Dynamical Systems on Spectral Metric Spaces
Kamran Reihani
University of Kansas
CIMPA-UNESCO-MICINN-THAILAND 2011 School
Spectral Triples and Their Applications
Chulalongkorn University, Bangkok, June 03
Joint work with Jean Bellissard and Matilde Marcolli
1 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Definition
A spectral triple for a C ∗ -algebra A is a triple X = (A, H, D),
where H is a Hilbert space and D is a densely defined self-adjoint
(hence closed) operator on H with the following properties:
2 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Definition
A spectral triple for a C ∗ -algebra A is a triple X = (A, H, D),
where H is a Hilbert space and D is a densely defined self-adjoint
(hence closed) operator on H with the following properties:
1. there is a (faithful) representation π : A 7→ B(H),
2 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Definition
A spectral triple for a C ∗ -algebra A is a triple X = (A, H, D),
where H is a Hilbert space and D is a densely defined self-adjoint
(hence closed) operator on H with the following properties:
1. there is a (faithful) representation π : A 7→ B(H),
2. D has compact resolvent: (1 + D 2 )−1 ∈ K(H),
2 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Definition
A spectral triple for a C ∗ -algebra A is a triple X = (A, H, D),
where H is a Hilbert space and D is a densely defined self-adjoint
(hence closed) operator on H with the following properties:
1. there is a (faithful) representation π : A 7→ B(H),
2. D has compact resolvent: (1 + D 2 )−1 ∈ K(H),
3. D has a core c(D), namely a subspace of the dom(D), dense
for the topology defined by the graph norm
kξkD := (kξk2 + kDξk2 )1/2 ,
ξ ∈ dom(D),
(so D|c(D) = D)
2 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Definition
A spectral triple for a C ∗ -algebra A is a triple X = (A, H, D),
where H is a Hilbert space and D is a densely defined self-adjoint
(hence closed) operator on H with the following properties:
1. there is a (faithful) representation π : A 7→ B(H),
2. D has compact resolvent: (1 + D 2 )−1 ∈ K(H),
3. D has a core c(D), namely a subspace of the dom(D), dense
for the topology defined by the graph norm
kξkD := (kξk2 + kDξk2 )1/2 ,
ξ ∈ dom(D),
(so D|c(D) = D) such that the set
C 1 (X ) := {a ∈ A ; π(a)c(D) ⊆ c(D), k[D, π(a)]k < ∞ on c(D)}
is dense in A.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Lemma
Let X = (A, H, D) be a spectral triple. Then the space C 1 (X ),
endowed with the norm
kakD := kak + k[D, π(a)]k
is a Banach ∗-algebra, which is closed under HFC (holomorphic
functional calculus).
We call C 1 (X ) the Lipschitz algebra of X .
3 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Example
(M, g ) = compact Rimannian (spinc ) manifold
A = C (M)
H = Hilbert space of L2 -sections of the spinor bundle
D = Dirac operator
4 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Example
(M, g ) = compact Rimannian (spinc ) manifold
A = C (M)
H = Hilbert space of L2 -sections of the spinor bundle
D = Dirac operator
(e.g., A = C (T), H = L2 (T), D = ∂ = −ιd/dθ, [∂, f ] = −ιf 0 )
4 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Example
(M, g ) = compact Rimannian (spinc ) manifold
A = C (M)
H = Hilbert space of L2 -sections of the spinor bundle
D = Dirac operator
(e.g., A = C (T), H = L2 (T), D = ∂ = −ιd/dθ, [∂, f ] = −ιf 0 )
Theorem (Connes)
The geodesic distance between x, y ∈ M can be calculated using
the above data:
4 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Example
(M, g ) = compact Rimannian (spinc ) manifold
A = C (M)
H = Hilbert space of L2 -sections of the spinor bundle
D = Dirac operator
(e.g., A = C (T), H = L2 (T), D = ∂ = −ιd/dθ, [∂, f ] = −ιf 0 )
Theorem (Connes)
The geodesic distance between x, y ∈ M can be calculated using
the above data:
dg (x, y ) = sup{|f (x) − f (y )| ; f ∈ A, k[D, π(f )]k ≤ 1}.
In fact, k[D, π(f )]k = k∇f k∞ = kf kLip . Hence, C 1 (X ) = Lip(M).
4 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: Let (A, H, D) be a spectral triple and α ∈ Aut(A). Is
b
there a canonical way of defining a spectral triple (A oα Z, K , D)
induced by (A, H, D) and α? If so, what are the metric properties
encoded by this triple?
5 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: Let (A, H, D) be a spectral triple and α ∈ Aut(A). Is
b
there a canonical way of defining a spectral triple (A oα Z, K , D)
induced by (A, H, D) and α? If so, what are the metric properties
encoded by this triple?
The answer is not always positive: actually it is positive if and only
if (A, H, D) represents the noncommutative analog of a metric
space for which α is represented by an equicontinuous family of
quasi-isometries. In particular, for uniformly hyperbolic actions
on a compact metric space, such a construction does not work.
5 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: Let (A, H, D) be a spectral triple and α ∈ Aut(A). Is
b
there a canonical way of defining a spectral triple (A oα Z, K , D)
induced by (A, H, D) and α? If so, what are the metric properties
encoded by this triple?
The answer is not always positive: actually it is positive if and only
if (A, H, D) represents the noncommutative analog of a metric
space for which α is represented by an equicontinuous family of
quasi-isometries. In particular, for uniformly hyperbolic actions
on a compact metric space, such a construction does not work.
Our main goals:
5 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: Let (A, H, D) be a spectral triple and α ∈ Aut(A). Is
b
there a canonical way of defining a spectral triple (A oα Z, K , D)
induced by (A, H, D) and α? If so, what are the metric properties
encoded by this triple?
