractions T1 T1 + T2 T2
The set of 1-dimensional representations (characters) of A (Df ) is
o
n
Df (C) = (λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 1 .
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Non-commutative domains
October 15, 2011
9 / 19
Example
g = X1 + X2 + X1 X2 . Then we obtain A (Dg )
fW1g , W2g g is the model for operators
T1 T1 + T2 T2 + T1 T2 T2 T1
I.
n
Dg (C) = (λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 + jλ1 λ2 j2
Alvaro Arias (Institute)
Non-commutative domains
o
1 .
October 15, 2011
10 / 19
Example
g = X1 + X2 + X1 X2 . Then we obtain A (Dg )
fW1g , W2g g is the model for operators
T1 T1 + T2 T2 + T1 T2 T2 T1
I.
n
Dg (C) = (λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 + jλ1 λ2 j2
o
1 .
Example
h = X1 + X2 + 12 X1 X2 + 12 X2 X1 . Then we obtain A (Dh )
W1h , W2h is the model for operators
T1 T1 + T2 T2 + 12 T1 T2 T2 T1 + 12 T2 T1 T1 T2
D
n h (C) =
I.
o
2 : λ 2+ λ 2+ 1 λ λ 2+ 1 λ λ 2
λ
,
λ
2
C
1
( 1 2)
j 1j
j 2j
2 j 1 2j
2 j 2 1j
o
n
D h ( C ) = ( λ 1 , λ 2 ) 2 C2 : j λ 1 j 2 + j λ 2 j 2 + j λ 1 λ 2 j 2 1 = D g ( C )
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
10 / 19
When are the non commutative domain algebras
isomorphic?
Isomorphism means completely isometric homomorphism.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
11 / 19
When are the non commutative domain algebras
isomorphic?
Isomorphism means completely isometric homomorphism.
Suppose that Φ : A (Df ) ! A (Dg ) is an isomorphism.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
11 / 19
When are the non commutative domain algebras
isomorphic?
Isomorphism means completely isometric homomorphism.
Suppose that Φ : A (Df ) ! A (Dg ) is an isomorphism.
For k
1, Dg Ck
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bk
Φ
! Df Ck
2
Cnk is biholomorphic.
Non-commutative domains
October 15, 2011
11 / 19
When are the non commutative domain algebras
isomorphic?
Isomorphism means completely isometric homomorphism.
Suppose that Φ : A (Df ) ! A (Dg ) is an isomorphism.
1, Dg Ck
bk
Φ
! Df Ck
Biholomorphic maps are “rigid”
For k
Alvaro Arias (Institute)
2
Cnk is biholomorphic.
Non-commutative domains
October 15, 2011
11 / 19
Theorem (Cartan’s Lemma, 1932)
Let D1 , D2 be bounded circular domains of Ck , k 2, containing the
origin. If ψ : D1 ! D2 is biholomorphic and ψ (0) = 0, then ψ is the
restriction of a linear map.
Theorem (Thullen, 1931)
Let D be a bounded Reinhardt domain of C2 . If there exists ψ 2 Aut (D )
such that ψ (0) 6= 0 (equivalently, Aut
n (D ) is not compact), theno D is a
ball, a polydisc, or a set of the form
Alvaro Arias (Institute)
(z, w ) : jz j2 + jw j2/p
Non-commutative domains
1 , p 6= 1.
October 15, 2011
12 / 19
f = X1 + X2 and g = X1 + X2 + X1 X2 .
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Non-commutative domains
October 15, 2011
13 / 19
f = X1 + X2 and g = X1 + X2 + X1 X2 .
Df (C) =
Dg (C) =
Alvaro Arias (Institute)
n
n
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2
1
o
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 + jλ1 λ2 j2
Non-commutative domains
1
October 15, 2011
o
13 / 19
f = X1 + X2 and g = X1 + X2 + X1 X2 .
Df (C) =
Dg (C) =
n
n
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2
1
o
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 + jλ1 λ2 j2
1
o
Since Df (C) is not biholomorphic to Dg (C) , we conclude that
A (Df ) is not isomorphic to A (Dg ) .
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Non-commutative domains
October 15, 2011
13 / 19
f = X1 + X2 and g = X1 + X2 + X1 X2 .
Df (C) =
Dg (C) =
n
n
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2
1
o
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 + jλ1 λ2 j2
1
o
Since Df (C) is not biholomorphic to Dg (C) , we conclude that
A (Df ) is not isomorphic to A (Dg ) .
How about A (Dg ) and A (Dh ) for h = X1 + X2 + 21 X1 X2 + 12 X2 X1 ?
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Non-commutative domains
October 15, 2011
13 / 19
f = X1 + X2 and g = X1 + X2 + X1 X2 .
Df (C) =
Dg (C) =
n
n
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2
1
o
(λ1 , λ2 ) 2 C2 : jλ1 j2 + jλ2 j2 + jλ1 λ2 j2
1
o
Since Df (C) is not biholomorphic to Dg (C) , we conclude that
A (Df ) is not isomorphic to A (Dg ) .
