Toward a Complete Pick Property in a W ∗ -Setting
Jenni Good
University of Iowa
FAS
Omaha, NE
March 28, 2015
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Jenni Good
Toward a W ∗ CPP
Classical Muhly/Solel
Definition
M, a W ∗ -algebra
E is a W ∗ -correspondence over M if
E is a Hilbert W ∗ -module over M
E is a left M-module via a unital W ∗ -homomorphism
ϕ : M → L (E ), i.e.
a · x = ϕ(a)(x)
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Jenni Good
∀a ∈ A, x ∈ E
Toward a W ∗ CPP
The Fock Space
Tensor Powers
E ⊗0 = M
E ⊗1 = E
E ⊗k = E ⊗ E ⊗k−1
∞
P
F (E ) =
⊕ E ⊗k
(Defined inductively for k ≥ 2).
(The Fock Space)
k=0
New Correspondences
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E ⊗k ,
left action: ϕk
F (E ),
left action: ϕ∞
Jenni Good
Toward a W ∗ CPP
Important elements of L (F (E ))
Left action maps, a ∈ M

∞
ϕ0 (a)
0
···


ϕ1 (a)
ϕ∞ (a) =  0

..
..
.
.
i,j=0
Creation operators, ξ ∈ E ⊗k , k ∈ N
Tξ (z) = ξ ⊗ z

0
T (0)
 ξ
Tξ = 
 0

0
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0
0
(1)
Tξ
0
···
···
···
..
.
Jenni Good
∞
0
0


0

0 i,j=0
(Case k = 1, ξ ∈ E )
Toward a W ∗ CPP
Important Operator Algebras
Subalgebras of L (F (E ))
T0+ (E ) (“algebraic tensor algebra”)
algebra generated by ϕ∞ (a), Tξ
T+ (E ): (“tensor algebra”)
norm-closure of T0+ (E )
H∞ (E ): (“Hardy algebra”)
ultraweak closure of T0+ (E )
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Jenni Good
Toward a W ∗ CPP
Generalization: Weights!
M: W ∗ -algebra
E : W ∗ -correspondence over M
The R-Sequence
Let {Rk }∞
k=0 be a family such that
Rk ∈ ϕk (M)c , ∀k ∈ N
Rk is positive and invertible ∀k ∈ N.
R0 = I .
supk∈N kRk−1 (IE ⊗ Rk−1 )k < ∞
lim supk→∞ kRk k1/k < ∞
The Z -Sequence
Define the “sequence of weights”, {Zk }∞
k=0 ,
(
IM ,
if k = 0
Zk =
−1
Rk (IE ⊗ Rk−1 ) if k > 0
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Jenni Good
Toward a W ∗ CPP
Put ’em Together...
Left action maps, a ∈ M

ϕ0 (a)
0
 0
ϕ
(a)
1
ϕ∞ (a) = 
∞
..


.
i,j=0
Weighted Creation Operators, ξ ∈ E ⊗k , k ∈ N

∞
0
Z T (0)