The answer is not always positive: actually it is positive if and only
if (A, H, D) represents the noncommutative analog of a metric
space for which α is represented by an equicontinuous family of
quasi-isometries. In particular, for uniformly hyperbolic actions
on a compact metric space, such a construction does not work.
Our main goals:
I
5 / 26
To characterize which automorphisms give rise to such
canonical constructions
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: Let (A, H, D) be a spectral triple and α ∈ Aut(A). Is
b
there a canonical way of defining a spectral triple (A oα Z, K , D)
induced by (A, H, D) and α? If so, what are the metric properties
encoded by this triple?
The answer is not always positive: actually it is positive if and only
if (A, H, D) represents the noncommutative analog of a metric
space for which α is represented by an equicontinuous family of
quasi-isometries. In particular, for uniformly hyperbolic actions
on a compact metric space, such a construction does not work.
Our main goals:
5 / 26
I
To characterize which automorphisms give rise to such
canonical constructions
I
To have a strategy to get around the situation for the other
cases: extending the algebra A to an algebra B representing
continuous functions on the “metric bundle”, and get a
canonical spectral triple for the crossed product B oᾱ Z, in
which A oα Z acts as a multiplier algebra.
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Let’s try to build a spectral triple for A oα Z, given one for A:
6 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Let’s try to build a spectral triple for A oα Z, given one for A:
Hilbert Space: K = H ⊗ `2 (Z) ⊗ C2 = `2 (Z, H ⊕ H)
f ∈ K ⇒ f = (fn )n∈Z ; fn = (fn+ , fn− ), fn,± ∈ H
6 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Let’s try to build a spectral triple for A oα Z, given one for A:
Hilbert Space: K = H ⊗ `2 (Z) ⊗ C2 = `2 (Z, H ⊕ H)
f ∈ K ⇒ f = (fn )n∈Z ; fn = (fn+ , fn− ), fn,± ∈ H
Representation: The left regular representation π̂ of A oα Z:
(π̂(a)f )n = π ◦ α−n (a)(fn )
(ûf )n = fn−1
û unitary on KPand (π̂, û) P
is a covariant pair ⇒ π̂ extends to
A oα Z by π̂( n an u n ) = n π̂(an )û n
6 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Let’s try to build a spectral triple for A oα Z, given one for A:
Hilbert Space: K = H ⊗ `2 (Z) ⊗ C2 = `2 (Z, H ⊕ H)
f ∈ K ⇒ f = (fn )n∈Z ; fn = (fn+ , fn− ), fn,± ∈ H
Representation: The left regular representation π̂ of A oα Z:
(π̂(a)f )n = π ◦ α−n (a)(fn )
(ûf )n = fn−1
û unitary on KPand (π̂, û) P
is a covariant pair ⇒ π̂ extends to
A oα Z by π̂( n an u n ) = n π̂(an )û n
Dirac Operator: Kasparov’s external product of the unbounded
Fredholm modules D and ∂ in the Baaj-Julg’s picture, namely:
6 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Let’s try to build a spectral triple for A oα Z, given one for A:
Hilbert Space: K = H ⊗ `2 (Z) ⊗ C2 = `2 (Z, H ⊕ H)
f ∈ K ⇒ f = (fn )n∈Z ; fn = (fn+ , fn− ), fn,± ∈ H
Representation: The left regular representation π̂ of A oα Z:
(π̂(a)f )n = π ◦ α−n (a)(fn )
(ûf )n = fn−1
û unitary on KPand (π̂, û) P
is a covariant pair ⇒ π̂ extends to
A oα Z by π̂( n an u n ) = n π̂(an )û n
Dirac Operator: Kasparov’s external product of the unbounded
Fredholm modules D and ∂ in the Baaj-Julg’s picture, namely:
0
D − ın
b
Df
=
fn
D + ın
0
n
6 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Let’s try to build a spectral triple for A oα Z, given one for A:
Hilbert Space: K = H ⊗ `2 (Z) ⊗ C2 = `2 (Z, H ⊕ H)
f ∈ K ⇒ f = (fn )n∈Z ; fn = (fn+ , fn− ), fn,± ∈ H
Representation: The left regular representation π̂ of A oα Z:
(π̂(a)f )n = π ◦ α−n (a)(fn )
(ûf )n = fn−1
û unitary on KPand (π̂, û) P
is a covariant pair ⇒ π̂ extends to
A oα Z by π̂( n an u n ) = n π̂(an )û n
Dirac Operator: Kasparov’s external product of the unbounded
Fredholm modules D and ∂ in the Baaj-Julg’s picture, namely:
0
D − ın
b
Df
=
fn
D + ın
0
n
b is self-dajoint with compact resolvent, because the eigenvalues
D
q
b are given by the ± λ2 + n2 and the eigenprojections are
of D
k
indeed finite dimensional
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
b
Commutators with D:
0 −ı
b
[D, û]f
=
fn−1
ı 0
n
7 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
b
Commutators with D:
0 −ı
b
[D, û]f
=
fn−1
ı 0
n
7 / 26
b π̂(a)]f
[D,
n
=
0 1
1 0
Kamran Reihani
[D, π ◦ α−n (a)]fn
Dynamical Systems on Spectral Metric Spaces
b
Commutators with D:
0 −ı
b
[D, û]f
=
fn−1
ı 0
n
b π̂(a)]f
[D,
n
=
0 1
1 0
[D, π ◦ α−n (a)]fn
Observation: Assuming
sup k[D, π ◦ α−n (a)k < ∞
n∈Z
b will be bounded operators
the commutators with D
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
b
Commutators with D:
0 −ı
b
[D, û]f
=
fn−1
ı 0
n
b π̂(a)]f
[D,
n
=
0 1
1 0
[D, π ◦ α−n (a)]fn
Observation: Assuming
sup k[D, π ◦ α−n (a)k < ∞
n∈Z
b will be bounded operators
the commutators with D
We say that the automorphism α generates an equicontinuous
subgroup hαi of Aut(A) if the condition above is fulfilled
7 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction works ,
8 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction works ,
I
8 / 26
Crossed product of an isometric diffeomorphism on a compact
Riemannian (spinc ) manifold, e.g., the rotation algebra
(motivating for our construction), and more generally:
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction works ,
8 / 26
I
Crossed product of an isometric diffeomorphism on a compact
Riemannian (spinc ) manifold, e.g., the rotation algebra
(motivating for our construction), and more generally:
I
NC or classical tori: by iteration of the crossed products, one
can exactly recover the so called canonical spectral triples for
the NC or classical tori
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction works ,
8 / 26
I
Crossed product of an isometric diffeomorphism on a compact
Riemannian (spinc ) manifold, e.g., the rotation algebra
(motivating for our construction), and more generally:
I
NC or classical tori: by iteration of the crossed products, one
can exactly recover the so called canonical spectral triples for
the NC or classical tori
I
Crossed product C (M) o Z of a diffeomorphism φ on a
compact Riemannian (spinc ) manifold (M, g ), with the
following condition
!
