On the existence of bases in the orbit of unitary group ...

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Transcript of On the existence of bases in the orbit of unitary group ...

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On the existence of bases in the orbit of

unitary group representations

Ulrik Enstad

Online workshop: C*-algebras and geometry of groups and semigroups

31st March 2021

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Families in the orbit of unitary group representations

• Let π : G → U(Hπ) be an irreducible, unitary group

representation of a locally compact group G on a Hilbert space

Hπ.

• Given a discrete subset Γ of G and a vector η ∈ Hπ, we are

interested in the Γ-indexed family

π(Γ)η = (π(γ)η)γ∈Γ.

Problem

Given Γ and η as above, what can we say about the “spanning”

and “linear independence” properties of the family π(Γ)η in Hπ?

• Intuition: For π(Γ)η to “span” Hπ, Γ must be “sufficiently

dense”, and for π(Γ)η to be “linearly independent”, Γ must be

“sufficiently sparse”.

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Page 5: On the existence of bases in the orbit of unitary group ...

Families in the orbit of unitary group representations

• Let π : G → U(Hπ) be an irreducible, unitary group

representation of a locally compact group G on a Hilbert space

Hπ.

• Given a discrete subset Γ of G and a vector η ∈ Hπ, we are

interested in the Γ-indexed family

π(Γ)η = (π(γ)η)γ∈Γ.

Problem

Given Γ and η as above, what can we say about the “spanning”

and “linear independence” properties of the family π(Γ)η in Hπ?

• Intuition: For π(Γ)η to “span” Hπ, Γ must be “sufficiently

dense”, and for π(Γ)η to be “linearly independent”, Γ must be

“sufficiently sparse”.

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Page 6: On the existence of bases in the orbit of unitary group ...

Families in the orbit of unitary group representations

• Let π : G → U(Hπ) be an irreducible, unitary group

representation of a locally compact group G on a Hilbert space

Hπ.

• Given a discrete subset Γ of G and a vector η ∈ Hπ, we are

interested in the Γ-indexed family

π(Γ)η = (π(γ)η)γ∈Γ.

Problem

Given Γ and η as above, what can we say about the “spanning”

and “linear independence” properties of the family π(Γ)η in Hπ?

• Intuition: For π(Γ)η to “span” Hπ, Γ must be “sufficiently

dense”, and for π(Γ)η to be “linearly independent”, Γ must be

“sufficiently sparse”.

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Page 7: On the existence of bases in the orbit of unitary group ...

Families in the orbit of unitary group representations

• Let π : G → U(Hπ) be an irreducible, unitary group

representation of a locally compact group G on a Hilbert space

Hπ.

• Given a discrete subset Γ of G and a vector η ∈ Hπ, we are

interested in the Γ-indexed family

π(Γ)η = (π(γ)η)γ∈Γ.

Problem

Given Γ and η as above, what can we say about the “spanning”

and “linear independence” properties of the family π(Γ)η in Hπ?

• Intuition: For π(Γ)η to “span” Hπ, Γ must be “sufficiently

dense”, and for π(Γ)η to be “linearly independent”, Γ must be

“sufficiently sparse”.

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Page 8: On the existence of bases in the orbit of unitary group ...

Families in the orbit of unitary group representations

• Let π : G → U(Hπ) be an irreducible, unitary group

representation of a locally compact group G on a Hilbert space

Hπ.

• Given a discrete subset Γ of G and a vector η ∈ Hπ, we are

interested in the Γ-indexed family

π(Γ)η = (π(γ)η)γ∈Γ.

Problem

Given Γ and η as above, what can we say about the “spanning”

and “linear independence” properties of the family π(Γ)η in Hπ?

• Intuition: For π(Γ)η to “span” Hπ, Γ must be “sufficiently

dense”, and for π(Γ)η to be “linearly independent”, Γ must be

“sufficiently sparse”. 1

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Spanning and linear independence

• ••

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• An orthonormal basis (ej)j∈J is characterized by the followingtwo properties:

• Parseval’s identity:

∀ ξ ∈ H :∑j∈J

|〈ξ, ej〉|2 = ‖ξ‖2.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• An orthonormal basis (ej)j∈J is characterized by the followingtwo properties:

• Parseval’s identity:

∀ ξ ∈ H :∑j∈J

|〈ξ, ej〉|2 = ‖ξ‖2.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• An orthonormal basis (ej)j∈J is characterized by the followingtwo properties:

• Parseval’s identity:

∀ ξ ∈ H :∑j∈J

|〈ξ, ej〉|2 = ‖ξ‖2.

• Orthonormality:

∀ (cj)j ∈ `2(J) :∥∥∥∑

j∈J

cjej

∥∥∥2

= ‖(cj)j‖22.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• Deforming Parseval’s identity and orthonormality, we get thefollowing definitions:

• Parseval’s identity:

∀ ξ ∈ H :∑j∈J

|〈ξ, ej〉|2 = ‖ξ‖2.

• Orthonormality:

∀ (cj)j ∈ `2(J) :∥∥∥∑

j∈J

cjej

∥∥∥2

= ‖(cj)j‖22.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• Deforming Parseval’s identity and orthonormality, we get thefollowing definitions:

• (ej)j is a frame if there exist A,B > 0 such that

∀ ξ ∈ H : A‖ξ‖2 ≤∑j∈J

|〈ξ, ej〉|2 ≤ B‖ξ‖2.

• Orthonormality:

∀ (cj)j ∈ `2(J) :∥∥∥∑

j∈J

cjej

∥∥∥2

= ‖(cj)j‖22.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• Deforming Parseval’s identity and orthonormality, we get thefollowing definitions:

• (ej)j is a frame if there exist A,B > 0 such that

∀ ξ ∈ H : A‖ξ‖2 ≤∑j∈J

|〈ξ, ej〉|2 ≤ B‖ξ‖2.

• (ej)j is a Riesz sequence if there exist A,B > 0 such that

∀ (cj)j ∈ `2(J) : A‖(cj)j‖22 ≤

∥∥∥∑j∈J

cjej

∥∥∥2

≤ B‖(cj)j‖22.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Spanning and linear independence

• Deforming Parseval’s identity and orthonormality, we get thefollowing definitions:

• (ej)j is a frame if there exist A,B > 0 such that

∀ ξ ∈ H : A‖ξ‖2 ≤∑j∈J

|〈ξ, ej〉|2 ≤ B‖ξ‖2.