0
 1 ξ



(1)
Wξ =  0

Z2 Tξ


..
.
i,j=0
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Jenni Good
Toward a W ∗ CPP
k = 1, ξ ∈ E ,
Weighted Operator Algebras
Subalgebras of L (F (E ))
T0+ (E , Z ): (“weighted algebraic tensor algebra”)
algebra generated by ϕ∞ (a), Wξ
T+ (E , Z ): (“weighted tensor algebra”)
norm-closure of T0+ (E , Z )
∞
H (E , Z ): (“weighted Hardy algebra”)
ultraweak closure of T0+ (E , Z )
Fundamental Idea
To understand an algebra, view it as an algebra of functions on its space
of representations.
Motivating Question:
Can we paramaterize the completely bounded ultraweakly continuous
representations of H∞ (E , Z ) on Hilbert space?
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Jenni Good
Toward a W ∗ CPP
Getting Those Representations
The Natural Choice
Fix σ : M → B(H) a normal unital ∗-hom.
Take z ∈ I(σ E ◦ ϕ, σ).
Fact: There is a representation, ρ : T0+ (E , Z ) → B(H) such that
ρ(ϕ∞ (a)) = σ(a)
ρ(Wξ )(h) = z(ξ ⊗ h).
Question:
Can we extend ρ to H∞ (E , Z )?
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Jenni Good
Toward a W ∗ CPP
Partial answer
Muhly, Solel: Matricial Function Theory and Weighted Shifts
∞
P
Suppose
z(k) (Rk2 ⊗ IH )z(k)∗ converges ultraweakly in B(H).
k=0
Define Rz : σ(M)0 → σ(M)0
∞
P
Rz (A) =
z(k) (Rk2 ⊗ A)z(k)∗
k=0
Then
Rz is a linear completely positive map.
IF
there exists a completely positive map, Xz : σ(M)0 → σ(M)0
such that kXz k ≤ 1, and
−1
Rz = (ι − Xz )
,
then ρ extends to an ultraweakly continuous representation of
H∞ (E , Z ),
In particular: we are very happy.
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Jenni Good
Toward a W ∗ CPP
A New Sequence, X
The X -Sequence