dg φk (P), φk (P 0 )
<∞
sup
sup
dg (P, P 0 )
k∈Z P6=P 0 ∈M
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction works ,
8 / 26
I
Crossed product of an isometric diffeomorphism on a compact
Riemannian (spinc ) manifold, e.g., the rotation algebra
(motivating for our construction), and more generally:
I
NC or classical tori: by iteration of the crossed products, one
can exactly recover the so called canonical spectral triples for
the NC or classical tori
I
Crossed product C (M) o Z of a diffeomorphism φ on a
compact Riemannian (spinc ) manifold (M, g ), with the
following condition
!
dg φk (P), φk (P 0 )
<∞
sup
sup
dg (P, P 0 )
k∈Z P6=P 0 ∈M
I
Bunce-Deddens algebras: odometer action on the Cantor set
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction does not work /
9 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction does not work /
I
9 / 26
Expansive diffeomorphisms
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction does not work /
9 / 26
I
Expansive diffeomorphisms
I
Uniformly hyperbolic diffeomorphisms, e.g., the Arnold Cat
map on T2 given by
2 1
M=
1 1
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction does not work /
9 / 26
I
Expansive diffeomorphisms
I
Uniformly hyperbolic diffeomorphisms, e.g., the Arnold Cat
map on T2 given by
2 1
M=
1 1
I
The group C ∗ algebra of the discrete Heisenberg group;
C ∗ (H3 ) ' C (T2 ) o Z with map on T2 given by
1 1
N=
0 1
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Some examples for which this construction does not work /
9 / 26
I
Expansive diffeomorphisms
I
Uniformly hyperbolic diffeomorphisms, e.g., the Arnold Cat
map on T2 given by
2 1
M=
1 1
I
The group C ∗ algebra of the discrete Heisenberg group;
C ∗ (H3 ) ' C (T2 ) o Z with map on T2 given by
1 1
N=
0 1
I
Lots of interesting examples! (with Bellissard and Marcolli)
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Wasserstein Metric:
10 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Wasserstein Metric:
Given a compact metric space (X , d), the Lipschitz seminorm
k · kLip on the set of functions f : X → R is defined by
kf kLip =
sup
x,y ∈X ;x6=y
10 / 26
Kamran Reihani
|f (x) − f (y )|
d(x, y )
Dynamical Systems on Spectral Metric Spaces
Wasserstein Metric:
Given a compact metric space (X , d), the Lipschitz seminorm
k · kLip on the set of functions f : X → R is defined by
kf kLip =
sup
x,y ∈X ;x6=y
|f (x) − f (y )|
d(x, y )
Then the Wasserstein metric on the set M1 (X ) of the probability
measures on X is defined by
Z
dW (µ, ν) = sup
f d(µ − ν),
µ, ν ∈ M1 (X )
kf kLip ≤1 X
10 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
dD (φ, ψ) := sup{|φ(a) − ψ(a)| : k[D, π(a)]k ≤ 1}
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
dD (φ, ψ) := sup{|φ(a) − ψ(a)| : k[D, π(a)]k ≤ 1}
I
11 / 26
Define the following objects
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
dD (φ, ψ) := sup{|φ(a) − ψ(a)| : k[D, π(a)]k ≤ 1}
I
Define the following objects
(i) the metric commutant; A0D := {a ∈ A : [D, π(a)] = 0}
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
dD (φ, ψ) := sup{|φ(a) − ψ(a)| : k[D, π(a)]k ≤ 1}
I
Define the following objects
(i) the metric commutant; A0D := {a ∈ A : [D, π(a)] = 0}
(ii) the Lipschitz ball; BLip (X ) := {a ∈ A : k[D, π(a)]k ≤ 1}
(always closed in norm)
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
dD (φ, ψ) := sup{|φ(a) − ψ(a)| : k[D, π(a)]k ≤ 1}
I
Define the following objects
(i) the metric commutant; A0D := {a ∈ A : [D, π(a)] = 0}
(ii) the Lipschitz ball; BLip (X ) := {a ∈ A : k[D, π(a)]k ≤ 1}
(always closed in norm)
Fact: If BLip has norm bounded image in A/A0D , then (S(A), dD ) is
a metric space with finite diameter.
