Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A...

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Hilbert C*-modules in harmonic analysis Nordfjordeid 2019 Ulrik Enstad 4th July 2019

Transcript of Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A...

Page 1: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modules in harmonicanalysisNordfjordeid 2019

Ulrik Enstad 4th July 2019

Page 2: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modules

Definition

Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:

1 For all ξ, η, γ ∈ E and a,b ∈ A we have that

•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .

2 For all ξ, η ∈ E we have that

•〈ξ, η〉∗ = •〈η, ξ〉 .

3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.

4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14

Page 3: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modulesDefinition

Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:

1 For all ξ, η, γ ∈ E and a,b ∈ A we have that

•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .

2 For all ξ, η ∈ E we have that

•〈ξ, η〉∗ = •〈η, ξ〉 .

3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.

4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14

Page 4: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modulesDefinition

Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:

1 For all ξ, η, γ ∈ E and a,b ∈ A we have that

•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .

2 For all ξ, η ∈ E we have that

•〈ξ, η〉∗ = •〈η, ξ〉 .

3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.

4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14

Page 5: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modulesDefinition

Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:

1 For all ξ, η, γ ∈ E and a,b ∈ A we have that

•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .

2 For all ξ, η ∈ E we have that

•〈ξ, η〉∗ = •〈η, ξ〉 .

3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.

4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14

Page 6: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modulesDefinition

Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:

1 For all ξ, η, γ ∈ E and a,b ∈ A we have that

•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .

2 For all ξ, η ∈ E we have that

•〈ξ, η〉∗ = •〈η, ξ〉 .

3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.

4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14

Page 7: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Hilbert C*-modulesDefinition

Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:

1 For all ξ, η, γ ∈ E and a,b ∈ A we have that

•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .

2 For all ξ, η ∈ E we have that

•〈ξ, η〉∗ = •〈η, ξ〉 .

3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.

4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14

Page 8: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Examples of Hilbert C*-modules

The left A-module Ak becomes a left Hilbert A-module with theinner product

•〈(a1, . . . , ak ), (b1, . . . , bk )〉 =k∑

j=1

ajb∗j .

For a countable index set J, define

`2(J ,A) = {(aj)j∈J ⊆ A :∑j∈J

aja∗j converges in A}.

This becomes a left Hilbert A-module with respect to

•⟨(aj), (bj)

⟩=∑j∈J

ajb∗j .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 2 / 14

Page 9: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Examples of Hilbert C*-modulesThe left A-module Ak becomes a left Hilbert A-module with theinner product

•〈(a1, . . . , ak ), (b1, . . . , bk )〉 =k∑

j=1

ajb∗j .

For a countable index set J, define

`2(J ,A) = {(aj)j∈J ⊆ A :∑j∈J

aja∗j converges in A}.

This becomes a left Hilbert A-module with respect to

•⟨(aj), (bj)

⟩=∑j∈J

ajb∗j .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 2 / 14

Page 10: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Examples of Hilbert C*-modulesThe left A-module Ak becomes a left Hilbert A-module with theinner product

•〈(a1, . . . , ak ), (b1, . . . , bk )〉 =k∑

j=1

ajb∗j .

For a countable index set J, define

`2(J ,A) = {(aj)j∈J ⊆ A :∑j∈J

aja∗j converges in A}.

This becomes a left Hilbert A-module with respect to

•⟨(aj), (bj)

⟩=∑j∈J

ajb∗j .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 2 / 14

Page 11: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Sections of vector bundles

If A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with

(f · s)(x) = f (x)s(x)

•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).

Theorem (Serre–Swan)

If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14

Page 12: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .

For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with

(f · s)(x) = f (x)s(x)

•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).

Theorem (Serre–Swan)

If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14

Page 13: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .

Γ(E) becomes a left Hilbert C(X )-module with

(f · s)(x) = f (x)s(x)

•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).

Theorem (Serre–Swan)

If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14

Page 14: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with

(f · s)(x) = f (x)s(x)

•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).

Theorem (Serre–Swan)

If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14

Page 15: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with

(f · s)(x) = f (x)s(x)

•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).