• (ej)j is a Riesz sequence if there exist A,B > 0 such that

∀ (cj)j ∈ `2(J) : A‖(cj)j‖22 ≤

∥∥∥∑j∈J

cjej

∥∥∥2

≤ B‖(cj)j‖22.

• (ej)j is a Riesz basis if it is both a frame and a Riesz sequence.

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Motivating example

1.

2.

3. In this context, families (π(γ)η)γ∈Γ for and a discrete subset

are known as Gabor systems, and have been extensively studied

in Gabor analysis.

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Motivating example

1. Let G = R2d ∼= Rd × Rd and let π : R2d → U(L2(Rd)) be the

Heisenberg representation given by

π(x , ω)ξ(t) = e2πiω·tξ(t−x) for (x , ω) ∈ R2d and ξ ∈ L2(Rd).

2. π is a σ-projective representation:

π(x , ω)π(x ′, ω′) = e−2πix ·ω′π(x + x ′, ω + ω′),

with associated 2-cocycle σ((x , ω), (x ′, ω′)) = e−2πix ·ω′.

3. In this context, families (π(γ)η)γ∈Γ for and a discrete subset

are known as Gabor systems, and have been extensively studied

in Gabor analysis.

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Motivating example

1. Let G = R2d ∼= Rd × Rd and let π : R2d → U(L2(Rd)) be the

Heisenberg representation given by

π(x , ω)ξ(t) = e2πiω·tξ(t−x) for (x , ω) ∈ R2d and ξ ∈ L2(Rd).

2. π is a σ-projective representation:

π(x , ω)π(x ′, ω′) = e−2πix ·ω′π(x + x ′, ω + ω′),

with associated 2-cocycle σ((x , ω), (x ′, ω′)) = e−2πix ·ω′.

3. In this context, families (π(γ)η)γ∈Γ for and a discrete subset

are known as Gabor systems, and have been extensively studied

in Gabor analysis.

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Page 20: On the existence of bases in the orbit of unitary group ...

Motivating example

1. Let G = R2d ∼= Rd × Rd and let π : R2d → U(L2(Rd)) be the

Heisenberg representation given by

π(x , ω)ξ(t) = e2πiω·tξ(t−x) for (x , ω) ∈ R2d and ξ ∈ L2(Rd).

2. π is a σ-projective representation:

π(x , ω)π(x ′, ω′) = e−2πix ·ω′π(x + x ′, ω + ω′),

with associated 2-cocycle σ((x , ω), (x ′, ω′)) = e−2πix ·ω′.

3. In this context, families (π(γ)η)γ∈Γ for η ∈ L2(Rd) and a

discrete subset Γ ⊆ R2d are known as Gabor systems, and

have been extensively studied in Gabor analysis.

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Motivating example

1. Let G = A× A for a locally compact abelian group A and let

π : G → U(L2(A)) be the Heisenberg representation given by

π(x , ω)ξ(t) = ω(t)ξ(x−1t) for (x , ω) ∈ G × G and ξ ∈ L2(A).

2. π is a σ-projective representation:

π(x , ω)π(x ′, ω′) = ω′(x)π(xx ′, ωω′),

with associated 2-cocycle σ((x , ω), (x ′, ω′)) = ω′(x).

3. In this context, families (π(γ)η)γ∈Γ for η ∈ L2(A) and a

discrete subset Γ ⊆ G × G are known as Gabor systems, and

have been extensively studied in Gabor analysis.

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Square-integrable representations

• For the rest of the talk, we assume that G is a unimodular,

second countable, locally compact group, and that π is a

σ-projective, irreducible, unitary representation of G which is

square-integrable, i.e., there exist nonzero ξ, η ∈ Hπ such that∫G|〈ξ, π(x)η〉|2 dx <∞.

• Under these assumptions, the following orthogonality relations

hold for all ξ, ξ′, η, η′ ∈ Hπ:∫G〈ξ, π(x)η〉〈ξ′, π(x)η′〉 dx = d−1

π 〈ξ, ξ′〉〈η, η′〉.

• The number dπ is called the formal dimension of π and

depends on the choice of Haar measure on G .

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Square-integrable representations

• For the rest of the talk, we assume that G is a unimodular,

second countable, locally compact group, and that π is a

σ-projective, irreducible, unitary representation of G which is

square-integrable, i.e., there exist nonzero ξ, η ∈ Hπ such that∫G|〈ξ, π(x)η〉|2 dx <∞.

• Under these assumptions, the following orthogonality relations

hold for all ξ, ξ′, η, η′ ∈ Hπ:∫G〈ξ, π(x)η〉〈ξ′, π(x)η′〉 dx = d−1

π 〈ξ, ξ′〉〈η, η′〉.

• The number dπ is called the formal dimension of π and

depends on the choice of Haar measure on G .

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Page 24: On the existence of bases in the orbit of unitary group ...

Square-integrable representations

• For the rest of the talk, we assume that G is a unimodular,

second countable, locally compact group, and that π is a

σ-projective, irreducible, unitary representation of G which is

square-integrable, i.e., there exist nonzero ξ, η ∈ Hπ such that∫G|〈ξ, π(x)η〉|2 dx <∞.

• Under these assumptions, the following orthogonality relations

hold for all ξ, ξ′, η, η′ ∈ Hπ:∫G〈ξ, π(x)η〉〈ξ′, π(x)η′〉 dx = d−1

π 〈ξ, ξ′〉〈η, η′〉.

• The number dπ is called the formal dimension of π and

depends on the choice of Haar measure on G .

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Page 25: On the existence of bases in the orbit of unitary group ...

Square-integrable representations

• For the rest of the talk, we assume that G is a unimodular,

second countable, locally compact group, and that π is a

σ-projective, irreducible, unitary representation of G which is

square-integrable, i.e., there exist nonzero ξ, η ∈ Hπ such that∫G|〈ξ, π(x)η〉|2 dx <∞.

• Under these assumptions, the following orthogonality relations

hold for all ξ, ξ′, η, η′ ∈ Hπ:∫G〈ξ, π(x)η〉〈ξ′, π(x)η′〉 dx = d−1

π 〈ξ, ξ′〉〈η, η′〉.

