extensions of hilbert modules over tensor algebrasadonsig1/nifas/1204-greene.pdf · modules modules...
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Extensions of Hilbert Modules over TensorAlgebras
Andrew Koichi Greene
Department of MathematicsUniversity of Iowa
Spring 2012
Greene, Andrew (University of Iowa) Ext of Hilbert Modules over Tensor Algebras 14 April 2012 1 / 32
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Outline
Outline of topics
1 Setup
2 Modules
3 Extensions
4 Derivations
5 Results
6 Future Work
7 References
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Setup
C ∗-correspondences
A - a unital C ∗ algebra
X a C ∗-correspondence. Recall that this means X is a certainkind of bimodule over A. Specifically,
X is a right Hilbert C∗-module over A.Its left A-action is given by a C∗-homomorphismφ : A→ L (X ).
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Setup
Tensor Powers
X⊗2 = X ⊗A X is a C ∗-correspondence satisfying
a · (x ⊗ y) := φ(a)x ⊗ y .
(x ⊗ y) · b := x ⊗ yb.
xa⊗ y := x ⊗ φ(a)y .
〈x1 ⊗ x2, y1 ⊗ y2〉 := 〈x2, φ(〈x1, y1〉)y2〉.Similarly, define X⊗3,X⊗4, . . .
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Setup
Constructing the Tensor Algebra
Form the Fock space:
F (X ) := A⊕ X ⊕ X⊗2 ⊕ X⊗3 ⊕ · · · .
Define φ∞ : A→ L (F (X )) by
φ∞(a) =
a
φ(a)φ2(a)
φ3(a). . .
where φn(a)(x1 ⊗ x2 ⊗ · · · ⊗ xn) = (φ(a)x1)⊗ x2 ⊗ · · · ⊗ xn.
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Setup
Constructing the Tensor Algebra
Fock space = F (X ) := A⊕ X ⊕ X⊗2 ⊕ X⊗3 ⊕ · · ·For each x ∈ X , we define the creation operator Tx ∈ L (F (X ))by
Tx =
0
T(1)x 0
T(2)x 0
T(3)x 0
. . .. . .
where T
(k)x : X⊗k → X⊗(k+1) is
T(k)x (x1 ⊗ · · · ⊗ xk) = x ⊗ x1 ⊗ · · · ⊗ xk .
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Setup
Constructing the Tensor Algebra
Definition
The tensor algebra of X , denoted T+(X ), is the norm closedsubalgebra of L (F (X )) generated by φ∞(A) and {Tx |x ∈ X}.
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Setup
Examples
1 A = X = C,T+(X ) = A(D) - classical disc algebra
2 A = C,X = Cd ,T+(X ) = Ad - Popescu’s noncommutativedisc algebra
3 Let α be an automorphism of a unital C ∗-algebra A. LetX = αA by defining
1 x · a := xa.2 φ(a)x = α(a)x .3 〈x , y〉 := x∗y .
φ : A→ L (A) equals α since L (A) = M(A) = A.F (X ) = `2(Z+; A)T+(X ) is generated by φ∞(A) and S = T1, a shift.T+(X ) = A×α Z+ is the analytic crossed product of A by Z+
determined by α.
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Modules
Modules
Definition
1 A Hilbert space H is a (c.b.) Hilbert module over an operatoralgebra B if the action of B on H is given by a completelybounded homomorphism π : B → B(H).
2 ϕ : H → H ′ is a Hilbert module map if it is a B-module mapbetween Hilbert modules that is bounded as a Hilbert spaceoperator.
Note: We will assume A ⊂ B is a C ∗-algebra, although B need notbe self-adjoint. Furthermore, the representation (π|A) : A→ B(H)is a C ∗-representation.
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Extensions
Extensions
Definition
An extension ξ is a short exact sequence
ξ : 0 −−−−→ Hϕ−−−−→ J
ψ−−−−→ K −−−−→ 0
where H, J, and K are Hilbert modules over an operator algebra Band ϕ and ψ are Hilbert-module maps.
Note: In particular, the range of ϕ equals the kernel of ψ. So ϕ isbounded below and ψ is bounded below on its initial space.
