Spectral triples for equicontinuous actions and
metrics on state spaces
Kamran Reihani
University of Kansas
Special Session on Recent Progress in Operator Algebras
AMS Central Section Meeting, University of Nebraska-Lincoln
October 14
Joint work with Jean Bellissard and Matilde Marcolli
“Dynamical systems on spectral metric spaces”
1 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral Triples
2 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral Triples
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 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral Triples
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 → B(H),
2 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral Triples
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 → B(H),
2. D has compact resolvent: (1 + D 2 )−1 ∈ K(H),
2 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral Triples
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 → B(H),
2. D has compact resolvent: (1 + D 2 )−1 ∈ K(H),
3. the set C 1 (X ) of elements a ∈ A such that
π(a) dom(D) ⊆ dom(D) and [D, π(a)] is a bounded operator
on dom(D) is dense in A.
2 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral Triples
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 → B(H),
2. D has compact resolvent: (1 + D 2 )−1 ∈ K(H),
3. the set C 1 (X ) of elements a ∈ A such that
π(a) dom(D) ⊆ dom(D) and [D, π(a)] is a bounded operator
on dom(D) is dense in A.
Lemma
C 1 (X ) with kakD := kak + k[D, π(a)]k is a Banach ∗-algebra,
which is closed under holomorphic functional calculus in A.
2 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 1
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Kamran Reihani
Spectral triples for equicontinuous actions
Example 1
(M, g ) = compact Rimannian (spinc ) manifold
A = C (M)
H = Hilbert space of L2 -sections of the spinor bundle
D = Dirac operator
3 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 1
(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 )
3 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 1
(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:
3 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 1
(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).
3 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
Γ = discrete group
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
Γ = discrete group
` : Γ → R+ a length function, i.e. `(e) = 0, `(γ −1 ) = `(γ),
`(γ1 γ2 ) ≤ `(γ1 ) + `(γ2 ) (for example, the word length associated
with a finite set of generators of a finitely generated group)
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
Γ = discrete group
` : Γ → R+ a length function, i.e. `(e) = 0, `(γ −1 ) = `(γ),
`(γ1 γ2 ) ≤ `(γ1 ) + `(γ2 ) (for example, the word length associated
with a finite set of generators of a finitely generated group)
λ = left regular representation on `2 (Γ) : (λ(g )ξ)(t) = ξ(g −1 t)
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
Γ = discrete group
` : Γ → R+ a length function, i.e. `(e) = 0, `(γ −1 ) = `(γ),
`(γ1 γ2 ) ≤ `(γ1 ) + `(γ2 ) (for example, the word length associated
with a finite set of generators of a finitely generated group)
λ = left regular representation on `2 (Γ) : (λ(g )ξ)(t) = ξ(g −1 t)
Cr∗ (Γ) = C ∗ (λ(Γ)) ⊂ B(`2 (Γ)), the reduced group C ∗ -algebra
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
Γ = discrete group
` : Γ → R+ a length function, i.e. `(e) = 0, `(γ −1 ) = `(γ),
`(γ1 γ2 ) ≤ `(γ1 ) + `(γ2 ) (for example, the word length associated
with a finite set of generators of a finitely generated group)
λ = left regular representation on `2 (Γ) : (λ(g )ξ)(t) = ξ(g −1 t)
Cr∗ (Γ) = C ∗ (λ(Γ)) ⊂ B(`2 (Γ)), the reduced group C ∗ -algebra
M` = the multiplication operator (M` ξ)(t) = `(t)ξ(t) on `2 (Γ)
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Example 2
Γ = discrete group
` : Γ → R+ a length function, i.e. `(e) = 0, `(γ −1 ) = `(γ),
`(γ1 γ2 ) ≤ `(γ1 ) + `(γ2 ) (for example, the word length associated
with a finite set of generators of a finitely generated group)
λ = left regular representation on `2 (Γ) : (λ(g )ξ)(t) = ξ(g −1 t)
Cr∗ (Γ) = C ∗ (λ(Γ)) ⊂ B(`2 (Γ)), the reduced group C ∗ -algebra
M` = the multiplication operator (M` ξ)(t) = `(t)ξ(t) on `2 (Γ)
Proposition
If limg →∞ `(g ) = ∞ then (Cr∗ (Γ), `2 (Γ), M` ) is a spectral triple.
Moreover, k[M` , λ(g )]k = `(g ), for all g ∈ Γ.
4 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Connes metric
5 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Connes metric
I
5 / 10
A spectral triple (A, H, D) defines the following Connes
pseudo-metric on the state space S(A) by setting
Kamran Reihani
Spectral triples for equicontinuous actions
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}
5 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
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
5 / 10
Define the following objects
Kamran Reihani
Spectral triples for equicontinuous actions
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}
5 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
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)
5 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
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.
5 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
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).
5 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with w ∗ -topology
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with 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.
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with 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).
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with 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),
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with 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 ,
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with 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)
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Compatibility of Connes metric with 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)
Caution: The conditions (ii) or (iii) are often very difficult to
check! Very different approaches were taken to treat the group
C ∗ -algebras of Zn and hyperbolic groups.