• The number dπ is called the formal dimension of π and

depends on the choice of Haar measure on G .

4

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A density theorem for lattices

• A lattice in G is a discrete subgroup Γ with finite covolume

vol(G/Γ).

Theorem (Romero–Van Velthoven 2020)

With π a σ-projective unitary representation of G as before, let Γ

be a lattice in G . Then the following hold for η ∈ Hπ:

1. If π(Γ)η is a frame for Hπ, then dπ vol(G/Γ) ≤ 1.

2. If π(Γ)η is a Riesz sequence for Hπ, then dπ vol(G/Γ) ≥ 1.

3. If π(Γ)η is a Riesz basis for Hπ, then dπ vol(G/Γ) = 1.

• The density theorem has a long history for the Heisenberg

representation of R2d .

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Page 27: On the existence of bases in the orbit of unitary group ...

A density theorem for lattices

• A lattice in G is a discrete subgroup Γ with finite covolume

vol(G/Γ).

Theorem (Romero–Van Velthoven 2020)

With π a σ-projective unitary representation of G as before, let Γ

be a lattice in G . Then the following hold for η ∈ Hπ:

1. If π(Γ)η is a frame for Hπ, then dπ vol(G/Γ) ≤ 1.

2. If π(Γ)η is a Riesz sequence for Hπ, then dπ vol(G/Γ) ≥ 1.

3. If π(Γ)η is a Riesz basis for Hπ, then dπ vol(G/Γ) = 1.

• The density theorem has a long history for the Heisenberg

representation of R2d .

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Page 28: On the existence of bases in the orbit of unitary group ...

A density theorem for lattices

• A lattice in G is a discrete subgroup Γ with finite covolume

vol(G/Γ).

Theorem (Romero–Van Velthoven 2020)

With π a σ-projective unitary representation of G as before, let Γ

be a lattice in G . Then the following hold for η ∈ Hπ:

1. If π(Γ)η is a frame for Hπ, then dπ vol(G/Γ) ≤ 1.

2. If π(Γ)η is a Riesz sequence for Hπ, then dπ vol(G/Γ) ≥ 1.

3. If π(Γ)η is a Riesz basis for Hπ, then dπ vol(G/Γ) = 1.

• The density theorem has a long history for the Heisenberg

representation of R2d .

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Page 29: On the existence of bases in the orbit of unitary group ...

A density theorem for lattices

• A lattice in G is a discrete subgroup Γ with finite covolume

vol(G/Γ).

Theorem (Romero–Van Velthoven 2020)

With π a σ-projective unitary representation of G as before, let Γ

be a lattice in G . Then the following hold for η ∈ Hπ:

1. If π(Γ)η is a frame for Hπ, then dπ vol(G/Γ) ≤ 1.

2. If π(Γ)η is a Riesz sequence for Hπ, then dπ vol(G/Γ) ≥ 1.

3. If π(Γ)η is a Riesz basis for Hπ, then dπ vol(G/Γ) = 1.

• The density theorem has a long history for the Heisenberg

representation of R2d .

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What about existence?

1. If e.g. dπ vol(G/Γ) ≤ 1, does there exist η ∈ Hπ such that

π(Γ)η is a frame for Hπ?

2. Not always: Counter-examples can be found e.g. in the

representation theory of SL2(R).

3. An element γ ∈ Γ is called σ-regular if σ(γ, γ′) = σ(γ′, γ)

whenever γγ′ = γ′γ. We say that (Γ, σ) satisfies Kleppner’s

condition if every nontrivial σ-regular conjugacy class is infinite.

Romero–Van Velthoven (2020)

If (Γ, σ) satisfies Kleppner’s condition, there exists η ∈ Hπ such

that π(Γ)η is a frame if and only if dπ vol(G/Γ) ≤ 1. Analagous

statements hold for Riesz sequences and Riesz bases.

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Page 31: On the existence of bases in the orbit of unitary group ...

What about existence?

1. If e.g. dπ vol(G/Γ) ≤ 1, does there exist η ∈ Hπ such that

π(Γ)η is a frame for Hπ?

2. Not always: Counter-examples can be found e.g. in the

representation theory of SL2(R).

3. An element γ ∈ Γ is called σ-regular if σ(γ, γ′) = σ(γ′, γ)

whenever γγ′ = γ′γ. We say that (Γ, σ) satisfies Kleppner’s

condition if every nontrivial σ-regular conjugacy class is infinite.

Romero–Van Velthoven (2020)

If (Γ, σ) satisfies Kleppner’s condition, there exists η ∈ Hπ such

that π(Γ)η is a frame if and only if dπ vol(G/Γ) ≤ 1. Analagous

statements hold for Riesz sequences and Riesz bases.

6

Page 32: On the existence of bases in the orbit of unitary group ...

What about existence?

1. If e.g. dπ vol(G/Γ) ≤ 1, does there exist η ∈ Hπ such that

π(Γ)η is a frame for Hπ?

2. Not always: Counter-examples can be found e.g. in the

representation theory of SL2(R).

3. An element γ ∈ Γ is called σ-regular if σ(γ, γ′) = σ(γ′, γ)

whenever γγ′ = γ′γ. We say that (Γ, σ) satisfies Kleppner’s

condition if every nontrivial σ-regular conjugacy class is infinite.

Romero–Van Velthoven (2020)

If (Γ, σ) satisfies Kleppner’s condition, there exists η ∈ Hπ such

that π(Γ)η is a frame if and only if dπ vol(G/Γ) ≤ 1. Analagous

statements hold for Riesz sequences and Riesz bases.

6

Page 33: On the existence of bases in the orbit of unitary group ...

What about existence?

1. If e.g. dπ vol(G/Γ) ≤ 1, does there exist η ∈ Hπ such that

π(Γ)η is a frame for Hπ?

2. Not always: Counter-examples can be found e.g. in the

representation theory of SL2(R).

3. An element γ ∈ Γ is called σ-regular if σ(γ, γ′) = σ(γ′, γ)

whenever γγ′ = γ′γ. We say that (Γ, σ) satisfies Kleppner’s

condition if every nontrivial σ-regular conjugacy class is infinite.

Romero–Van Velthoven (2020)

If (Γ, σ) satisfies Kleppner’s condition, there exists η ∈ Hπ such

that π(Γ)η is a frame if and only if dπ vol(G/Γ) ≤ 1. Analagous

statements hold for Riesz sequences and Riesz bases.