0,
k
Xk = P
P
 (−1)l+1
if k = 0
l
2
⊗ Rα(i)
α i=1
l=1
(summing over α : {1, . . . , l} → N such that
l
P
if k ≥ 1
α(i) = k )
i=1
Idea behind the Formula
∞
P
k=0
rk2 t k =
1−
1
∞
P
xk t k
k=1
Recall:
IF
there exists a completely positive map, Xz : σ(M)0 → σ(M)0 such that
kXz k ≤ 1, and
Rz = (ι − Xz )−1 ,
...
then ρ extends to an ultraweakly continuous representation of H∞ (E , Z ),
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Jenni Good
Toward a W ∗ CPP
Here Comes the Complete Pick Property
The X -map
Suppose that Xz : σ(M)0 → σ(M)0
Xz (A) =
∞
X
z(k) (Xk ⊗ A)z(k)∗
k=0
is well-defined. Ignoring convergence issues, Rz = (ι − Xz )−1 !!
Questions:
Is Xz well-defined?
Is kXz k ≤ 1 ?
Is Xz completely positive? If all Xk are positive: yes!!
Answer in Case M = E = C (McCullough-Quiggin, Agler, McCarthy)
Xk ≥ 0, ∀k ≥ 1
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⇐⇒
Jenni Good
the associated kernel has the
“Complete Pick Property”
Toward a W ∗ CPP
Crash Course in RKHS’s
Definition
A reproducing kernel Hilbert space, RKHS, is a Hilbert space, H, of
functions on some set, Ω such that evaluation at each point of Ω is a
non-zero continuous linear functional on H.
Definition
The reproducing kernel at w ∈ Ω: kw ∈ H such that
f (w ) = hf , kw i
∀f ∈ H
Definition
The reproducing kernel associated with H: K : Ω × Ω → C,
K (w , z) = hkz , kw i
Definition
φ : Ω → C is a multiplier of H if its pointwise product with any element
of H also belongs to H.
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Jenni Good
Toward a W ∗ CPP
Classical Interpolation Problem
Important Example: the Hardy Space, H 2 (D)
Kernel: K : D × D → C, K (w , z) =
1
1−w z
Multipliers: H ∞ (D)
Question:
Fix k ∈ N ; {ωi }ki=1 , {λi }ki=1 ⊂ D.
When is there a function, φ, in H ∞ (D) of norm at most 1 that
interpolates this data; i.e. when 1 ≤ i ≤ k,
φ(ωi ) = λi ?
Answer: (Nevanlinna/Pick (1915)) Happy 100th Anniversary!
Such a φ exists iff
h
ik
k
1−λ λ
K (ωi , ωj )(1 − λi λj ) i,j=1 = 1−ωii ωjj
i,j=1
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Jenni Good
Toward a W ∗ CPP
≥0
Generalized Operator-Theoretic Interpolation Problem
Question:
H: RKHS on Ω;
K : reproducing kernel
k ∈ N; {ωi }ki=1 ⊂ Ω, {λi }ki=1 ⊂ D.
When is there a multiplier, φ, of H of norm at most 1 such that
φ(ωi ) = λi
∀1 ≤ i ≤ k?
Definition
H has the Pick property if
k
K (ωi , ωj )(1 − λi λj ) i,j=1 ≥ 0
implies the existance of an interpolating multiplier of norm ≤ 1.
Example: The Hardy Space, H 2 (D)
Non-Example: The Bergman space
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Jenni Good
Toward a W ∗ CPP
The Complete Pick Property
Ms×t Pick Property
K , a kernel, on Ω
s, t ∈ N
K has the Ms×t Pick Property if whenever
k ∈ N, {ωi }ki=1 ⊆ Ω, {Wi }ki=1 ⊆ Ms×t (C)
[K (ωi , ωj )(I − Wi Wj∗ )]ki,j=1 ≥ 0
t
(in Mk (Ms (C)))
s
there exists Φ ∈ Mult(H ⊗ C , H ⊗ C ), such that
Φ(ωi ) = Wi
∀1 ≤ i ≤ k
Definition
K has the Complete Pick Property if it has the Ms×t property
∀s, t ∈ N.
Example: Hardy Space
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Jenni Good
Toward a W ∗ CPP
What Does That Have to Do with the Price of Eggs?
Theorem (Agler, McCarthy)
An (irreducible) kernel, K , is a Complete Pick kernel if and only if
K (ζ, λ) =
δ(ζ)δ(λ)
1 − F (ζ, λ)
for a positive semi-definite function F : Ω × Ω → D and a nowhere
vanishing function δ : Ω → C
Recall:
. . . IF there exists a completely positive map, Xz : σ(M)0 → σ(M)0 such
that kXz k ≤ 1, and
−1
Rz = (ι − Xz ) ,
then ρ extends to an ultraweakly continuous representation of H∞ (E , Z )
Conclusion
In the one-dimensional case, we can use the Complete Pick Property.
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Jenni Good
Toward a W ∗ CPP
The Goal
Formulate a W ∗ -Complete Pick Property
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Jenni Good
Toward a W ∗ CPP
Generalizing the Classical Theory
Reproducing Kernel Hilbert
Space (RKHS)
Reproducing Kernel W ∗ -Correspondence
(RKWC)
E : an (M, N) W ∗ -correspondence, for
W ∗ -algebras, M, N.
H: a Hilbert space
H is a subspace of
F (Ω, C), for a set, Ω.
E is a sub N-module of F (M × Ω, N),
for a set, Ω.
(Positive Kernel)
K : Ω × Ω → C such
that
(Normal Completely Positive Kernel)
K : Ω × Ω → Buw (M, N) such that
[K (ωi , ωj )(ai aj∗ )]nij=1 ≥ 0,
[K (ωi , ωj )]nij=1 ≥ 0
for any n ∈ N, {ωi }ni=1 ⊆ Ω,
{ai }ni=1 ⊆ M.
for any n ∈ N,
{ωi }ni ⊆ Ω.
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Jenni Good
Toward a W ∗ CPP
Generalizing the Classical Theory
W ∗ -Multipliers
Multipliers
φ : Ω → C is a multiplier of
H if φ · f ∈ H, ∀f ∈ H,
φ · f (ω) = φ(ω)f (ω)
φ : Ω → N is a multiplier of E
if φ · f ∈ E , ∀f ∈ E ,
φ · f (a, ω) = φ(ω)f (a, ω)
Mult(H): the multipliers of H
Mult(E ): the multipliers of E .
Matrix-Valued Multipliers
Matrix-Valued W ∗ -Multipliers (G)
Φ : Ω → Ms×t (C) is an
(s, t)-multiplier if
∃{φmn } ⊂ Mult(H), ∀ω ∈ Ω,
Φ : Ω → Ms×t (N) is an
(s, t)-multiplier if
∃{φmn } ⊂ Mult(E ), ∀ω ∈ Ω,
t
t
Φ(ω) = [φmn (ω)]sm=1 n=1
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Φ(ω) = [φmn (ω)]sm=1 n=1
Jenni Good
Toward a W ∗ CPP
Generalizing the Classical Theory
Recall: Ms×t Pick Property, s, t ∈ N
K : Ω × Ω → C, a positive kernel, has the Ms×t Pick Property if
k ∈ N, {ωi }ki=1 ⊆ Ω, {Wi }ki=1 ⊆ Ms×t (C)
[K (ωi , ωj )(I − Wi Wj∗ )]ki,j=1 ≥ 0
(in Mk (Ms (C)))
implies there exists an (s, t) multiplier, Φ, of norm ≤ 1 such that
Φ(ωi ) = Wi
∀1 ≤ i ≤ k
Observation
[K (ωi , ωj )(I − Wi Wj∗ )]ki,j=1 =
""K (ω , ω )
#
i
"K (ω , ω )
j
.
.
i
− Wi
.
.
.
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K (ωi , ωj )
(t × t)
Jenni Good
Wj∗
.
K (ωi , ωj )
(s × s)
#k
#
j
Toward a W ∗ CPP
i,j=1
A W ∗ -formulation of the Complete Pick Property
W ∗ -Ms×t Pick Property, s, t ∈ N (G)
K : Ω × Ω → Buw (M, N), NCP kernel, has the Ms×t Pick Property if
whenever
k ∈ N,
{ωi }ki=1 ⊆ Ω,
{Bi }ki=1 ⊆ Ms×s (N),
{Di }ki=1 ⊆ Ms×t (N)
A : Mk (M) → Mk (Ms (N)) is completely positive, where
A
[aij ]k
ij =