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Connes metric:
I A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
dD (φ, ψ) := sup{|φ(a) − ψ(a)| : k[D, π(a)]k ≤ 1}
I
Define the following objects
(i) the metric commutant; A0D := {a ∈ A : [D, π(a)] = 0}
(ii) the Lipschitz ball; BLip (X ) := {a ∈ A : k[D, π(a)]k ≤ 1}
(always closed in norm)
Fact: If BLip has norm bounded image in A/A0D , then (S(A), dD ) is
a metric space with finite diameter.
Result (Rennie-Varilly)
Let (A, H, D) be a spectral triple such that A is a separable unital
C ∗ -algebra and 1A acts as the identity operator on H (i.e. the
representation is non-degenerate). Assume that the metric
commutant A0D = C1A . Then dD is a metric on S(A).
11 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
12 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
If A is a unital C ∗ -algebra then the state space S(A) is a closed
(and convex) subset of the unit ball of A∗ in the w ∗ -topology
σ(A∗ , A), so S(A) is w ∗ -compact.
12 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
If A is a unital C ∗ -algebra then the state space S(A) is a closed
(and convex) subset of the unit ball of A∗ in the w ∗ -topology
σ(A∗ , A), so S(A) is w ∗ -compact.
Result (Pavlovich-Rieffel-Ozawa)
Let X = (A, H, D) be a spectral triple with A unital. Assume that
the metric commutatnt A0D is trivial (= C1).
12 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
If A is a unital C ∗ -algebra then the state space S(A) is a closed
(and convex) subset of the unit ball of A∗ in the w ∗ -topology
σ(A∗ , A), so S(A) is w ∗ -compact.
Result (Pavlovich-Rieffel-Ozawa)
Let X = (A, H, D) be a spectral triple with A unital. Assume that
the metric commutatnt A0D is trivial (= C1). T.F.A.E:
(i) dD induces the w ∗ -topology on S(A),
12 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
If A is a unital C ∗ -algebra then the state space S(A) is a closed
(and convex) subset of the unit ball of A∗ in the w ∗ -topology
σ(A∗ , A), so S(A) is w ∗ -compact.
Result (Pavlovich-Rieffel-Ozawa)
Let X = (A, H, D) be a spectral triple with A unital. Assume that
the metric commutatnt A0D is trivial (= C1). T.F.A.E:
(i) dD induces the w ∗ -topology on S(A),
(ii) BLip (X ) := {a ∈ A : k[D, π(a)]k ≤ 1} has precompact image
in A/A0D ,
12 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
If A is a unital C ∗ -algebra then the state space S(A) is a closed
(and convex) subset of the unit ball of A∗ in the w ∗ -topology
σ(A∗ , A), so S(A) is w ∗ -compact.
Result (Pavlovich-Rieffel-Ozawa)
Let X = (A, H, D) be a spectral triple with A unital. Assume that
the metric commutatnt A0D is trivial (= C1). T.F.A.E:
(i) dD induces the w ∗ -topology on S(A),
(ii) BLip (X ) := {a ∈ A : k[D, π(a)]k ≤ 1} has precompact image
in A/A0D ,
(iii) ker(φ) ∩ BLip (X ) is precompact in A for some φ ∈ S(A),
12 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Compatibility with the w ∗ -topology:
If A is a unital C ∗ -algebra then the state space S(A) is a closed
(and convex) subset of the unit ball of A∗ in the w ∗ -topology
σ(A∗ , A), so S(A) is w ∗ -compact.
Result (Pavlovich-Rieffel-Ozawa)
Let X = (A, H, D) be a spectral triple with A unital. Assume that
the metric commutatnt A0D is trivial (= C1). T.F.A.E:
(i) dD induces the w ∗ -topology on S(A),
(ii) BLip (X ) := {a ∈ A : k[D, π(a)]k ≤ 1} has precompact image
in A/A0D ,
(iii) ker(φ) ∩ BLip (X ) is precompact in A for some φ ∈ S(A),
(iv) ker(φ) ∩ BLip (X ) is precompact in A for all φ ∈ S(A).
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Dynamical Systems on Spectral Metric Spaces
For a non-unital C ∗ -algebra, the state space is never locally
compact, but it is a Gδ in the w ∗ -compact set Q(A) = positive
functionals with norm at most 1. So, S(A) with the w ∗ -topology is
always a Polish space, i.e. metrizable and complete in the metric.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
For a non-unital C ∗ -algebra, the state space is never locally
compact, but it is a Gδ in the w ∗ -compact set Q(A) = positive
functionals with norm at most 1. So, S(A) with the w ∗ -topology is
always a Polish space, i.e. metrizable and complete in the metric.
Definition
A strictly positive element h ∈ A is a positive element such that
hAh = A.
Equivalently, φ(h) > 0 for all nonzero states 0 6= φ ∈ S(A).
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
For a non-unital C ∗ -algebra, the state space is never locally
compact, but it is a Gδ in the w ∗ -compact set Q(A) = positive
functionals with norm at most 1. So, S(A) with the w ∗ -topology is
always a Polish space, i.e. metrizable and complete in the metric.
Definition
A strictly positive element h ∈ A is a positive element such that
hAh = A.
Equivalently, φ(h) > 0 for all nonzero states 0 6= φ ∈ S(A).