6

Page 34: On the existence of bases in the orbit of unitary group ...

What about existence?

1. If e.g. dπ vol(G/Γ) ≤ 1, does there exist η ∈ Hπ such that

π(Γ)η is a frame for Hπ?

2. Not always: Counter-examples can be found e.g. in the

representation theory of SL2(R).

3. An element γ ∈ Γ is called σ-regular if σ(γ, γ′) = σ(γ′, γ)

whenever γγ′ = γ′γ. We say that (Γ, σ) satisfies Kleppner’s

condition if every nontrivial σ-regular conjugacy class is infinite.

Romero–Van Velthoven (2020)

If (Γ, σ) satisfies Kleppner’s condition, there exists η ∈ Hπ such

that π(Γ)η is a frame if and only if dπ vol(G/Γ) ≤ 1. Analagous

statements hold for Riesz sequences and Riesz bases.

6

Page 35: On the existence of bases in the orbit of unitary group ...

Characterization of existence

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Characterization of existence

Theorem (Bekka, 2004)

Let G be a unimodular, second countable, locally compact group

and let π be a square-integrable, irreducible, unitary representation

of G . Let Γ be a lattice in G . Let η ∈ Hπ be a unit vector. Define

a function φ ∈ `∞(Γ) by

φ(γ) =dπ|Cγ |

∫G/Γγ

〈η, π(y−1γy)η〉 d(yΓγ)

if the conjugacy class Cγ is finite, and φ(γ) = 0 otherwise. Then:

1. There exists η ∈ Hπ such that π(Γ)η is a frame if and only if

δe − φ is positive definite.

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Page 37: On the existence of bases in the orbit of unitary group ...

Characterization of existence

Theorem (Bekka, 2004)

Let G be a unimodular, second countable, locally compact group

and let π be a square-integrable, irreducible, unitary representation

of G . Let Γ be a lattice in G . Let η ∈ Hπ be a unit vector. Define

a function φ ∈ `∞(Γ) by

φ(γ) =dπ|Cγ |

∫G/Γγ

〈η, π(y−1γy)η〉 d(yΓγ)

if the conjugacy class Cγ is finite, and φ(γ) = 0 otherwise. Then:

1. There exists η ∈ Hπ such that π(Γ)η is a frame if and only if

δe − φ is positive definite.

7

Page 38: On the existence of bases in the orbit of unitary group ...

Characterization of existence

Theorem (E.)

Let G be a unimodular, second countable, locally compact group

with 2-cocycle σ and let π be a σ-projective, square-integrable,

irreducible, unitary representation of G . Let Γ be a lattice in G .

Let η ∈ Hπ be a unit vector. Define a function φ ∈ `∞(Γ) by

φ(γ) =dπ|Cγ |

∫G/Γγ

σ(γ, y)σ(y , γ)〈η, π(y−1γy)η〉 d(yΓγ)

if the conjugacy class Cγ is σ-regular and finite, and φ(γ) = 0otherwise. Then:

1. ∃ η ∈ Hπ: π(Γ)η is a frame iff δe − φ is σ-positive definite.

2. ∃ η ∈ Hπ: π(Γ)η is a Riesz sequence iff φ− δe is σ-positive definite.

3. ∃ η ∈ Hπ: π(Γ)η is a Riesz basis iff φ = δe .

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Page 39: On the existence of bases in the orbit of unitary group ...

Characterization of existence

Theorem (E.)

Let G be a unimodular, second countable, locally compact group

with 2-cocycle σ and let π be a σ-projective, square-integrable,

irreducible, unitary representation of G . Let Γ be a lattice in G .

Let η ∈ Hπ be a unit vector. Define a function φ ∈ `∞(Γ) by

φ(γ) =dπ|Cγ |

∫G/Γγ

σ(γ, y)σ(y , γ)〈η, π(y−1γy)η〉 d(yΓγ)

if the conjugacy class Cγ is σ-regular and finite, and φ(γ) = 0otherwise. Then:

1. ∃ η ∈ Hπ: π(Γ)η is a frame iff δe − φ is σ-positive definite.

2. ∃ η ∈ Hπ: π(Γ)η is a Riesz sequence iff φ− δe is σ-positive definite.

3. ∃ η ∈ Hπ: π(Γ)η is a Riesz basis iff φ = δe .

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Page 40: On the existence of bases in the orbit of unitary group ...

Characterization of existence

Theorem (E.)

Let G be a unimodular, second countable, locally compact group

with 2-cocycle σ and let π be a σ-projective, square-integrable,

irreducible, unitary representation of G . Let Γ be a lattice in G .

Let η ∈ Hπ be a unit vector. Define a function φ ∈ `∞(Γ) by

φ(γ) =dπ|Cγ |

∫G/Γγ

σ(γ, y)σ(y , γ)〈η, π(y−1γy)η〉 d(yΓγ)

if the conjugacy class Cγ is σ-regular and finite, and φ(γ) = 0otherwise. Then:

1. ∃ η ∈ Hπ: π(Γ)η is a frame iff δe − φ is σ-positive definite.

2. ∃ η ∈ Hπ: π(Γ)η is a Riesz sequence iff φ− δe is σ-positive definite.

3. ∃ η ∈ Hπ: π(Γ)η is a Riesz basis iff φ = δe .

7

Page 41: On the existence of bases in the orbit of unitary group ...

Characterization of existence

Theorem (E.)

Let G be a unimodular, second countable, locally compact group

with 2-cocycle σ and let π be a σ-projective, square-integrable,

irreducible, unitary representation of G . Let Γ be a lattice in G .

Let η ∈ Hπ be a unit vector. Define a function φ ∈ `∞(Γ) by

φ(γ) =dπ|Cγ |

∫G/Γγ

σ(γ, y)σ(y , γ)〈η, π(y−1γy)η〉 d(yΓγ)

if the conjugacy class Cγ is σ-regular and finite, and φ(γ) = 0otherwise. Then:

1. ∃ η ∈ Hπ: π(Γ)η is a frame iff δe − φ is σ-positive definite.

2. ∃ η ∈ Hπ: π(Γ)η is a Riesz sequence iff φ− δe is σ-positive definite.

3. ∃ η ∈ Hπ: π(Γ)η is a Riesz basis iff φ = δe .