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Extensions
Equivalence of Extensions
Two extensions ξ and ξ′ are equivalent if and only if there exist aHilbert-module map θ : J → J ′ making the following diagramcommute:
ξ : 0 −−−−→ Hϕ−−−−→ J
ψ−−−−→ K −−−−→ 0∥∥∥ yθ ∥∥∥ξ′ : 0 −−−−→ H ′
ϕ′−−−−→ J ′ψ′−−−−→ K ′ −−−−→ 0
The collection (in fact group) of equivalence classes of extensionsis denoted Ext1(K ,H).
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Derivations
Hilbert Space Decomposition
ξ : 0 −−−−→ Hϕ−−−−→ J
ψ−−−−→ K −−−−→ 0
As Hilbert spaces, J ∼= H⊕K (but not neccesarily as B-modules.)
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Derivations
Cocycles
ξ : 0 −−−−→ Hϕ−−−−→ H⊕K
ψ−−−−→ K −−−−→ 0
Let π : B → B(H) and ρ : B → B(K ) be the representations of Bon H and K , respectively.
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Derivations
Derivations
The B-module action on H⊕K , is given by(π(·) δ(·)
0 ρ(·)
): B → B(H⊕K )
where δ : B → B(K ,H) is a completely bounded A-derivation
1 δ(fg) = δ(f )ρ(g)k + π(f )δ(g) ∀f , g ∈ B
2 δ(a) = 0 for all a ∈ A.
Note: δ is, technically, a φ∞(A)-derivation).
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Derivations
Equivalence of Extensions
If the derivations δ and δ′ correspond, respectively, to extensions ξand ξ′, then ξ ≈ ξ′ if and only if δ− δ′ is an inner derivation: thereexists L ∈ B(K ,H) such that
(δ − δ′)(f ) = π(f )L− Lρ(f )∀f ∈ B.
An inner derivation is A-linear iff π(a)L = Lρ(a) ∀a ∈ A.
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Derivations
Cocycles
Alternatively, we can describe extensions in terms of cocycles:
Definition
A cocycle is a bilinear map σ : B × K → H satisfying
σ(fg , k) = π(f )σ(g , k) + σ(f , ρ(g)k).
which is completely bounded when H and K are given theircolumn Hilbert space structure.
Derivations and cocycles are related via the equation
σ(f , k) = δ(f )k .
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Derivations
Extension Equivalence
ξ ≈ ξ′ if and only if
σ(f , k)− σ′(f , k) = π(f )Lk − Lρ(f )k .
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Derivations
Product Rule
Proposition
Suppose H and K are Hilbert modules over B with representationsπ : T+(αA)→ B(H) and ρ : T+(αA)→ B(K ), respectively. Ifσ : T+(αA)× K → H is a cocycle, then
σ(Sn+1, k) =n∑
j=0
π(Sn−j)σ(S , ρ(S j)k)
for every n ≥ 0, S ∈ B, k ∈ K .
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Derivations
Induced Representation
Let ψ : A→ B(E ) be a representation and let {em}m≥0 be anorthonormal basis for E .
From now on, we only consider B = T+(αA) andH = `2(Z+; A)⊗ψ E .
{δn ⊗ em}n,m≥0 is an orthonormal basis for `2(Z+; A)⊗ψ E .,where δn(k) = δnk1A.
π : T+(αA)→ B(`2(Z+; A)⊗ψ E ) is given byπ|A = φ∞ ⊗ idE and π(T1) = U+ ⊗ idE .
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Derivations
Cocycles Defined by Vectors
Definition
We say a sequence of vectors in K , {km} define a cocycle σ ifσ(S , k) =
∑m〈k , km〉δ0 ⊗ em.
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Results
Motivation
Theorem (Carlson & Clark, 1995)
Let K be a Hilbert A(D)-module. Then a vector k0 ∈ K defines acocycle σ : A(D)× K → H2 if and only if
∞∑n=0
|〈ρ(Sn)k , k0〉|2 <∞
for all k ∈ K .
Note: H2 is the classical Hardy space and σ(S , k) = 〈k , k0〉 ∈ H2.
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Results
Boundedness Criterion
Theorem (Greene, 2011)
Let K be a Hilbert T+(αA)-module. Then a sequence inK , {km}∞m=0 defines a cocycle σ : T+(αA)× K → `2(Z+; A)⊗ψ Eif and only if
1∞∑n=0
∞∑m=0
|〈ρ(Sn)k , km〉|2 <∞ ∀k ∈ K
2
π(α(a))km =∑m′
〈ψ(a)em, em′〉km′
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Results
Corollary
Corollary
If N = dim(E ) <∞ and sp(ρ(S)) ⊂ D, then any {km}1≤m≤Nsatisfying (2) defines a cocycle σ.