Result (Latrémolière)
Let X = (A, H, D) be a spectral triple with A non-unital and
separable. If the D-commutant is trivial and the representation of
A is non-degenerate, the Connes distance defines the w ∗ -topology
on the state space if and only if there is a strictly positive element
h ∈ A such that hBLip h is compact. In such a case the Connes
metric is a path metric and the state space is complete (and with
finite diameter) for this metric.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Spectral metric spaces:
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Spectral metric spaces:
Definition
A bounded spectral metric space is a spectral triple
X = (A, H, D) such that
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Spectral metric spaces:
Definition
A bounded spectral metric space is a spectral triple
X = (A, H, D) such that
(i) the representation of A on H is non-degenerate,
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Spectral metric spaces:
Definition
A bounded spectral metric space is a spectral triple
X = (A, H, D) such that
(i) the representation of A on H is non-degenerate,
(ii) the metric commutant A0D = {a ∈ A ; [D, a] = 0} is trivial,
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Spectral metric spaces:
Definition
A bounded spectral metric space is a spectral triple
X = (A, H, D) such that
(i) the representation of A on H is non-degenerate,
(ii) the metric commutant A0D = {a ∈ A ; [D, a] = 0} is trivial,
(iii) there is a strictly positive element h ∈ A such that the
h-compressed Lipschitz Ball hBLip h has precompact image in the
normed space A/A0D .
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Spectral metric spaces:
Definition
A bounded spectral metric space is a spectral triple
X = (A, H, D) such that
(i) the representation of A on H is non-degenerate,
(ii) the metric commutant A0D = {a ∈ A ; [D, a] = 0} is trivial,
(iii) there is a strictly positive element h ∈ A such that the
h-compressed Lipschitz Ball hBLip h has precompact image in the
normed space A/A0D .
A spectral metric space X = (A, H, D) will be called compact if A
is unital. (It can be shown that in this case, the inclusion
ι : C 1 (X ) 7→ A is a compact ∗-homomorphism.)
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
A non-compact example: c0 (Z)
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
A non-compact example: c0 (Z)
Let dZ be a metric on Z.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
A non-compact example: c0 (Z)
Let dZ be a metric on Z.
C ∗ -algebra. A := c0 (Z); elements a ∈ A are sequences
a = (an )n∈Z , such that limn→∞ = 0 and kak := supn∈Z |an |
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
A non-compact example: c0 (Z)
Let dZ be a metric on Z.
C ∗ -algebra. A := c0 (Z); elements a ∈ A are sequences
a = (an )n∈Z , such that limn→∞ = 0 and kak := supn∈Z |an |
Hilbert space. `2 (Z × N∗ ) ⊗ E, where N∗ := the set of nonzero
natural integers, and E is a 4-dimensional Hilbert space carrying 4
Dirac matrices γ1 , . . . , γ4 ; each vector f ∈ `2 (Z × N∗ ) ⊗ E can be
seen as a double sequence f = (fn,r )(n,r )∈Z×N∗ with fn,r ∈ E.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
A non-compact example: c0 (Z)
Let dZ be a metric on Z.
C ∗ -algebra. A := c0 (Z); elements a ∈ A are sequences
a = (an )n∈Z , such that limn→∞ = 0 and kak := supn∈Z |an |
Hilbert space. `2 (Z × N∗ ) ⊗ E, where N∗ := the set of nonzero
natural integers, and E is a 4-dimensional Hilbert space carrying 4
Dirac matrices γ1 , . . . , γ4 ; each vector f ∈ `2 (Z × N∗ ) ⊗ E can be
seen as a double sequence f = (fn,r )(n,r )∈Z×N∗ with fn,r ∈ E.
Representation. The C ∗ -algebra c0 (Z) is represented on
`2 (Z × N∗ ) ⊗ E by (af )n,r = an fn,r .
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Dirac operator. Will have compact resolvent!
D=
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γ1 + ıγ2
γ1 − ıγ2 ∗
∇+
∇ + λ(X + R) .
2
2
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Dirac operator. Will have compact resolvent!
D=
γ1 + ıγ2
γ1 − ıγ2 ∗
∇+
∇ + λ(X + R) .
2
2
where λ ∈ R \ {0} and
(∇f )n,r =
fn,r − fn−r ,r
, (Xf )n,r = cn cn+r γ3 fn,r , (Rf )n,r = r γ4 fn,r ,
dZ (n, n − r )
and the sequence c = (cn )n∈Z is required to satisfy
lim cn = +∞,
|n|→∞
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Kamran Reihani
cn dZ (n, n − 1) ≥ 1 .
Dynamical Systems on Spectral Metric Spaces
Dirac operator. Will have compact resolvent!
D=
γ1 + ıγ2
γ1 − ıγ2 ∗
∇+
∇ + λ(X + R) .
2
2
where λ ∈ R \ {0} and
(∇f )n,r =
fn,r − fn−r ,r
, (Xf )n,r = cn cn+r γ3 fn,r , (Rf )n,r = r γ4 fn,r ,
dZ (n, n − r )
and the sequence c = (cn )n∈Z is required to satisfy
lim cn = +∞,
|n|→∞
cn dZ (n, n − 1) ≥ 1 .
Proposition
Let dZ be a metric on Z. Then (c0 (Z), `2 (Z × N∗ ) ⊗ E, D) is a
spectral triple. It is a spectral metric space if and only if the metric
dZ is bounded.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Group Actions: Let X = (A, H, D) be a spectral triple and
G ⊆ Aut(A) equipped with norm-pointwise topology, namely a
basis of neighborhoods of an element α ∈ Aut(A) is the family of
sets of the form U(α; a, ) = {β ∈ Aut(A) ; kβ(a) − α(a)k < },
where a ∈ A and > 0.
Definition
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Group Actions: Let X = (A, H, D) be a spectral triple and
G ⊆ Aut(A) equipped with norm-pointwise topology, namely a
basis of neighborhoods of an element α ∈ Aut(A) is the family of
sets of the form U(α; a, ) = {β ∈ Aut(A) ; kβ(a) − α(a)k < },
where a ∈ A and > 0.