B
 i






K (wi , wj )(aij )

.
.
.

K (wi , wj )(aij )


 ∗

B − D 
i 
 j


K (wi , wj )(aij )

.
.
.
There exists an (s, t) multiplier, Φ, of norm ≤ 1 such that
Bi Φ(ωi ) = Di ,
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Jenni Good
1 ≤ i ≤ k.
Toward a W ∗ CPP
k


 ∗
D 
 j 


K (wi , wj )(aij )
ij
A W ∗ -formulation of the Complete Pick Property
Complete Pick NCP Kernel
K : Ω × Ω → Buw (M, N), an NCP kernel, is a Complete Pick NCP
Kernel if it has the Ms×t Pick Property ∀s, t ∈ N
Question:
Are there any examples?
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Jenni Good
Toward a W ∗ CPP
Finding a Complete Pick NCP Kernel
Recall:
. . . IF there exists a completely positive map, Xz : σ(M)0 → σ(M)0 such
that kXz k ≤ 1, and
−1
Rz = (ι − Xz ) ,
then ρ extends to an ultraweakly continuous representation of H∞ (E , Z )
Can’t win the game? Change the Rules
Our approach: R → X .
In fact: R X .
If we start with X , we can rig it so the above condition above is
satisfied.
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Jenni Good
Toward a W ∗ CPP
Finding a Complete Pick NCP Kernel
The Setup
M: a W ∗ -algebra;
E : a W ∗ -correspondence over M
X : an “admissible” sequence
R: constructed from X , with sequence of weights, Z
σ : M → B(H) faithful, unital, W ∗ -homomorphism
∞
P
E
(k)
(k)∗ D(X , σ) = z ∈ I(σ ◦ ϕ, σ) : z (Xk ⊗ IH )z < 1 .
k=1
z ∈ D(X , σ) =⇒ “ρ” extends to (σ × z) : H∞ (E , Z ) → B(H).
Y ∈ H∞ (E , Z ) =⇒ Yb : D(X , σ) → B(H)
Yb (z) = (σ × z)(Y )
R : D(X , σ) × D(X , σ) → Buw (σ(M)0 , B(H)), (NCP kernel)
∞
P
R(w, z)(A) =
w(k) (Rk2 ⊗ A)z(k)∗
k=0
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Jenni Good
Toward a W ∗ CPP
Weighted Nevanlinna Pick
Theorem (G, 2014)
s, t ∈ N.
k ∈ N.
k points, {zi }ki=1 ⊆ D(X , σ)
{Bi }ki=1 ⊆ Ms×s (B (H)),
{Di }ki=1 ⊆ Ms×t (B (H)).
A : Mk (σ(M)0 ) → Mk (Ms (B (H)))
A
[aij ]kij

