universal continuous calculus for su...

IntroductionOverview

Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras

Continuous Calculi

Universal Continuous Calculus for Su∗-AlgebrasBased on arXiv preprints 1811.04878 and 1901.04076

Matthias Schotz

16.04.2019

Freiburg 2019

Algebraic and geometric aspects in Quantum Field Theory

Matthias Schotz Universal Continuous Calculus for Su∗-Algebras

Continuous Calculi

Introduction

Consider the following (unital) ∗-algebras,which one is the “most unbounded”?

C (R), the continuous complex-valued functions on R.

Cpol(R) :={f ∈ C (R)

∣∣ ∃p∈R[x]∀t∈R : |f (t)| ≤ p(t)}

,its (unital) ∗-subalgebra of polynomially bounded functions.

Cbd(R) :={f ∈ C (R)

∣∣ ∃λ∈[0,∞[∀t∈R : |f (t)| ≤ λ}

,its (unital) ∗-subalgebra of uniformly bounded functions.

It’s Cpol(R)...One might want to have an abstract approach to such ∗-algebras.

Continuous Calculi

Introduction

Consider the following (unital) ∗-algebras,which one is the “most unbounded”?

C (R), the continuous complex-valued functions on R.

Cpol(R) :={f ∈ C (R)

∣∣ ∃p∈R[x]∀t∈R : |f (t)| ≤ p(t)}

,its (unital) ∗-subalgebra of polynomially bounded functions.

Cbd(R) :={f ∈ C (R)

∣∣ ∃λ∈[0,∞[∀t∈R : |f (t)| ≤ λ}

,its (unital) ∗-subalgebra of uniformly bounded functions.

It’s Cpol(R)...One might want to have an abstract approach to such ∗-algebras.

Continuous Calculi

Overview

How ist Cpol(R) the “most unbounded” one?

Why is this important?

Ordered ∗-Algebras and Su∗-Algebras

Continuous Calculi

Recall: A positive linear functional on a ∗-algebra A is alinear ω : A → C fulfilling ω(a∗a) ≥ 0 for all a ∈ A.

Let ω be a positive linear functional on Cbd(R).Cbd(R) is C∗-Algebra with supremums-norm ‖ · ‖∞,R,

so |ω(f n)| ≤ ω(1)‖f ‖n∞,R for all f ∈ Cbd(R), n ∈ N.

Let ω be a positive linear functional on C (R).Then ω is integral over a Borel measure on some compact K ⊆ R,so |ω(f n)| ≤ ω(1)‖f ‖n∞,K for all f ∈ C (R), n ∈ N.

Let ω be the positive linear functional on Cpol(R) given by

ω(f ) :=

f (x) e−πx2

dx , f ∈ Cpol(R),

and f ∈ Cpol(R), x 7→ f (x) := x2. Then

ω(f n)

= Γ(n + 1/2)

for all n ∈ N grows faster than exponentially with n.

Continuous Calculi

ω(f ) :=

f (x) e−πx2

dx , f ∈ Cpol(R),

ω(f n)

= Γ(n + 1/2)

Continuous Calculi

ω(f ) :=

f (x) e−πx2

dx , f ∈ Cpol(R),

ω(f n)

= Γ(n + 1/2)

Continuous Calculi

ω(f ) :=

f (x) e−πx2

dx , f ∈ Cpol(R),

ω(f n)

= Γ(n + 1/2)

Continuous Calculi

More general:

All GNS-representations of Cbd(R) and C (R) are by bounded operators,but Cpol(R) has unbounded GNS-representations.

That’s important:

Similarly:

If A is a unital ∗-algebra and P,Q ∈ A fulfil

[P , Q ] := PQ − QP = λ1 ,

then A has only unbounded GNS-representations(except for the trivial one).

Continuous Calculi

More general:

All GNS-representations of Cbd(R) and C (R) are by bounded operators,but Cpol(R) has unbounded GNS-representations.

That’s important:

Similarly:

If A is a unital ∗-algebra and P,Q ∈ A fulfil

[P , Q ] := PQ − QP = λ1 ,

then A has only unbounded GNS-representations(except for the trivial one).

Continuous Calculi

Generalizing C ∗-algebras

As complete locally convex ∗-algebras:

C∗-algebras ⊆

Banach∗-algebras

pro-C∗-algebras

⊆ complete lmc∗-algebras

But their (continuous) GNS-representations are all bounded.

As ordered ∗-algebras:

C∗-algebras ⊆ Su∗-algebras ⊆ Archimedean ordered ∗-algebras

Example: Cpol(R) is a Su∗-algebra.