7

Page 42: On the existence of bases in the orbit of unitary group ...

Consequences

• With φ as on the last slide, we have

φ(e) = dπ vol(G/Γ).

Hence we recover the density theorem.

• If

φ = dπ vol(G/Γ)δe

then the condition dπ vol(G/Γ) ≤ 1 (resp. dπ vol(G/Γ) ≥ 1) is

sufficient for the existence of a frame (resp. Riesz sequence)

π(Γ)η for some η ∈ Hπ.

• If (Γ, σ) satisfies Kleppner’s condition, then φ = dπ vol(G/Γ)δe .

Corollary (E.)

If G is abelian and (G , σ) satisfies Kleppner’s condition, then

φ = dπ vol(G/Γ)δe .

8

Page 43: On the existence of bases in the orbit of unitary group ...

Consequences

• With φ as on the last slide, we have

φ(e) = dπ vol(G/Γ).

Hence we recover the density theorem.

• If

φ = dπ vol(G/Γ)δe

then the condition dπ vol(G/Γ) ≤ 1 (resp. dπ vol(G/Γ) ≥ 1) is

sufficient for the existence of a frame (resp. Riesz sequence)

π(Γ)η for some η ∈ Hπ.

• If (Γ, σ) satisfies Kleppner’s condition, then φ = dπ vol(G/Γ)δe .

Corollary (E.)

If G is abelian and (G , σ) satisfies Kleppner’s condition, then

φ = dπ vol(G/Γ)δe .

8

Page 44: On the existence of bases in the orbit of unitary group ...

Consequences

• With φ as on the last slide, we have

φ(e) = dπ vol(G/Γ).

Hence we recover the density theorem.

• If

φ = dπ vol(G/Γ)δe

then the condition dπ vol(G/Γ) ≤ 1 (resp. dπ vol(G/Γ) ≥ 1) is

sufficient for the existence of a frame (resp. Riesz sequence)

π(Γ)η for some η ∈ Hπ.

• If (Γ, σ) satisfies Kleppner’s condition, then φ = dπ vol(G/Γ)δe .

Corollary (E.)

If G is abelian and (G , σ) satisfies Kleppner’s condition, then

φ = dπ vol(G/Γ)δe .

8

Page 45: On the existence of bases in the orbit of unitary group ...

Consequences

• With φ as on the last slide, we have

φ(e) = dπ vol(G/Γ).

Hence we recover the density theorem.

• If

φ = dπ vol(G/Γ)δe

then the condition dπ vol(G/Γ) ≤ 1 (resp. dπ vol(G/Γ) ≥ 1) is

sufficient for the existence of a frame (resp. Riesz sequence)

π(Γ)η for some η ∈ Hπ.

• If (Γ, σ) satisfies Kleppner’s condition, then φ = dπ vol(G/Γ)δe .

Corollary (E.)

If G is abelian and (G , σ) satisfies Kleppner’s condition, then

φ = dπ vol(G/Γ)δe .

8

Page 46: On the existence of bases in the orbit of unitary group ...

Consequences

• With φ as on the last slide, we have

φ(e) = dπ vol(G/Γ).

Hence we recover the density theorem.

• If

φ = dπ vol(G/Γ)δe

then the condition dπ vol(G/Γ) ≤ 1 (resp. dπ vol(G/Γ) ≥ 1) is

sufficient for the existence of a frame (resp. Riesz sequence)

π(Γ)η for some η ∈ Hπ.

• If (Γ, σ) satisfies Kleppner’s condition, then φ = dπ vol(G/Γ)δe .

Corollary (E.)

If G is abelian and (G , σ) satisfies Kleppner’s condition, then

φ = dπ vol(G/Γ)δe .

8

Page 47: On the existence of bases in the orbit of unitary group ...

Twisted group von Neumann algebras

1. Let Γ be a discrete group with 2-cocycle σ. The σ-twisted left

regular representation λσ of Γ on `2(Γ) is given by

λσ(γ)f (γ′) = σ(γ, γ−1γ′)f (γ−1γ′) for γ, γ′ ∈ Γ and f ∈ `2(Γ).

2. The σ-twisted group von Neumann algebra of Γ is

L(Γ, σ) = {λσ(γ) : γ ∈ Γ}′′ .

3. The action of Γ on Hπ via π|Γ extends to give Hπ the

structure of a left Hilbert L(Γ, σ)-module.

9

Page 48: On the existence of bases in the orbit of unitary group ...

Twisted group von Neumann algebras

1. Let Γ be a discrete group with 2-cocycle σ. The σ-twisted left

regular representation λσ of Γ on `2(Γ) is given by

λσ(γ)f (γ′) = σ(γ, γ−1γ′)f (γ−1γ′) for γ, γ′ ∈ Γ and f ∈ `2(Γ).

2. The σ-twisted group von Neumann algebra of Γ is

L(Γ, σ) = {λσ(γ) : γ ∈ Γ}′′ .

3. The action of Γ on Hπ via π|Γ extends to give Hπ the

structure of a left Hilbert L(Γ, σ)-module.

9

Page 49: On the existence of bases in the orbit of unitary group ...

Twisted group von Neumann algebras

1. Let Γ be a discrete group with 2-cocycle σ. The σ-twisted left

regular representation λσ of Γ on `2(Γ) is given by

λσ(γ)f (γ′) = σ(γ, γ−1γ′)f (γ−1γ′) for γ, γ′ ∈ Γ and f ∈ `2(Γ).

2. The σ-twisted group von Neumann algebra of Γ is

L(Γ, σ) = {λσ(γ) : γ ∈ Γ}′′ .

3. The action of Γ on Hπ via π|Γ extends to give Hπ the

structure of a left Hilbert L(Γ, σ)-module.

9

Page 50: On the existence of bases in the orbit of unitary group ...

Twisted group von Neumann algebras

1. Let Γ be a discrete group with 2-cocycle σ. The σ-twisted left

regular representation λσ of Γ on `2(Γ) is given by

λσ(γ)f (γ′) = σ(γ, γ−1γ′)f (γ−1γ′) for γ, γ′ ∈ Γ and f ∈ `2(Γ).

2. The σ-twisted group von Neumann algebra of Γ is

L(Γ, σ) = {λσ(γ) : γ ∈ Γ}′′ .