Definition
I
C 1 (G , X ) ⊆ C 1 (X ) will denote the set of elements a ∈ C 1 (X )
for which
(i) g (a) ∈ C 1 (X ) for all g ∈ G ,
(ii) the map g ∈ G 7→ [D, π ◦ g (a)] ∈ B(H) is continuous.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Group Actions: Let X = (A, H, D) be a spectral triple and
G ⊆ Aut(A) equipped with norm-pointwise topology, namely a
basis of neighborhoods of an element α ∈ Aut(A) is the family of
sets of the form U(α; a, ) = {β ∈ Aut(A) ; kβ(a) − α(a)k < },
where a ∈ A and > 0.
Definition
I
C 1 (G , X ) ⊆ C 1 (X ) will denote the set of elements a ∈ C 1 (X )
for which
(i) g (a) ∈ C 1 (X ) for all g ∈ G ,
(ii) the map g ∈ G 7→ [D, π ◦ g (a)] ∈ B(H) is continuous.
I
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Cb1 (G , X ) ⊆ C 1 (G , X ) will denote the set of elements
a ∈ C 1 (G , X ) for which, supg ∈G k[D, π ◦ g (a)k < ∞.
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Group Actions: Let X = (A, H, D) be a spectral triple and
G ⊆ Aut(A) equipped with norm-pointwise topology, namely a
basis of neighborhoods of an element α ∈ Aut(A) is the family of
sets of the form U(α; a, ) = {β ∈ Aut(A) ; kβ(a) − α(a)k < },
where a ∈ A and > 0.
Definition
I
C 1 (G , X ) ⊆ C 1 (X ) will denote the set of elements a ∈ C 1 (X )
for which
(i) g (a) ∈ C 1 (X ) for all g ∈ G ,
(ii) the map g ∈ G 7→ [D, π ◦ g (a)] ∈ B(H) is continuous.
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I
Cb1 (G , X ) ⊆ C 1 (G , X ) will denote the set of elements
a ∈ C 1 (G , X ) for which, supg ∈G k[D, π ◦ g (a)k < ∞.
I
G (or the action of G ) is said to be quasi-isometric if
C 1 (G , X ) = C 1 (X ).
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Group Actions: Let X = (A, H, D) be a spectral triple and
G ⊆ Aut(A) equipped with norm-pointwise topology, namely a
basis of neighborhoods of an element α ∈ Aut(A) is the family of
sets of the form U(α; a, ) = {β ∈ Aut(A) ; kβ(a) − α(a)k < },
where a ∈ A and > 0.
Definition
I
C 1 (G , X ) ⊆ C 1 (X ) will denote the set of elements a ∈ C 1 (X )
for which
(i) g (a) ∈ C 1 (X ) for all g ∈ G ,
(ii) the map g ∈ G 7→ [D, π ◦ g (a)] ∈ B(H) is continuous.
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I
Cb1 (G , X ) ⊆ C 1 (G , X ) will denote the set of elements
a ∈ C 1 (G , X ) for which, supg ∈G k[D, π ◦ g (a)k < ∞.
I
G (or the action of G ) is said to be quasi-isometric if
C 1 (G , X ) = C 1 (X ).
I
G (or the action of G ) is said to be equicontinuous if
Cb1 (G , X ) = C 1 (X ).
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
I
G (or the action of G ) is said to be isometric if it is
quasi-isometric and k[D, π ◦ g (a)]k = k[D, a]k, for all
a ∈ C 1 (G , X ) = C 1 (X ) and all g ∈ G .
Remark
A quasi-isometric automorphism of a compact spectral triple
X = (A, H, D) is an isometry if and only if it preserves the Connes
distance on the state space S(A).
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
I
G (or the action of G ) is said to be isometric if it is
quasi-isometric and k[D, π ◦ g (a)]k = k[D, a]k, for all
a ∈ C 1 (G , X ) = C 1 (X ) and all g ∈ G .
Remark
A quasi-isometric automorphism of a compact spectral triple
X = (A, H, D) is an isometry if and only if it preserves the Connes
distance on the state space S(A).
Proof: “if” part: a 7→ k[D, a]k defines a lower semicontinuous
Lipschitz seminorm on C 1 (X ), therefore, following Rieffel, it can be
recovered via Connes distance by
k[D, a]k =
|φ(a) − ψ(a)|
dD (φ, ψ)
φ6=ψ∈S(A)
sup
The “only if” part is obvious from the definition of an isometry. 2
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Similarly, quasi-isometries on X correspond to bi-Lipschitz
transformations on S(A). In fact, we have the following.
Lemma
Let X = (A, H, D) be a spectral triple. A ∗-automorphism
α ∈ Aut(A) is quasi-isometric if and only if one of the two
following conditions hold
(i) α defines a bounded ∗-automorphism of C 1 (X ) (with k · kD ),
(ii) the action of α on the state space of A is bi-Lipschitz for the
Connes metric.
Proof: (i) If α : C 1 (X ) → C 1 (X ) is bounded, it follows, from the
definition kakD := kak + k[D, a]k that α is quasi-isometric.
Conversely if α is quasi-isometric, it is a linear map
α : C 1 (X ) → C 1 (X ) which is defined everywhere. By the closed
graph theorem it is bounded. That it is a ∗-homomorphism is
obvious from the definition. The same can be said for α−1 .