Continuous Calculi

Generalizing C ∗-algebras

As complete locally convex ∗-algebras:

C∗-algebras ⊆

Banach∗-algebras

pro-C∗-algebras

⊆ complete lmc∗-algebras

But their (continuous) GNS-representations are all bounded.

As ordered ∗-algebras:

C∗-algebras ⊆ Su∗-algebras ⊆ Archimedean ordered ∗-algebras

Example: Cpol(R) is a Su∗-algebra.

Continuous Calculi

(Archimedean) ordered ∗-algebras

Definition

An ordered ∗-algebra is a (unital) ∗-algebra A together with a partialorder ≤ on the Hermitian elements, i.e. on AH :=

{a ∈ A

∣∣ a = a∗}

,such that

a + c ≤ b + c , d∗a d ≤ d∗b d and 0 ≤ 1

for all a, b, c ∈ AH with a ≤ b and all d ∈ A.

An ordered ∗-algebra is called Archimedean if the following holds:Whenever a ≤ εb for two a, b ∈ AH, b ≥ 0 and all ε ∈ ]0,∞[, then a ≤ 0.

Examples of Archimedean ordered ∗-algebras:

∗-algebras of complex-valued functions like C (R), Cpol(R), Cbd(R)

O∗-algebras, i.e. ∗-algebras of (possibly unbounded) operators

This especially includes C∗-algebras.

Continuous Calculi

Definition

{a ∈ A

∣∣ a = a∗}

,such that

a + c ≤ b + c , d∗a d ≤ d∗b d and 0 ≤ 1

This especially includes C∗-algebras.

Continuous Calculi

Definition

{a ∈ A

∣∣ a = a∗}

,such that

a + c ≤ b + c , d∗a d ≤ d∗b d and 0 ≤ 1

This especially includes C∗-algebras.Matthias Schotz Universal Continuous Calculus for Su∗-Algebras

Continuous Calculi

Relation to C ∗-algebras

Let A be an Archimedean ordered ∗-algebra, then define

‖ · ‖∞ : A → [0,∞]

a 7→ ‖a‖∞ := min{λ ∈ [0,∞]

∣∣ a∗a ≤ λ21} .

Proposition

Abd :={a ∈ A

∣∣ ‖a‖∞ <∞}

, the set of uniformly bounded elements, isa unital ∗-subalgebra of A and ‖ · ‖∞ is a C∗-norm on Abd.

Example: C (R)bd = Cbd(R).If A is a C∗-algebra with norm ‖ · ‖, then ‖ · ‖∞ = ‖ · ‖ and Abd = A.

Continuous Calculi

‖ · ‖∞ : A → [0,∞]

a 7→ ‖a‖∞ := min{λ ∈ [0,∞]

∣∣ a∗a ≤ λ21} .Proposition

Abd :={a ∈ A

∣∣ ‖a‖∞ <∞}

Continuous Calculi

‖ · ‖∞ : A → [0,∞]

a 7→ ‖a‖∞ := min{λ ∈ [0,∞]

∣∣ a∗a ≤ λ21} .Proposition

Abd :={a ∈ A

∣∣ ‖a‖∞ <∞}

Continuous Calculi

Su*-Algebras

Definition

An Archimedean ordered ∗-algebra A is uniformly complete if it iscomplete with respect to the metric

d∞ : A×A → [0,∞[

(a, b) 7→ d∞(a, b) := min{‖a− b‖∞, 1

Definition

Let A be an ordered ∗-algebra. A Hermitian element a ∈ AH is calledcoercive if a ≥ ε1 for some ε ∈ ]0,∞[.Moreover, A is called symmetric if every coercive a ∈ AH has amultiplicative inverse.

Examples: C (R), C (R)pol and Cbd(R) as well as all C∗-algebras aresymmetric and uniformly complete Archimedean ordered ∗-algebras.

Continuous Calculi

Su*-Algebras

Definition

An Archimedean ordered ∗-algebra A is uniformly complete if it iscomplete with respect to the metric

d∞ : A×A → [0,∞[

(a, b) 7→ d∞(a, b) := min{‖a− b‖∞, 1

Definition

Let A be an ordered ∗-algebra. A Hermitian element a ∈ AH is calledcoercive if a ≥ ε1 for some ε ∈ ]0,∞[.Moreover, A is called symmetric if every coercive a ∈ AH has amultiplicative inverse.

Examples: C (R), C (R)pol and Cbd(R) as well as all C∗-algebras aresymmetric and uniformly complete Archimedean ordered ∗-algebras.

Continuous Calculi

Su*-Algebras

Definition

A Su∗-algebra is a symmetric and uniformly complete Archimedeanordered ∗-algebra.