Theorem (Serre–Swan)

If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14

Page 16: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Orthonormal bases in HilbertC*-modules?

Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑

j;finite

ajsj : aj ∈ A, sj ∈ E

= E.

Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14

Page 17: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Orthonormal bases in HilbertC*-modules?

Any separable Hilbert space H admits a countable orthonormalbasis.

A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑

j;finite

ajsj : aj ∈ A, sj ∈ E

= E.

Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14

Page 18: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Orthonormal bases in HilbertC*-modules?

Any separable Hilbert space H admits a countable orthonormalbasis.

A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑

j;finite

ajsj : aj ∈ A, sj ∈ E

= E.

Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14

Page 19: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Orthonormal bases in HilbertC*-modules?

Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑

j;finite

ajsj : aj ∈ A, sj ∈ E

= E.

Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14

Page 20: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Orthonormal bases in HilbertC*-modules?

Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑

j;finite

ajsj : aj ∈ A, sj ∈ E

= E.

Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?

No!

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14

Page 21: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Orthonormal bases in HilbertC*-modules?

Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑

j;finite

ajsj : aj ∈ A, sj ∈ E

= E.

Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14

Page 22: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Frames

Definition (Frank–Larson)

Let J be a countable index set. A sequence (ηj)j∈J ⊆ E is called aframe if there exists C,D >0 such that

C•〈ξ, ξ〉 ≤∑j∈J•⟨ξ, ηj

⟩•⟨ξ, ηj

⟩∗ ≤ D•〈ξ, ξ〉

for all ξ ∈ E .

The frame is normalized tight if one can choose C = D = 1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 5 / 14

Page 23: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

FramesDefinition (Frank–Larson)

Let J be a countable index set. A sequence (ηj)j∈J ⊆ E is called aframe if there exists C,D >0 such that

C•〈ξ, ξ〉 ≤∑j∈J•⟨ξ, ηj

⟩•⟨ξ, ηj

⟩∗ ≤ D•〈ξ, ξ〉

for all ξ ∈ E .

The frame is normalized tight if one can choose C = D = 1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 5 / 14

Page 24: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

FramesDefinition (Frank–Larson)

Let J be a countable index set. A sequence (ηj)j∈J ⊆ E is called aframe if there exists C,D >0 such that

C•〈ξ, ξ〉 ≤∑j∈J•⟨ξ, ηj

⟩•⟨ξ, ηj

⟩∗ ≤ D•〈ξ, ξ〉

for all ξ ∈ E .

The frame is normalized tight if one can choose C = D = 1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 5 / 14

Page 25: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Dual frames

Theorem (Frank–Larson)

If (ηj)j∈J is a frame in E , there exists another frame (γj)j∈J such that

ξ =∑j∈J•⟨ξ, ηj

⟩γj =

∑j∈J•⟨xi , γj

⟩ηj

for all ξ ∈ E .

Theorem (Frank–Larson)

If E is a countably generated Hilbert A-module, then it admits a (count-able) frame.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 6 / 14

Page 26: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Dual framesTheorem (Frank–Larson)

If (ηj)j∈J is a frame in E , there exists another frame (γj)j∈J such that

ξ =∑j∈J•⟨ξ, ηj

⟩γj =

∑j∈J•⟨xi , γj

⟩ηj

for all ξ ∈ E .

Theorem (Frank–Larson)

If E is a countably generated Hilbert A-module, then it admits a (count-able) frame.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 6 / 14

Page 27: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Dual framesTheorem (Frank–Larson)

If (ηj)j∈J is a frame in E , there exists another frame (γj)j∈J such that

ξ =∑j∈J•⟨ξ, ηj

⟩γj =

∑j∈J•⟨xi , γj

⟩ηj

for all ξ ∈ E .

Theorem (Frank–Larson)

If E is a countably generated Hilbert A-module, then it admits a (count-able) frame.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 6 / 14

Page 28: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Properties of frames

A finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,

span{s1(x), . . . , sk (x)} = Ex .

In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14

Page 29: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Properties of framesA finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.

If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,

span{s1(x), . . . , sk (x)} = Ex .