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
(ii) Let α ∈ Aut(A) be quasi-isometric. Then α : C 1 (X ) → C 1 (X )
is a bounded ∗-isomorphism. Then, if φ, ψ are two states on A
dD (φ ◦ α, ψ ◦ α) = sup{|φ(a) − ψ(a)| ; k[D, α−1 (a)]k ≤ 1}
Since α is bounded on C 1 (X ) it follows that there is K > 0 such
that k[D, α−1 (a)]k ≤ 1 ⇒ k[D, a]k ≤ K . Therefore
dD (φ ◦ α, ψ ◦ α) ≤ KdD (φ, ψ). Replacing α by its inverse leads to
1
dD (φ, ψ) ≤ dD (φ ◦ α, ψ ◦ α) ≤ KdD (φ, ψ) .
K
(1)
Conversely, if the above equation holds, it follows that both α, α−1
leave C 1 (X ) invariant, showing that α is quasi-isometric.
2
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Theorem (Arzelà-Ascoli)
Let X = (A, H, D) be a compact spectral metric space. Let
G ⊆ Aut(A) be a quasi-isometric subgroup. Then G is
equicontinuous if and only if it has a compact closure.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Theorem (Arzelà-Ascoli)
Let X = (A, H, D) be a compact spectral metric space. Let
G ⊆ Aut(A) be a quasi-isometric subgroup. Then G is
equicontinuous if and only if it has a compact closure.
Theorem (Bohr)
A group G ⊆ Aut(A) has compact closure in Aut(A) (with
norm-pointwise topology) if and only if the action of G on A is
almost periodic, i.e., for each a ∈ A, its orbit G (a) := {g (a)}g ∈G
has compact closure in A.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Theorem
Let A be a unital, separable C ∗ -algebra and α ∈ Aut(A). Let u
denote the unitary implementing α in the crossed product algebra
A oα Z. Then there is a compact spectral metric space
X = (A, H, D) based on A with α implementing an equicontinuous
Z-action, if and only if, there is a compact spectral metric space
b based on the crossed product algebra, such
Y = (A oα Z, K , D),
that
22 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Theorem
Let A be a unital, separable C ∗ -algebra and α ∈ Aut(A). Let u
denote the unitary implementing α in the crossed product algebra
A oα Z. Then there is a compact spectral metric space
X = (A, H, D) based on A with α implementing an equicontinuous
Z-action, if and only if, there is a compact spectral metric space
b based on the crossed product algebra, such
Y = (A oα Z, K , D),
that
(i) the dual action on A oα Z is quasi-isometric,
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Theorem
Let A be a unital, separable C ∗ -algebra and α ∈ Aut(A). Let u
denote the unitary implementing α in the crossed product algebra
A oα Z. Then there is a compact spectral metric space
X = (A, H, D) based on A with α implementing an equicontinuous
Z-action, if and only if, there is a compact spectral metric space
b based on the crossed product algebra, such
Y = (A oα Z, K , D),
that
(i) the dual action on A oα Z is quasi-isometric,
b u] = u ∗ Du
b −D
b is bounded and commutes with
(ii) γ̂u := u ∗ [D,
the elements of A,
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Theorem
Let A be a unital, separable C ∗ -algebra and α ∈ Aut(A). Let u
denote the unitary implementing α in the crossed product algebra
A oα Z. Then there is a compact spectral metric space
X = (A, H, D) based on A with α implementing an equicontinuous
Z-action, if and only if, there is a compact spectral metric space
b based on the crossed product algebra, such
Y = (A oα Z, K , D),
that
(i) the dual action on A oα Z is quasi-isometric,
b u] = u ∗ Du
b −D
b is bounded and commutes with
(ii) γ̂u := u ∗ [D,
the elements of A,
(iii) the Connes metrics induced on the state space of A associated
with the two spectral metric spaces X and Y are equivalent (A as
a subalgebra of A oα Z).
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: What if the family {αn }n∈Z is not equicontinuous?
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: What if the family {αn }n∈Z is not equicontinuous?
Short Answer: Maybe we should replace A oα Z by a naturally
related B oᾱ Z which fits in the previous construction. B resembles
the metric bundle on a manifold (following the idea of Connes and
Moscovici in the ninties; the so called Diff-Invariant Geometry)
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Question: What if the family {αn }n∈Z is not equicontinuous?
Short Answer: Maybe we should replace A oα Z by a naturally
related B oᾱ Z which fits in the previous construction. B resembles
the metric bundle on a manifold (following the idea of Connes and
Moscovici in the ninties; the so called Diff-Invariant Geometry)
Theorem (“metric bundle”)
Let X = (A, H, D) be a spectral metric space and let α ∈ Aut(A)
be quasi-isometric. If α is not equicontinuous, there is a spectral
metric space Y based on A ⊗ c0 (Z), induced by X , such that
(i) A acts as a multiplier algebra on Y
(ii) α implements an isometric action α∗ on Y defined by
(α∗ (b))n = (α(bn−1 )), for b = (bn ) ∈ c0 (Z, A) = A ⊗ c0 (Z).
(iii) there is a spectral metric space based on (A ⊗ c0 (Z)) oα∗ Z,
on which A oα Z acts as a multiplier algebra.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Remark
The only disadvantage of the previous theorem is that with
ᾱ ∈ Aut(B) defined by (ᾱ(b))n = α(bn−1 ), we have
B oᾱ Z = (A ⊗ c0 (Z)) oᾱ Z ' K ⊗ A
so we loose the action through a “Takai duality” phenomena!
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Remark
The only disadvantage of the previous theorem is that with
ᾱ ∈ Aut(B) defined by (ᾱ(b))n = α(bn−1 ), we have
B oᾱ Z = (A ⊗ c0 (Z)) oᾱ Z ' K ⊗ A
so we loose the action through a “Takai duality” phenomena!
Maybe A ⊗ c0 (Z) is somewhat “small” for our purpose.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Remark
The only disadvantage of the previous theorem is that with
ᾱ ∈ Aut(B) defined by (ᾱ(b))n = α(bn−1 ), we have
B oᾱ Z = (A ⊗ c0 (Z)) oᾱ Z ' K ⊗ A
so we loose the action through a “Takai duality” phenomena!