How about A (Dg ) and A (Dh ) for h = X1 + X2 + 21 X1 X2 + 12 X2 X1 ?
Dg (C) = Dh (C) !
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
13 / 19
Theorem (Main result of [AL1])
If Φ : A (Df ) 7! A (Dg ) is a unital completely contractive isomorphism
b 1 (0) = 0, then there exist an invertible matrix M = [mij ] such that
and Φ
for i
n,
Φ Wif
Alvaro Arias (Institute)
= mi 1 W1g + mi 2 W2g +
Non-commutative domains
min Wng .
October 15, 2011
14 / 19
Theorem (Main result of [AL1])
If Φ : A (Df ) 7! A (Dg ) is a unital completely contractive isomorphism
b 1 (0) = 0, then there exist an invertible matrix M = [mij ] such that
and Φ
for i
n,
Φ Wif
= mi 1 W1g + mi 2 W2g +
min Wng .
Idea of the proof:
Φ Wif =[constant] +[linear]+[quadratic]+[cubic]+
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
14 / 19
Theorem (Main result of [AL1])
If Φ : A (Df ) 7! A (Dg ) is a unital completely contractive isomorphism
b 1 (0) = 0, then there exist an invertible matrix M = [mij ] such that
and Φ
for i
n,
Φ Wif
= mi 1 W1g + mi 2 W2g +
min Wng .
Idea of the proof:
Φ Wif =[constant] +[linear]+[quadratic]+[cubic]+
b 1 (0) = 0.
The constant term is zero because Φ
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
14 / 19
Theorem (Main result of [AL1])
If Φ : A (Df ) 7! A (Dg ) is a unital completely contractive isomorphism
b 1 (0) = 0, then there exist an invertible matrix M = [mij ] such that
and Φ
for i
n,
Φ Wif
= mi 1 W1g + mi 2 W2g +
min Wng .
Idea of the proof:
Φ Wif =[constant] +[linear]+[quadratic]+[cubic]+
b 1 (0) = 0.
The constant term is zero because Φ
The quadratic term is zero - uses Cartan’s Lemma applied to
representations on C2 .
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
14 / 19
Theorem (Main result of [AL1])
If Φ : A (Df ) 7! A (Dg ) is a unital completely contractive isomorphism
b 1 (0) = 0, then there exist an invertible matrix M = [mij ] such that
and Φ
for i
n,
Φ Wif
= mi 1 W1g + mi 2 W2g +
min Wng .
Idea of the proof:
Φ Wif =[constant] +[linear]+[quadratic]+[cubic]+
b 1 (0) = 0.
The constant term is zero because Φ
The quadratic term is zero - uses Cartan’s Lemma applied to
representations on C2 .
The cubic term is zero - uses Cartan’s Lemma applied to
representation on C3 , etc. etc.
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Non-commutative domains
October 15, 2011
14 / 19
Corollary
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Non-commutative domains
October 15, 2011
15 / 19
Corollary
g = X1 + X2 + X1 X2
h = X1 + X2 + 12 X1 X2 + 12 X2 X1
Recall that Dg (C) = Dh (C)
C2
Assume Φ : A (Dg ) ! A (Dh ) is an isomorphism
b : Dh (C) ! Dg (C) is biholomorphic
Then Φ
b (0) = 0
By Thullen Φ
By the Main Theorem, Φ is very simple and after some work we reach
a contradiction. A (Dg ) and A (Dh ) are not isomorphic!
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Non-commutative domains
October 15, 2011
15 / 19
Corollary
g = X1 + X2 + X1 X2
h = X1 + X2 + 12 X1 X2 + 12 X2 X1
Recall that Dg (C) = Dh (C)
C2
Assume Φ : A (Dg ) ! A (Dh ) is an isomorphism
b : Dh (C) ! Dg (C) is biholomorphic
Then Φ
b (0) = 0
By Thullen Φ
By the Main Theorem, Φ is very simple and after some work we reach
a contradiction. A (Dg ) and A (Dh ) are not isomorphic!
However, the method was ad hoc and we could not generalize it until
[AL2]
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Non-commutative domains
October 15, 2011
15 / 19
n
Df (C) = (λ1 , . . . , λn ) 2 Cn : ∑jαj
2
1 aα j λ α j
Df (C) is a Reinhardt domain. That is, whenever
1
o
(λ1 , . . . , λn ) 2 Df (C) , and θ 1 , . . . , θ n 2 R, then
e i θ 1 λ1 , . . . , e i θ n λn 2 Df (C) .
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Non-commutative domains
October 15, 2011
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n
Df (C) = (λ1 , . . . , λn ) 2 Cn : ∑jαj
2
1 aα j λ α j
Df (C) is a Reinhardt domain. That is, whenever
1
o
(λ1 , . . . , λn ) 2 Df (C) , and θ 1 , . . . , θ n 2 R, then
e i θ 1 λ1 , . . . , e i θ n λn 2 Df (C) .