3. The action of Γ on Hπ via π|Γ extends to give Hπ the

structure of a left Hilbert L(Γ, σ)-module.

9

Page 51: On the existence of bases in the orbit of unitary group ...

Idea of proof

• If π(Γ)η is a frame, then the operator C : Hπ → `2(Γ) given by

Cξ = (〈ξ, π(γ)η〉)γ∈Γ

is bounded and injective, with closed range. It also intertwines

π|Γ and λσ|Γ.

• Conversely, if C : Hπ → `2(Γ) is an isometry that intertwines

π|Γ and λσ|Γ, then π(Γ)η is a frame, where η = Pδe (P

projection of `2(Γ) onto Hπ).

• Conclusion: There exists a frame π(Γ)η for some η ∈ Hπ if

and only if Hπ is a subrepresentation of `2(Γ). This extends to

an inclusion of Hilbert L(Γ, σ)-modules: Hπ ≤ `2(Γ).

• Hilbert modules over finite von Neumann algebras are entirely

determined by their center-valued dimension: Hπ ≤ `2(Γ) if

and only if cdimHπ ≤ cdim `2(Γ) = I .

10

Page 52: On the existence of bases in the orbit of unitary group ...

Idea of proof

• If π(Γ)η is a frame, then the operator C : Hπ → `2(Γ) given by

Cξ = (〈ξ, π(γ)η〉)γ∈Γ

is bounded and injective, with closed range. It also intertwines

π|Γ and λσ|Γ.

• Conversely, if C : Hπ → `2(Γ) is an isometry that intertwines

π|Γ and λσ|Γ, then π(Γ)η is a frame, where η = Pδe (P

projection of `2(Γ) onto Hπ).

• Conclusion: There exists a frame π(Γ)η for some η ∈ Hπ if

and only if Hπ is a subrepresentation of `2(Γ). This extends to

an inclusion of Hilbert L(Γ, σ)-modules: Hπ ≤ `2(Γ).

• Hilbert modules over finite von Neumann algebras are entirely

determined by their center-valued dimension: Hπ ≤ `2(Γ) if

and only if cdimHπ ≤ cdim `2(Γ) = I .

10

Page 53: On the existence of bases in the orbit of unitary group ...

Idea of proof

• If π(Γ)η is a frame, then the operator C : Hπ → `2(Γ) given by

Cξ = (〈ξ, π(γ)η〉)γ∈Γ

is bounded and injective, with closed range. It also intertwines

π|Γ and λσ|Γ.

• Conversely, if C : Hπ → `2(Γ) is an isometry that intertwines

π|Γ and λσ|Γ, then π(Γ)η is a frame, where η = Pδe (P

projection of `2(Γ) onto Hπ).

• Conclusion: There exists a frame π(Γ)η for some η ∈ Hπ if

and only if Hπ is a subrepresentation of `2(Γ). This extends to

an inclusion of Hilbert L(Γ, σ)-modules: Hπ ≤ `2(Γ).

• Hilbert modules over finite von Neumann algebras are entirely

determined by their center-valued dimension: Hπ ≤ `2(Γ) if

and only if cdimHπ ≤ cdim `2(Γ) = I .

10

Page 54: On the existence of bases in the orbit of unitary group ...

Idea of proof

• If π(Γ)η is a frame, then the operator C : Hπ → `2(Γ) given by

Cξ = (〈ξ, π(γ)η〉)γ∈Γ

is bounded and injective, with closed range. It also intertwines

π|Γ and λσ|Γ.

• Conversely, if C : Hπ → `2(Γ) is an isometry that intertwines

π|Γ and λσ|Γ, then π(Γ)η is a frame, where η = Pδe (P

projection of `2(Γ) onto Hπ).

• Conclusion: There exists a frame π(Γ)η for some η ∈ Hπ if

and only if Hπ is a subrepresentation of `2(Γ). This extends to

an inclusion of Hilbert L(Γ, σ)-modules: Hπ ≤ `2(Γ).

• Hilbert modules over finite von Neumann algebras are entirely

determined by their center-valued dimension: Hπ ≤ `2(Γ) if

and only if cdimHπ ≤ cdim `2(Γ) = I .

10

Page 55: On the existence of bases in the orbit of unitary group ...

Idea of proof

• If π(Γ)η is a frame, then the operator C : Hπ → `2(Γ) given by

Cξ = (〈ξ, π(γ)η〉)γ∈Γ

is bounded and injective, with closed range. It also intertwines

π|Γ and λσ|Γ.

• Conversely, if C : Hπ → `2(Γ) is an isometry that intertwines

π|Γ and λσ|Γ, then π(Γ)η is a frame, where η = Pδe (P

projection of `2(Γ) onto Hπ).

• Conclusion: There exists a frame π(Γ)η for some η ∈ Hπ if

and only if Hπ is a subrepresentation of `2(Γ). This extends to

an inclusion of Hilbert L(Γ, σ)-modules: Hπ ≤ `2(Γ).

• Hilbert modules over finite von Neumann algebras are entirely

determined by their center-valued dimension: Hπ ≤ `2(Γ) if

and only if cdimHπ ≤ cdim `2(Γ) = I .

10

Page 56: On the existence of bases in the orbit of unitary group ...

Additional regularity of η

• So far, we have only considered the question of when there

exists η ∈ Hπ such that π(Γ)η is a frame.

• Often, one wants additional regularity of η, e.g., through decay

and/or smoothness of the matrix coefficients of η.

• Set

H1π =

{ξ ∈ Hπ :

∫G|〈ξ, π(x)ξ〉| dx <∞

}.

• What can we say about existence of a frame of the form π(Γ)η

for some η ∈ H1π?

11

Page 57: On the existence of bases in the orbit of unitary group ...

Additional regularity of η

• So far, we have only considered the question of when there

exists η ∈ Hπ such that π(Γ)η is a frame.

• Often, one wants additional regularity of η, e.g., through decay

and/or smoothness of the matrix coefficients of η.

• Set

H1π =

{ξ ∈ Hπ :

∫G|〈ξ, π(x)ξ〉| dx <∞

}.

• What can we say about existence of a frame of the form π(Γ)η

for some η ∈ H1π?

11

Page 58: On the existence of bases in the orbit of unitary group ...

Additional regularity of η

• So far, we have only considered the question of when there

exists η ∈ Hπ such that π(Γ)η is a frame.

