universal continuous calculus for su...
TRANSCRIPT
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
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and 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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and 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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
Continuous Calculi
Overview
How ist Cpol(R) the “most unbounded” one?
Why is this important?
Ordered ∗-Algebras and Su∗-Algebras
Continuous Calculi
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-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 ) :=
∫R
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-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 ) :=
∫R
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-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 ) :=
∫R
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-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 ) :=
∫R
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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).
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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).
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
Continuous Calculi
Generalizing C ∗-algebras
As complete locally convex ∗-algebras:
C∗-algebras ⊆
Banach∗-algebras
or
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
Continuous Calculi
Generalizing C ∗-algebras
As complete locally convex ∗-algebras:
C∗-algebras ⊆
Banach∗-algebras
or
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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:
C
∗-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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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:
C
∗-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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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:
C
∗-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.Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and 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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and 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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and 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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
Continuous Calculi
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}.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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(
spec(a1, . . . , aN),F(a1, . . . , aN), Γa1,...,aN
)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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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(
spec(a1, . . . , aN),F(a1, . . . , aN), Γa1,...,aN
)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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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(
spec(a1, . . . , aN),F(a1, . . . , aN), Γa1,...,aN
)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.
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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!
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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!
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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!
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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!
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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!
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras
IntroductionOverview
Positive Linear Functionals on ∗-AlgebrasOrdered ∗-Algebras and Su∗-Algebras
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!
Matthias Schotz Universal Continuous Calculus for Su∗-Algebras