Note: The C∗-algebras are exactly those Su∗-algebras in which allelements are uniformly bounded.

Continuous Calculi

Definition

Let A be an Archimedean ordered ∗-algebra and a1, . . . , aN ∈ AH withN ∈ N. A continuous calculus for a1, . . . , aN is a triple (X , I,Φ) of:

a closed subset X of RN ,

a unital ∗-subalgebra I of C (X ) that contains all uniformly boundedfunctions of C (X ), i.e.

C (X )bd ⊆ I ⊆ C (X ) ,

as well as all the projections prn : X → R,

(x1, . . . , xN) 7→ prn(x1, . . . , xN) := xn ,

a unital ∗-homomorphism Φ: I → A such that Φ(prn) = an holdsfor all n ∈ {1, . . . ,N}.

Continuous Calculi

Universal Continuous Calculus

Definition

Let A be an Archimedean ordered ∗-algebra and a1, . . . , aN ∈ AH withN ∈ N. The universal continuous calculus for a1, . . . , aN (if it exists) isthe continuous calculus(

spec(a1, . . . , aN),F(a1, . . . , aN), Γa1,...,aN

)with the following property:

Whenever (X , I,Φ) is a continuous calculus for a1, . . . , aN , then:

X ⊇ spec(a1, . . . , aN),

f |spec(a1,...,aN ) ∈ F(a1, . . . , aN) for all f ∈ I,

Φ(f ) = Γa1,...,aN

(f |spec(a1,...,aN )

)for all f ∈ I.

Continuous Calculi

The main result

Theorem

Let A be a Su∗-algebra and a1, . . . , aN ∈ AH with N ∈ N pairwisecommuting. Then the universal continuous calculus(

)for a1, . . . , aN exists.

Moreover, Γa1,...,aN maps into the bicommutant of {a1, . . . , aN} and it isan order embedding, i.e. it is injective and

f ≤ g ⇐⇒ Γa1,...,aN (f ) ≤ Γa1,...,aN (g)

holds for all f , g ∈ F(a1, . . . , aN).

This universal continuous calculus e.g. yields inverses of coerciveelements, square roots of positive Hermitian elements or absolute valuesof Hermitian elements.

Continuous Calculi

The main result

Theorem

)for a1, . . . , aN exists.Moreover, Γa1,...,aN maps into the bicommutant of {a1, . . . , aN} and it isan order embedding, i.e. it is injective and

f ≤ g ⇐⇒ Γa1,...,aN (f ) ≤ Γa1,...,aN (g)

Continuous Calculi

The main result

Theorem

)for a1, . . . , aN exists.Moreover, Γa1,...,aN maps into the bicommutant of {a1, . . . , aN} and it isan order embedding, i.e. it is injective and

f ≤ g ⇐⇒ Γa1,...,aN (f ) ≤ Γa1,...,aN (g)

Continuous Calculi

Sketch of the proof

Start with polynomial calculus for a1, . . . , aN .

Using the existence of inverses and the completeness of A, constructa suitable unital ∗-homomorphism Ψ′ : C

(RN ∪ {∞}

)→ A, where

RN ∪ {∞} is the 1-point-compactification of RN .

Extend Ψ′ to a continuous calculus(RN , I,Ψ

)for a1, . . . , aN with

I ⊆ C (RN) as large as possible by setting

Ψ(f ) := Ψ′(f −1)−1

for all coercive f ∈ IH.

Show that the kernel of Ψ is the vanishing ideal of a closed subset Xof RN .

Set spec(a1, . . . , aN) := X , F(a1, . . . , aN) := I/kerΨ andΓa1,...,aN

([f ])

:= Ψ(f ) for all [f ] ∈ F(a1, . . . , aN) with representantf ∈ I.

Thank you for your attention!

Continuous Calculi

Sketch of the proof

(RN ∪ {∞}

)→ A, where

Ψ(f ) := Ψ′(f −1)−1

([f ])

Continuous Calculi

Sketch of the proof

(RN ∪ {∞}

)→ A, where

Ψ(f ) := Ψ′(f −1)−1

([f ])

Continuous Calculi

Sketch of the proof

(RN ∪ {∞}

)→ A, where

Ψ(f ) := Ψ′(f −1)−1

([f ])

Continuous Calculi

Sketch of the proof

(RN ∪ {∞}

)→ A, where

Ψ(f ) := Ψ′(f −1)−1

([f ])

Continuous Calculi

Sketch of the proof

(RN ∪ {∞}

)→ A, where

Ψ(f ) := Ψ′(f −1)−1

([f ])

universal continuous calculus for su...

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