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  • Hilbert C*-modules in harmonic analysis Nordfjordeid 2019

    Ulrik Enstad 4th July 2019

  • Hilbert C*-modules

    Definition

    Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module E 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 implies ξ = 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

  • Hilbert C*-modules Definition

    Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module E 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 implies ξ = 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

  • Hilbert C*-modules Definition

    Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module E 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 implies ξ = 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

  • Hilbert C*-modules Definition

    Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module E 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 implies ξ = 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

  • Hilbert C*-modules Definition

    Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module E 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 implies ξ = 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

  • Hilbert C*-modules Definition

    Let A be a unital C*-algebra. A left Hilbert A-module is a left A-module E 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 implies ξ = 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

  • Examples of Hilbert C*-modules

    The left A-module Ak becomes a left Hilbert A-module with the inner 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

  • Examples of Hilbert C*-modules The left A-module Ak becomes a left Hilbert A-module with the inner 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

  • Examples of Hilbert C*-modules The left A-module Ak becomes a left Hilbert A-module with the inner 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

  • Sections of vector bundles

    If A is commutative, then A ∼= C(X ) for a compact Hausdorff space X . For a Hermitian vector bundle E → X , denote by Γ(E) the set of continuous 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)〉x

    for s, t ∈ Γ(E), f ∈ C(X ).

    Theorem (Serre–Swan)

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

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

  • Sections of vector bundles If A is commutative, then A ∼= C(X ) for a compact Hausdorff space X .

    For a Hermitian vector bundle E → X , denote by Γ(E) the set of continuous 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)〉x

    for s, t ∈ Γ(E), f ∈ C(X ).

    Theorem (Serre–Swan)

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

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

  • Sections of vector bundles If A is commutative, then A ∼= C(X ) for a compact Hausdorff space X . For a Hermitian vector bundle E → X , denote by Γ(E) the set of continuous 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)〉x

    for s, t ∈ Γ(E), f ∈ C(X ).

    Theorem (Serre–Swan)

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

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

  • Sections of vector bundles If A is commutative, then A ∼= C(X ) for a compact Hausdorff space X . For a Hermitian vector bundle E → X , denote by Γ(E) the set of continuous 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)〉x

    for s, t ∈ Γ(E), f ∈ C(X ).

    Theorem (Serre–Swan)

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

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

  • Sections of vector bundles If A is commutative, then A ∼= C(X ) for a compact Hausdorff space X . For a Hermitian vector bundle E → X , denote by Γ(E) the set of continuous 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)〉x

    for s, t ∈ Γ(E), f ∈ C(X ).

    Theorem (Serre–Swan)

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

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

  • Orthonormal bases in Hilbert C*-modules?

    Any separable Hilbert space H admits a countable orthonormal basis. A left Hilbert A-module E is called countably generated if there exists a countable set S ⊆ E such that∑

    j;finite

    ajsj : aj ∈ A, sj ∈ E

     = E. Does any countably generated Hilbert C∗-module admit a countable “orthonormal basis”? No!

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

  • Orthonormal bases in Hilbert C*-modules?

    Any separable Hilbert space H admits a countable orthonormal basis.

    A left Hilbert A-module E is called countably generated if there exists a countable set S ⊆ E such that∑

    j;finite

    ajsj : a