• Often, one wants additional regularity of η, e.g., through decay

and/or smoothness of the matrix coefficients of η.

• Set

H1π =

{ξ ∈ Hπ :

∫G|〈ξ, π(x)ξ〉| dx <∞

}.

• What can we say about existence of a frame of the form π(Γ)η

for some η ∈ H1π?

11

Page 59: On the existence of bases in the orbit of unitary group ...

Additional regularity of η

• So far, we have only considered the question of when there

exists η ∈ Hπ such that π(Γ)η is a frame.

• Often, one wants additional regularity of η, e.g., through decay

and/or smoothness of the matrix coefficients of η.

• Set

H1π =

{ξ ∈ Hπ :

∫G|〈ξ, π(x)ξ〉| dx <∞

}.

• What can we say about existence of a frame of the form π(Γ)η

for some η ∈ H1π?

11

Page 60: On the existence of bases in the orbit of unitary group ...

Additional regularity of η

• So far, we have only considered the question of when there

exists η ∈ Hπ such that π(Γ)η is a frame.

• Often, one wants additional regularity of η, e.g., through decay

and/or smoothness of the matrix coefficients of η.

• Set

H1π =

{ξ ∈ Hπ :

∫G|〈ξ, π(x)ξ〉| dx <∞

}.

• What can we say about existence of a frame of the form π(Γ)η

for some η ∈ H1π?

11

Page 61: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 62: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 63: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 64: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 65: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 66: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 67: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

Consequently, there are no Riesz bases of the form π(Γ)η for

η ∈ H1π.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland).

12

Page 68: On the existence of bases in the orbit of unitary group ...

The Balian–Low Theorem

• Let π be the Heisenberg representation of G = R2d .

• The space H1π is in this setting known as the Feichtinger

algebra.

Theorem (Feichtinger–Kaiblinger 2002)

Let Γ be a lattice in R2d , and let η ∈ H1π. Then the following

hold:

1. If π(Γ)η is a frame, then vol(R2d/Γ) < 1.

2. If π(Γ)η is a Riesz sequence, then vol(R2d/Γ) > 1.

Consequently, there are no Riesz bases of the form π(Γ)η for

η ∈ H1π.

• Generalizes to locally compact abelian groups A with

noncompact identity component (E., Jakobsen, Luef, Omland). 12

Page 69: On the existence of bases in the orbit of unitary group ...

Converses to the Balian–Low Theorem

Theorem (Jakobsen–Luef 2018)

Let Γ = MZ2d be a lattice in R2d for M ∈ GL2d(R) such that the

matrix MtJM contains at least one irrational entry. Here J

denotes the standard symplectic 2n × 2n matrix. Then the

following hold:

1. If vol(R2d/Γ) < 1, then there exists η ∈ H1π such that π(Γ)η is

a frame.

2. If vol(R2d/Γ) > 1, then there exists η ∈ H1π such that π(Γ)η is

a Riesz sequence.

• Frames π(Γ)η with η ∈ H1π can be interpreted as single

generators of Rieffel’s Heisenberg module over C ∗(Γ, σ) ∼= AΘ,

Θ = MtJM.

13

Page 70: On the existence of bases in the orbit of unitary group ...

Converses to the Balian–Low Theorem

Theorem (Jakobsen–Luef 2018)

Let Γ = MZ2d be a lattice in R2d for M ∈ GL2d(R) such that the

matrix MtJM contains at least one irrational entry. Here J

denotes the standard symplectic 2n × 2n matrix. Then the

following hold:

1. If vol(R2d/Γ) < 1, then there exists η ∈ H1π such that π(Γ)η is

a frame.

2. If vol(R2d/Γ) > 1, then there exists η ∈ H1π such that π(Γ)η is

a Riesz sequence.

• Frames π(Γ)η with η ∈ H1π can be interpreted as single

generators of Rieffel’s Heisenberg module over C ∗(Γ, σ) ∼= AΘ,

Θ = MtJM.

13

Page 71: On the existence of bases in the orbit of unitary group ...

Converses to the Balian–Low Theorem

Theorem (Jakobsen–Luef 2018)

Let Γ = MZ2d be a lattice in R2d for M ∈ GL2d(R) such that the

matrix MtJM contains at least one irrational entry. Here J

denotes the standard symplectic 2n × 2n matrix. Then the

following hold:

1. If vol(R2d/Γ) < 1, then there exists η ∈ H1π such that π(Γ)η is

a frame.

2. If vol(R2d/Γ) > 1, then there exists η ∈ H1π such that π(Γ)η is

a Riesz sequence.

• Frames π(Γ)η with η ∈ H1π can be interpreted as single

generators of Rieffel’s Heisenberg module over C ∗(Γ, σ) ∼= AΘ,

Θ = MtJM.

13

Page 72: On the existence of bases in the orbit of unitary group ...

Converses to the Balian–Low Theorem

Theorem (Jakobsen–Luef 2018)

Let Γ = MZ2d be a lattice in R2d for M ∈ GL2d(R) such that the

matrix MtJM contains at least one irrational entry. Here J

denotes the standard symplectic 2n × 2n matrix. Then the

following hold:

1. If vol(R2d/Γ) < 1, then there exists η ∈ H1π such that π(Γ)η is

a frame.

2. If vol(R2d/Γ) > 1, then there exists η ∈ H1π such that π(Γ)η is

a Riesz sequence.

• Frames π(Γ)η with η ∈ H1π can be interpreted as single

generators of Rieffel’s Heisenberg module over C ∗(Γ, σ) ∼= AΘ,

Θ = MtJM.

13

Page 73: On the existence of bases in the orbit of unitary group ...

Converses to the Balian–Low Theorem

Theorem (Jakobsen–Luef 2018)

Let Γ = MZ2d be a lattice in R2d for M ∈ GL2d(R) such that the

matrix MtJM contains at least one irrational entry. Here J

denotes the standard symplectic 2n × 2n matrix. Then the

following hold:

1. If vol(R2d/Γ) < 1, then there exists η ∈ H1π such that π(Γ)η is

a frame.

2. If vol(R2d/Γ) > 1, then there exists η ∈ H1π such that π(Γ)η is

a Riesz sequence.

• Frames π(Γ)η with η ∈ H1π can be interpreted as single

generators of Rieffel’s Heisenberg module over C ∗(Γ, σ) ∼= AΘ,

Θ = MtJM.