Theorem (Sunada)
Let D be a bounded Reinhardt domains of Ck , k 2,that contains the
origin. Then after rescaling and permuting, we can …nd several indexes
such that
D \ (Cp
f0g) is a product of balls
D \ (f0g Cn p ) = f0g D1 , and
(z1 , . . . , zr , zr +1 , . . . , zs ) 2 D i¤ jz1 j < 1, . . . , jzr j < 1 and
0
1
zr +1
zs
A 2 D1
@
,
q r +1,j ,
q
2
2 s ,j
r
r
Πj =1 1 jzj j
Πj =1 1 jzj j
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
16 / 19
We …rst show that Df (C) is not a product of balls
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
17 / 19
We …rst show that Df (C) is not a product of balls
Then we show that when n = 2, then Df (C) is not a Thullen domain
(boundary analysis)
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Non-commutative domains
October 15, 2011
17 / 19
We …rst show that Df (C) is not a product of balls
Then we show that when n = 2, then Df (C) is not a Thullen domain
(boundary analysis)
Then we conclude that all automorphisms of Df (C) …x the origin or
that Df (C) is a ball
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
17 / 19
We …rst show that Df (C) is not a product of balls
Then we show that when n = 2, then Df (C) is not a Thullen domain
(boundary analysis)
Then we conclude that all automorphisms of Df (C) …x the origin or
that Df (C) is a ball
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
17 / 19
We …rst show that Df (C) is not a product of balls
Then we show that when n = 2, then Df (C) is not a Thullen domain
(boundary analysis)
Then we conclude that all automorphisms of Df (C) …x the origin or
that Df (C) is a ball
Theorem ([AL2])
If f and g are aspherical (not balls) then all biholomorphic maps from
Df (C) to Dg (C) …x the origin.
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Non-commutative domains
October 15, 2011
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Theorem ([AL2])
b 1 (0) = 0 then f and g
If Φ : A (Df ) ! A (Dg ) is an isomorphism and Φ
are permutation-rescaling equivalent.
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
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Theorem ([AL2])
b 1 (0) = 0 then f and g
If Φ : A (Df ) ! A (Dg ) is an isomorphism and Φ
are permutation-rescaling equivalent.
Φ Wif = ui 1 W1f + ui 2 W2f +
Alvaro Arias (Institute)
+ uin Wnf for i
Non-commutative domains
n
October 15, 2011
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Theorem ([AL2])
b 1 (0) = 0 then f and g
If Φ : A (Df ) ! A (Dg ) is an isomorphism and Φ
are permutation-rescaling equivalent.
Φ Wif = ui 1 W1f + ui 2 W2f +
+ uin Wnf for i n
Rescale so that aif = aig = 1. This implies that U = [uij ] is unitary
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
18 / 19
Theorem ([AL2])
b 1 (0) = 0 then f and g
If Φ : A (Df ) ! A (Dg ) is an isomorphism and Φ
are permutation-rescaling equivalent.
Φ Wif = ui 1 W1f + ui 2 W2f +
+ uin Wnf for i n
Rescale so that aif = aig = 1. This implies that U = [uij ] is unitary
Wαf
2
=
1
b αf
∑jαj=k cα Wαf
Alvaro Arias (Institute)
and
2
= ∑jαj=k jcα j2 Wαf
2
= ∑jαj=k jcα j2
Non-commutative domains
1
b αf
October 15, 2011
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Theorem ([AL2])
b 1 (0) = 0 then f and g
If Φ : A (Df ) ! A (Dg ) is an isomorphism and Φ
are permutation-rescaling equivalent.
Φ Wif = ui 1 W1f + ui 2 W2f +
+ uin Wnf for i n
Rescale so that aif = aig = 1. This implies that U = [uij ] is unitary
Wαf
2
=
1
b αf
and
∑jαj=k cα Wαf
bαf
! aαf
Alvaro Arias (Institute)
2
= ∑jαj=k jcα j2 Wαf
2
= ∑jαj=k jcα j2
Non-commutative domains
1
b αf
October 15, 2011
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1
b ijf
= Wijf
2
= Φ Wijf
Φ Wif Φ Wjf
Alvaro Arias (Institute)
2
= Φ Wif Wjf
2
=
2
Non-commutative domains
October 15, 2011
19 / 19
1
b ijf
= Wijf
2
= Φ Wijf
= Φ Wif Wjf
2
=
2
Φ Wif Φ Wjf
1
b ijf
2
2
2
= k∑k ∑l uik ujl Wkg Wlg k = ∑k ∑l juik j2 jujl j2 kWklg k =
2
2
∑k ∑l juik j jujl j
Alvaro Arias (Institute)
1
g
b kl
Non-commutative domains
October 15, 2011
19 / 19
1
b ijf
= Wijf
2
= Φ Wijf
= Φ Wif Wjf
2
=
2
Φ Wif Φ Wjf
1
b ijf
2
2
2
= k∑k ∑l uik ujl Wkg Wlg k = ∑k ∑l juik j2 jujl j2 kWklg k =
2
2
∑k ∑l juik j jujl j
1
g
b kl
And this is a convex combination
Alvaro Arias (Institute)
Non-commutative domains
October 15, 2011
19 / 19