6 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral metric spaces
7 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral metric spaces
Definition
A (compact) spectral metric space is a spectral triple
X = (A, H, D) for a unital separable C ∗ -algebra A such that
7 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral metric spaces
Definition
A (compact) spectral metric space is a spectral triple
X = (A, H, D) for a unital separable C ∗ -algebra A such that
(i) the representation of A on H is non-degenerate (unital),
7 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral metric spaces
Definition
A (compact) spectral metric space is a spectral triple
X = (A, H, D) for a unital separable C ∗ -algebra A such that
(i) the representation of A on H is non-degenerate (unital),
(ii) A0D = {a ∈ A ; [D, a] = 0} is trivial (= C1),
7 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Spectral metric spaces
Definition
A (compact) spectral metric space is a spectral triple
X = (A, H, D) for a unital separable C ∗ -algebra A such that
(i) the representation of A on H is non-degenerate (unital),
(ii) A0D = {a ∈ A ; [D, a] = 0} is trivial (= C1),
(iii) the Lipschitz Ball BLip (X ) has precompact image in the
normed space A/A0D .
7 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
spectral triples for crossed products
8 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
spectral triples for crossed products
Assume X = (A, H, D) is a spectral triple. we will try to build a
spectral triple for A oα Z:
8 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
spectral triples for crossed products
Assume X = (A, H, D) is a spectral triple. we will try to build a
spectral triple for A oα Z:
Hilbert Space: K = H ⊗ `2 (Z) ⊗ C2 = `2 (Z, H ⊕ H)
f ∈ K ⇒ f = (fn )n∈Z ; fn = (fn+ , fn− ), fn,± ∈ H
8 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
spectral triples for crossed products
Assume X = (A, H, D) is a spectral triple. we will try to build a
spectral triple for A oα Z:
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
n
A oα Z by π̂( n an u ) = n π̂(an )û n
8 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
spectral triples for crossed products
Assume X = (A, H, D) is a spectral triple. we will try to build a
spectral triple for A oα Z:
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
n
A oα Z by π̂( n an u ) = n π̂(an )û n
Dirac Operator: Kasparov’s external product of D and ∂
0
D
−
ın
b
Df
=
fn
D + ın
0
n
8 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
spectral triples for crossed products
Assume X = (A, H, D) is a spectral triple. we will try to build a
spectral triple for A oα Z:
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
n
A oα Z by π̂( n an u ) = n π̂(an )û n
Dirac Operator: Kasparov’s external product of D and ∂
0
D
−
ın
b
Df
=
fn
D + ın
0
n
b is (essentially) self-dajoint with compact resolvent
D
8 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
b
Commutators with D:
0 −ı
b
[D, û]f
=
fn−1
ı 0
n
9 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
b
Commutators with D:
0 −ı
b
[D, û]f
=
fn−1
ı 0
n
9 / 10
b π̂(a)]f
[D,
n
=
0 1
1 0
Kamran Reihani
[D, π ◦ α−n (a)]fn
Spectral triples for equicontinuous actions
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
Definition
Let X = (A, H, D) be a spectral triple. We say that the
automorphism α ∈ Aut(A) is equicontinuous if
9 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
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
Definition
Let X = (A, H, D) be a spectral triple. We say that the
automorphism α ∈ Aut(A) is equicontinuous if
I
9 / 10
a ∈ C 1 (X ) ⇔ α(a) ∈ C 1 (X )
Kamran Reihani
Spectral triples for equicontinuous actions
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
Definition
Let X = (A, H, D) be a spectral triple. We say that the
automorphism α ∈ Aut(A) is equicontinuous if
9 / 10
I
a ∈ C 1 (X ) ⇔ α(a) ∈ C 1 (X )
I
supn∈Z k[D, π ◦ α−n (a)k < ∞ for all a ∈ C 1 (X ).
Kamran Reihani
Spectral triples for equicontinuous actions
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
Definition
Let X = (A, H, D) be a spectral triple. We say that the
automorphism α ∈ Aut(A) is equicontinuous if
I
a ∈ C 1 (X ) ⇔ α(a) ∈ C 1 (X )
I
supn∈Z k[D, π ◦ α−n (a)k < ∞ for all a ∈ C 1 (X ).
Theorem (Bellissard-Marcolli-R)
b is a spectral triple. Moreover, if (A, H, D) is a
(A oα Z, K , D)
spectral metric space and α is an equicontinuous automorphism of
b is a spectral metric space.
A, then (A oα Z, K , D)
9 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Some examples ,
10 / 10
Kamran Reihani
Spectral triples for equicontinuous actions
Some examples ,
I
10 / 10
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
Spectral triples for equicontinuous actions
Some examples ,
10 / 10
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
Spectral triples for equicontinuous actions
Some examples ,
10 / 10
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
Spectral triples for equicontinuous actions
Some examples ,
10 / 10
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
Spectral triples for equicontinuous actions