In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14

Page 30: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Properties of framesA finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,

span{s1(x), . . . , sk (x)} = Ex .

In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14

Page 31: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Properties of framesA finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,

span{s1(x), . . . , sk (x)} = Ex .

In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14

Page 32: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Projective modules

Recall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix

P = (•⟨ηi , ηj

⟩)ki ,j=1

is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C

τ̃([E ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14

Page 33: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.

Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix

P = (•⟨ηi , ηj

⟩)ki ,j=1

is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C

τ̃([E ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14

Page 34: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.

If (η1, . . . , ηk ) is a frame for E , then the matrix

P = (•⟨ηi , ηj

⟩)ki ,j=1

is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C

τ̃([E ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14

Page 35: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix

P = (•⟨ηi , ηj

⟩)ki ,j=1

is a projection in Mk (A), and E ∼= AkP.

In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C

τ̃([E ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14

Page 36: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix

P = (•⟨ηi , ηj

⟩)ki ,j=1

is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.

If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C

τ̃([E ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14

Page 37: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix

P = (•⟨ηi , ηj

⟩)ki ,j=1

is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C

τ̃([E ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14

Page 38: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Gabor frames

Given θ ∈ R \ {0}, define operators Tθ,M : L2(R)→ L2(R) by

Tθξ(t) = ξ(t − θ)

Mξ(t) = e2πitξ(t).

These two operators satisfy

MTθ = e2πiθTθM.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 9 / 14

Page 39: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Gabor framesGiven θ ∈ R \ {0}, define operators Tθ,M : L2(R)→ L2(R) by

Tθξ(t) = ξ(t − θ)

Mξ(t) = e2πitξ(t).

These two operators satisfy

MTθ = e2πiθTθM.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 9 / 14

Page 40: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Gabor framesGiven θ ∈ R \ {0}, define operators Tθ,M : L2(R)→ L2(R) by

Tθξ(t) = ξ(t − θ)

Mξ(t) = e2πitξ(t).

These two operators satisfy

MTθ = e2πiθTθM.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 9 / 14

Page 41: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Problem

Given η1, . . . , ηk ∈ L2(R) and θ ∈ R \ {0}, when is the set

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

an orthonormal basis for L2(R), or more generally a frame for L2(R)?

A Balian–Low Theorem (Battle)

If η ∈ S(R), then{MnT m

θ η : m,n ∈ Z}

cannot be an orthonormal basis for L2(R).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 10 / 14

Page 42: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Problem

Given η1, . . . , ηk ∈ L2(R) and θ ∈ R \ {0}, when is the set

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

an orthonormal basis for L2(R), or more generally a frame for L2(R)?

A Balian–Low Theorem (Battle)

If η ∈ S(R), then{MnT m

θ η : m,n ∈ Z}

cannot be an orthonormal basis for L2(R).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 10 / 14

Page 43: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Problem

Given η1, . . . , ηk ∈ L2(R) and θ ∈ R \ {0}, when is the set

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

an orthonormal basis for L2(R), or more generally a frame for L2(R)?

A Balian–Low Theorem (Battle)

If η ∈ S(R), then{MnT m

θ η : m,n ∈ Z}

cannot be an orthonormal basis for L2(R).

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 10 / 14

Page 44: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The noncommutative torus

Define

Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).

The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .Aθ is equipped with a trace τ : Aθ → C given by

τ(∑

m,nam,nMnT m

θ

)= a0,0.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14

Page 45: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The noncommutative torusDefine

Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).

The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .Aθ is equipped with a trace τ : Aθ → C given by

τ(∑

m,nam,nMnT m

θ

)= a0,0.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14

Page 46: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The noncommutative torusDefine

Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).

The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .

Aθ is equipped with a trace τ : Aθ → C given by

τ(∑

m,nam,nMnT m

θ

)= a0,0.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14

Page 47: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The noncommutative torusDefine

Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).

The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .Aθ is equipped with a trace τ : Aθ → C given by

τ(∑

m,nam,nMnT m

θ

)= a0,0.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14

Page 48: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The Heisenberg module

for a =∑

m,n am,nMnT nθ ∈ Aθ and ξ ∈ S(R), define(∑

m,namnMnT n

θ

)ξ =

∑m,n

amnMnT nθ ξ.