Maybe A ⊗ c0 (Z) is somewhat “small” for our purpose.
Some other candidates are the reduced and generalized Toeplitz
algebras associated with the crossed product.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Remark
The only disadvantage of the previous theorem is that with
ᾱ ∈ Aut(B) defined by (ᾱ(b))n = α(bn−1 ), we have
B oᾱ Z = (A ⊗ c0 (Z)) oᾱ Z ' K ⊗ A
so we loose the action through a “Takai duality” phenomena!
Maybe A ⊗ c0 (Z) is somewhat “small” for our purpose.
Some other candidates are the reduced and generalized Toeplitz
algebras associated with the crossed product.
The following proposition suggests a “larger” algebra as the right
“measure bundle” at the measure-theoretic level.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Proposition (Paschke-R)
Let (X , µ) be a σ-finite measure space, and let Γ be a group of
measurable transformations of X leaving µ quasi-invariant (i.e.
mapping µ-null sets to µ-null sets).
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Proposition (Paschke-R)
Let (X , µ) be a σ-finite measure space, and let Γ be a group of
measurable transformations of X leaving µ quasi-invariant (i.e.
mapping µ-null sets to µ-null sets). Let Y = X × R+
∗ and let
ν = µ × λ, where λ is Lebesgue (not Haar) measure on R+
∗ :
λ([a, b]) = b − a.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Proposition (Paschke-R)
Let (X , µ) be a σ-finite measure space, and let Γ be a group of
measurable transformations of X leaving µ quasi-invariant (i.e.
mapping µ-null sets to µ-null sets). Let Y = X × R+
∗ and let
ν = µ × λ, where λ is Lebesgue (not Haar) measure on R+
∗ :
λ([a, b]) = b − a. Then Γ acts on Y leaving ν invariant by defining
dµ
is the
φ̄(x, s) = (φ(x), hφ (x)s) for φ ∈ Γ, where hφ := d(µ◦φ)
Radon-Nikodym derivative of µ with respect to µ ◦ φ.
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Proposition (Paschke-R)
Let (X , µ) be a σ-finite measure space, and let Γ be a group of
measurable transformations of X leaving µ quasi-invariant (i.e.
mapping µ-null sets to µ-null sets). Let Y = X × R+
∗ and let
ν = µ × λ, where λ is Lebesgue (not Haar) measure on R+
∗ :
λ([a, b]) = b − a. Then Γ acts on Y leaving ν invariant by defining
dµ
is the
φ̄(x, s) = (φ(x), hφ (x)s) for φ ∈ Γ, where hφ := d(µ◦φ)
Radon-Nikodym derivative of µ with respect to µ ◦ φ.
∞
Moreover,
R if ω is the fns trace on L (X , µ) defined by
ω(f ) = X f dµ inducing the dual fns weight ω̄ and the modular
automorphism group {σtω̄ } on L∞ (X , µ) o Γ,
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Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Proposition (Paschke-R)
Let (X , µ) be a σ-finite measure space, and let Γ be a group of
measurable transformations of X leaving µ quasi-invariant (i.e.
mapping µ-null sets to µ-null sets). Let Y = X × R+
∗ and let
ν = µ × λ, where λ is Lebesgue (not Haar) measure on R+
∗ :
λ([a, b]) = b − a. Then Γ acts on Y leaving ν invariant by defining
dµ
is the
φ̄(x, s) = (φ(x), hφ (x)s) for φ ∈ Γ, where hφ := d(µ◦φ)
Radon-Nikodym derivative of µ with respect to µ ◦ φ.
∞
Moreover,
R if ω is the fns trace on L (X , µ) defined by
ω(f ) = X f dµ inducing the dual fns weight ω̄ and the modular
automorphism group {σtω̄ } on L∞ (X , µ) o Γ, then we have the
isomorphism
(L∞ (X , µ) oα Γ) oσω̄ R ' L∞ (Y , ν) oᾱ Γ,
where αφ (f ) = f ◦ φ−1 for f ∈ L∞ (X , µ) and ᾱφ (F ) = F ◦ φ̄−1 for
F ∈ L∞ (Y , ν).
25 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Q: What can be said if the Connes metric on the state space is not
bounded (e.g. Z with its usual metric)?
26 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Q: What can be said if the Connes metric on the state space is not
bounded (e.g. Z with its usual metric)?
Result (Dobrushin)
Let (X , d) be a complete separable metric space and let M1 (X ) be
the set of probability measures on X . A subset F ⊂ M1 (X ) is
called D-tight if Rthere is x0 ∈ X such that for all > 0 there is
r > 0 for which d(x0 ,x)≥r d(x0 , x) dµ(x) ≤ for all µ ∈ F . Then
the Wasserstein distance generates the w ∗ -topology on each
w ∗ -closed D-tight set.
26 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces
Q: What can be said if the Connes metric on the state space is not
bounded (e.g. Z with its usual metric)?
Result (Dobrushin)
Let (X , d) be a complete separable metric space and let M1 (X ) be
the set of probability measures on X . A subset F ⊂ M1 (X ) is
called D-tight if Rthere is x0 ∈ X such that for all > 0 there is
r > 0 for which d(x0 ,x)≥r d(x0 , x) dµ(x) ≤ for all µ ∈ F . Then
the Wasserstein distance generates the w ∗ -topology on each
w ∗ -closed D-tight set.
Aim: To generalize this result for spectral triples to get
“unbounded” spectral metric spaces.
26 / 26
Kamran Reihani
Dynamical Systems on Spectral Metric Spaces