Proof.
Define the functions hm(z) = 〈∑
n(zρ(S))nk , km〉. By hypothesishm(z) =
⟨(idK − zρ(S))−1k , km
⟩for |z | < ‖ρ(S)‖−1 and hm(z)
are analytic across the unit circle.
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Results
Proof Continued
Continuation of proof.
N∑m=1
∞∑n=0
|〈zρ(Sn)k , km〉|2 = ‖∑n,m
〈zρ(S)nk , km〉δn ⊗ em‖
≤∑m
‖〈(idK − zρ(S))−1k , km〉‖
≤N∑
m=1
‖hm(z)‖
< ∞.
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Results
Corollary
Corollary
If ρ(S) = idK , then {km} defines a cocycle σ only if km = 0 forevery m. It follows that Ext(K , `2(Z+; A)⊗ψ E ) = 0.
Proof.
∑n,m
|〈ρ(Sn)k, km〉|2 =∑n,m
|〈k , km〉|2 <∞ ⇐⇒ km = 0∀m.
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Results
Characterization of Cocycles
Theorem (Greene, 2011)
Every cocycle σ is equivalent to a cocycle defined by some {km}.
Proof.
1 Let σ be a cocycle.
2 By the Riesz Representation theorem, there exist Kn,m ∈ Kwith
σ(S , k) =∑n,m
〈k,Kn,m〉δn ⊗ em.
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Results
Characterization of Cocycles
Proof.
3 By the product formula,
σ(SN+1, k) =N∑j=0
π(SN−j)σ(S , ρ(S j)k)
=N∑j=0
π(SN−j)∞∑n=0
∞∑m=0
〈ρ(S j)k ,Kn,m〉δn ⊗ em
=N∑j=0
∞∑n=0
∞∑m=0
〈k , ρ(S)∗jKn,m〉δN+n−j ⊗ em
=∞∑n=0
N∑j=0
∞∑m=0
〈k , ρ(S)∗jKn,m〉δN+n−j ⊗ em
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Results
Characterization of Cocycles
Proof.
4 The coefficient of the δν ⊗ em term of σ(SN+1, k) is{∑Nj=0〈k , ρ(S)∗jKν+j−N,m〉 for ν ≥ N∑νj=0〈k , ρ(S)∗N−ν+jKj ,m〉 for ν < N.
5 Therefore,{⟨
k ,∑N
j=1 ρ(S)∗jKj+p,m
⟩}∞N=1
is a bounded
sequence in N.
6 Letting Lim be a Banach limit on `∞, we define kp,m ∈ K by
〈k, kp,m〉 = LimN→∞
⟨k ,
N∑j=0
ρ(S)∗jKj+p,m
⟩.
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Results
Characterization of Cocycles
Proof.
7 Define σ0 by σ0(S , k) =∑
m〈k , k0,m〉δ0 ⊗ em.Note: σ0 is A-linear iff π(α(a))k0,m =
∑p〈ψ(a)em, ep〉k0,p.
8 Define L : K → `2(Z+; A)⊗ψ E byLk =
∑j ,m〈k , kj+1,m〉δj ⊗ em.
9 σ(S , k)− σ0(S , k) = (π(S)L− Lρ(S))k .
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Future Work
Ongoing and Future Work
1 Characterize the coboundaries.
2 Calculuate Ext1(K , `2(Z+; A)⊗ψ E ).
3 Study the more general setting with α ∈ End(A).
4 Generalize to T+(X ).
5 Study projectivity and injectivity in terms of Ext.
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References
References
J. F. Carlson and D. N. Clark
Cohmology and Extensions of Hilbert Modules,
J. Funct. Anal. 128 (1995), 278–306.
S. Ling and P.S Muhly
An Automorphic Form of Ando’s Theorem,
Integral Equations Operator Theory 12 (1989), 424-434.
P. S. Muhly and B. Solel
Tensor Algebras over C∗-Correspondences: Representations, Dilations,and C∗-Envelopes,
J. Funct. Anal. 158 (1998), 389–457.
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References
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Greene, Andrew (University of Iowa) Ext of Hilbert Modules over Tensor Algebras 14 April 2012 32 / 32