For ξ, η ∈ S(R), define

•〈ξ, η〉 =∑m,n〈ξ,MnT m

θ η〉MnT mθ ∈ Aθ.

We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14

Page 49: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The Heisenberg modulefor a =

∑m,n am,nMnT n

θ ∈ Aθ and ξ ∈ S(R), define(∑m,n

amnMnT nθ

)ξ =

∑m,n

amnMnT nθ ξ.

For ξ, η ∈ S(R), define

•〈ξ, η〉 =∑m,n〈ξ,MnT m

θ η〉MnT mθ ∈ Aθ.

We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14

Page 50: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The Heisenberg modulefor a =

∑m,n am,nMnT n

θ ∈ Aθ and ξ ∈ S(R), define(∑m,n

amnMnT nθ

)ξ =

∑m,n

amnMnT nθ ξ.

For ξ, η ∈ S(R), define

•〈ξ, η〉 =∑m,n〈ξ,MnT m

θ η〉MnT mθ ∈ Aθ.

We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14

Page 51: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

The Heisenberg modulefor a =

∑m,n am,nMnT n

θ ∈ Aθ and ξ ∈ S(R), define(∑m,n

amnMnT nθ

)ξ =

∑m,n

amnMnT nθ ξ.

For ξ, η ∈ S(R), define

•〈ξ, η〉 =∑m,n〈ξ,MnT m

θ η〉MnT mθ ∈ Aθ.

We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14

Page 52: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Luef)

A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

is a (normalized tight) frame for L2(R).

If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then

τ̃([Eθ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩) =

k∑j=1

‖ηj‖2.

If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal

basis, thenτ̃([Eθ]) = k .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14

Page 53: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Luef)

A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

is a (normalized tight) frame for L2(R).

If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then

τ̃([Eθ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩) =

k∑j=1

‖ηj‖2.

If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal

basis, thenτ̃([Eθ]) = k .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14

Page 54: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Luef)

A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

is a (normalized tight) frame for L2(R).

If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then

τ̃([Eθ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩) =

k∑j=1

‖ηj‖2.

If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal

basis, thenτ̃([Eθ]) = k .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14

Page 55: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Luef)

A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if

{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}

is a (normalized tight) frame for L2(R).

If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then

τ̃([Eθ]) =k∑

j=1

τ(•⟨ηj , ηj

⟩) =

k∑j=1

‖ηj‖2.

If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal

basis, thenτ̃([Eθ]) = k .

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14

Page 56: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Rieffel)

Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then

τ̃([Eθ]) = θ.

If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then

θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that

Eθ ∼= Γ(E).

One can show that this is a line bundle with Chern class −1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14

Page 57: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Rieffel)

Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then

τ̃([Eθ]) = θ.

If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then

θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that

Eθ ∼= Γ(E).

One can show that this is a line bundle with Chern class −1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14

Page 58: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Rieffel)

Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then

τ̃([Eθ]) = θ.

If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then

θ = 1.

Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that

Eθ ∼= Γ(E).

One can show that this is a line bundle with Chern class −1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14

Page 59: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Rieffel)

Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then

τ̃([Eθ]) = θ.

If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then

θ = 1.Consequently Aθ ∼= C(T2).

By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that

Eθ ∼= Γ(E).

One can show that this is a line bundle with Chern class −1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14

Page 60: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Rieffel)

Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then

τ̃([Eθ]) = θ.

If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then

θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that

Eθ ∼= Γ(E).

One can show that this is a line bundle with Chern class −1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14

Page 61: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Theorem (Rieffel)

Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then

τ̃([Eθ]) = θ.

If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then

θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that

Eθ ∼= Γ(E).

One can show that this is a line bundle with Chern class −1.

Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14

Page 62: Hilbert C*-modules in harmonic analysis - Nordfjordeid 2019 · Hilbert C*-modules Definition Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module Etogether with

Ulrik Enstad

Hilbert C*-modules inharmonic analysisNordfjordeid 2019