13

Page 74: On the existence of bases in the orbit of unitary group ...

Beyond the lattice case

• State of the art: Discrete subsets Γ ⊆ G without any group

structure.

• Suppose G is compactly generated. We define the lower and

upper Beurling densities of Γ to be

D−(Γ) = lim infr→∞

infx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

D+(Γ) = lim supr→∞

supx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

where the balls Br (x) are defined using a choice of word metric

on G .

• If Γ is a lattice in G , then D−(Γ) = D+(Γ) = 1/ vol(G/Γ).

14

Page 75: On the existence of bases in the orbit of unitary group ...

Beyond the lattice case

• State of the art: Discrete subsets Γ ⊆ G without any group

structure.

• Suppose G is compactly generated. We define the lower and

upper Beurling densities of Γ to be

D−(Γ) = lim infr→∞

infx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

D+(Γ) = lim supr→∞

supx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

where the balls Br (x) are defined using a choice of word metric

on G .

• If Γ is a lattice in G , then D−(Γ) = D+(Γ) = 1/ vol(G/Γ).

14

Page 76: On the existence of bases in the orbit of unitary group ...

Beyond the lattice case

• State of the art: Discrete subsets Γ ⊆ G without any group

structure.

• Suppose G is compactly generated. We define the lower and

upper Beurling densities of Γ to be

D−(Γ) = lim infr→∞

infx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

D+(Γ) = lim supr→∞

supx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

where the balls Br (x) are defined using a choice of word metric

on G .

• If Γ is a lattice in G , then D−(Γ) = D+(Γ) = 1/ vol(G/Γ).

14

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Beyond the lattice case

• State of the art: Discrete subsets Γ ⊆ G without any group

structure.

• Suppose G is compactly generated. We define the lower and

upper Beurling densities of Γ to be

D−(Γ) = lim infr→∞

infx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

D+(Γ) = lim supr→∞

supx∈G

|Γ ∩ Br (x)|µ(Br (x))

,

where the balls Br (x) are defined using a choice of word metric

on G .

• If Γ is a lattice in G , then D−(Γ) = D+(Γ) = 1/ vol(G/Γ).

14

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Density and Balian–Low Theorems for irregular point sets

Theorem (Fuhr–Grochenig–Haimi–Klotz–Romero 2017)

Let G be a compactly generated, locally compact group with

polynomial growth, and let π be a square-integrable, irreducible,

unitary representation of G . Let Γ ⊆ G be discrete and η ∈ Hπ.

Then:

1. If π(Γ)η is a frame for Hπ, then D−(Γ) ≥ dπ.

2. If π(Γ)η is a Riesz sequence for Hπ, then D+(Γ) ≤ dπ.

Theorem (Grochenig–Romero–Van Velthoven 2019)

Let G be a homogeneous Lie group and let π be a

square-integrable, irreducible, unitary representation of G . Let

Γ ⊆ G be discrete and η ∈ H1π. Then:

1. If π(Γ)η is a frame for Hπ, then D−(Γ) > dπ.

2. If π(Γ)η is a Riesz sequence for Hπ, then D+(Γ) < dπ.

15

Page 79: On the existence of bases in the orbit of unitary group ...

Density and Balian–Low Theorems for irregular point sets

Theorem (Fuhr–Grochenig–Haimi–Klotz–Romero 2017)

Let G be a compactly generated, locally compact group with

polynomial growth, and let π be a square-integrable, irreducible,

unitary representation of G . Let Γ ⊆ G be discrete and η ∈ Hπ.

Then:

1. If π(Γ)η is a frame for Hπ, then D−(Γ) ≥ dπ.

2. If π(Γ)η is a Riesz sequence for Hπ, then D+(Γ) ≤ dπ.

Theorem (Grochenig–Romero–Van Velthoven 2019)

Let G be a homogeneous Lie group and let π be a

square-integrable, irreducible, unitary representation of G . Let

Γ ⊆ G be discrete and η ∈ H1π. Then:

1. If π(Γ)η is a frame for Hπ, then D−(Γ) > dπ.

2. If π(Γ)η is a Riesz sequence for Hπ, then D+(Γ) < dπ.

15

Page 80: On the existence of bases in the orbit of unitary group ...

Density and Balian–Low Theorems for irregular point sets

Theorem (Fuhr–Grochenig–Haimi–Klotz–Romero 2017)

Let G be a compactly generated, locally compact group with

polynomial growth, and let π be a square-integrable, irreducible,

unitary representation of G . Let Γ ⊆ G be discrete and η ∈ Hπ.

Then:

1. If π(Γ)η is a frame for Hπ, then D−(Γ) ≥ dπ.

2. If π(Γ)η is a Riesz sequence for Hπ, then D+(Γ) ≤ dπ.

Theorem (Grochenig–Romero–Van Velthoven 2019)

Let G be a homogeneous Lie group and let π be a

square-integrable, irreducible, unitary representation of G . Let

Γ ⊆ G be discrete and η ∈ H1π. Then:

1. If π(Γ)η is a frame for Hπ, then D−(Γ) > dπ.

2. If π(Γ)η is a Riesz sequence for Hπ, then D+(Γ) < dπ. 15

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References

1. B. Bekka: Square-integrable representations, von Neumann algebras and

an application to Gabor analysis, J. Fourier Anal. Appl. 10 (2004), no. 4,

325–349.

2. U. Enstad: The density theorem for projective representations via twisted

group von Neumann algebras, preprint, arXiv:2103.14467.

3. H. Fuhr, K. Grochenig, A. Haimi, A. Klotz, J. L. Romero: Sampling and

interpolation in reproducing kernel Hilbert spaces, J. Lond. Math. Soc. (2)

96 (2017), no. 3, 663–686.

4. K. Grochenig, J. L. Romero, J. T. van Velthoven: Balian–Low Type

theorems on homogeneous groups, Anal. Math. 46 (2020), no. 3, 483–515.

5. M. S. Jakobsen, F. Luef: Duality of Gabor frames and Heisenberg modules,

J. Noncommut. Geom. 14 (2020), no. 4, 1445–1550.

6. J. L. Romero, J. T. van Velthoven: The density theorem for discrete series

representations restricted to lattices, preprint, arXiv:2003.08347.

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