=
R(z , z )(a
Bi 
i
j

ij )
.
.
 B∗
j
.
R(z , z )(a
i
j

ij )
.
− Di 
.
.
R(zi , zj )(aij )
R(zi , zj )(aij )
Then A is completely positive iff there exists
t
Y = [Ymn ]sm=1 n=1 ∈ Ms×t (H∞ (E , Z )) such that kY k ≤ 1 and
h
is
d
Bi ◦ Y
mn (zi )
t
m=1 n=1
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Jenni Good
= Di
1≤i ≤k
Toward a W ∗ CPP
k
 D∗
j
ij
Multiplier Theorem
Theorem (G, 2014)
s, t ∈ N.
Φ : D(X , σ) → Ms×t (B(H)) any function.
Then Φ is an (s, t) multiplier of norm at most 1 iff there exists
t
s
Y = [Ymn ]m=1 n=1 ∈ Ms×t (H∞ (E , Z )) such that kY k ≤ 1 and
h
is
d
Φ(z) = Y
mn (z)
t
m=1 n=1
∀z ∈ D(X , σ)
Corollary 1
If Y ∈ H∞ (E , Z ) then Yb is a multiplier of the RKWC , and every
multiplier is obtained in this fashion.
(s = t = 1)
Corollary 2
R is an NCP Complete Pick Kernel. Whoohoo!
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Jenni Good
Toward a W ∗ CPP
Is this the “Right” Complete Pick Property?
ANOTHER Equivalent Condition (McCullough-Quiggin)
An irreducible kernel, K , is a Complete Pick kernel iff ∀N ∈ N and
distinct {ωi }N
i=1 ⊂ Ω,
kiN kNj
FN = 1 −
kij kNN
N−1
≥0
where
i,j=1
kij = K (ωi , ωj ) ∈ C
One of These Things Is Not Like the Other...
N−1
0
N ∈ N, {wi }N
i=1 ⊆ Ω, {Ai }i=1 ⊆ σ(M)
h
iN−1
Ai A∗j − Rij−1 RiN (Ai )(RNN (I ))−1 RNj (A∗j )
i,j
Rij = R(ωi , ωj ) ∈ CB(σ(M)0 )
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Jenni Good
Toward a W ∗ CPP
An “Inner Multiplier”
Ω: a set
M, N: W ∗ -algebras
E : an (Ω, M, N) RKWC
The Old
φ : Ω → N is an “Outer multiplier” of E if φ · f ∈ E , ∀f ∈ E ,
φ · f (a, ω) = φ(ω)f (a, ω)
The New
µ : Ω → CB(M) is an “Inner multiplier” of E if µ · f ∈ E , ∀f ∈ E ,
µ · f (a, ω) = f (µ(ω)(a), ω)
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Jenni Good
Toward a W ∗ CPP
Inner (s, t) Multipliers
Inner (s, t) Multiplier
t
Given a family, {µmn }sm=1 n=1 of inner multipliers, define
M : Ω → F (Ms×s (M), Ms×t (M))
M (z) ([amn ]smn=1 )
=
" s
X
∗
µpn (z)(amp )
p=1
Jenni Good
t
m=1 n=1
(an Inner (s, t) Multiplier).
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#s
Toward a W ∗ CPP
Recall
The “Outer” W ∗ -Ms×t Pick Property, s, t ∈ N (G)
K : Ω × Ω → Buw (M, N), NCP kernel, has the Ms×t Pick Property if
whenever
k ∈ N,
{ωi }ki=1 ⊆ Ω,
{Bi }ki=1 ⊆ Ms×s (N),
{Di }ki=1 ⊆ Ms×t (N)
A : Mk (M) → Mk (Ms (N)) is completely positive, where
A
[aij ]k
ij =




B
 i






K (wi , wj )(aij )

.
.
.

K (wi , wj )(aij )


 ∗

B − D 
i 
 j


K (wi , wj )(aij )

.
.
.
there exists an (s, t) multiplier, Φ, of norm ≤ 1 such that
Bi Φ(ωi ) = Di ,
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Jenni Good
1 ≤ i ≤ k.
Toward a W ∗ CPP
k


 ∗
D 
 j 


K (wi , wj )(aij )
ij
A Different Complete Pick Property:
Definition: The “Inner” W ∗ -Ms×t Pick Property, s, t ∈ N (G)
K : Ω × Ω → Buw (M, N), NCP kernel, has the “Inner” Ms×t Pick
Property if whenever
k ∈N
{z}ki=1 ⊆ Ω
{Bi }ki=1 ⊆ Ms×s (M)
{Di }ki=1 ⊆ Ms×t (M)
k
A = K (zi , zj ) s (Bi Bj∗ − Di Dj∗ ) i,j=1 ≥ 0
where K (zi , zj ) s : Ms×s (M) → Ms×s (M) is given by
K (zi , zj ) s ([amn ]mn ) = [K (zi , zj )(amn )]mn
There exists an inner s × t multiplier, M : Ω → F (Ms×s (M), Ms×t (M))
such that
M (zi )(Bi ) = Di whenever 1 ≤ i ≤ k
kM k ≤ 1
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Jenni Good
Toward a W ∗ CPP
Future Work
Definition: The “Inner” Complete Pick Property, s, t ∈ N (G)
A NCP kernel, K : Ω × Ω → Buw (M, N) has the Inner W ∗ -Complete
Pick Property if it has the Inner W ∗ -Ms×t Pick Property ∀s, t ∈ N.
Recall:
. . . IF there exists a completely positive map, Xz : σ(M)0 → σ(M)0 such
that kXz k ≤ 1, and
−1
Rz = (ι − Xz ) ,
then ρ extends to an ultraweakly continuous representation of H∞ (E , Z )
A New Toy to Play With
Hypothesis:
R : Ω × Ω → CBuw (σ(M)0 ) has the Inner W ∗ -Complete Pick
Property if and only if there is an NCP kernel,
X : Ω × Ω → CCuw (σ(M)0 ) such that ∀w, z ∈ Ω.
R(w, z) = (ι − X (w, z))
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Jenni Good
−1
Toward a W ∗ CPP
,
Thanks!
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Jenni Good
Toward a W ∗ CPP
Further Reading
Jim Agler and John E. McCarthy.
Pick interpolation and Hilbert function spaces, volume 44 of
Graduate Studies in Mathematics.
American Mathematical Society, Providence, RI, 2002.
Jonas R. Meyer.
Noncommutative Hardy Algebras, Multipliers, and Quotients.
PhD thesis, The University of Iowa, 2010.
Paul S. Muhly and Baruch Solel.
Matricial function theory and weighted shifts.
In preparation.
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Jenni Good
Toward a W ∗ CPP