Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Noncommutative Sierpinski gasket andspectral triples
Daniele Guido
Università di Roma Tor Vergata-
collaboration with F.Cipriani, T.Isola, J-L.Sauvageot
Cornell U., June 15, 2017
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Outline
1 A functional approach to the Sierpinski gasketFunctional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
2 The noncommutative Sierpinski gasketThe C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Outline
1 A functional approach to the Sierpinski gasketFunctional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
2 The noncommutative Sierpinski gasketThe C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Classical description: points, edges and cells.
Sierpinski gasket: K =⋃
i=1,2,3 wi(K )Similarities: wσ = wσn · ... · wσ1 , |σ| = n.Cells: K =
⋃|σ|=n Kσ, Kσ = wσ(K )
Edges: E =⋃
n En, E0 = {e1,e2,e3},En = {wσ(e), |σ| = n,e ∈ E0},e−,e+ end points of e.
Vertices: V =⋃
n Vn, V0 = {x1, x2, x3},Vn = {wσ(x), |σ| = n, x ∈ V0}.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Classical description: points, edges and cells.
KΣ
Sierpinski gasket: K =⋃
i=1,2,3 wi(K )Similarities: wσ = wσn · ... · wσ1 , |σ| = n.Cells: K =
⋃|σ|=n Kσ, Kσ = wσ(K )
Edges: E =⋃
n En, E0 = {e1,e2,e3},En = {wσ(e), |σ| = n,e ∈ E0},e−,e+ end points of e.
Vertices: V =⋃
n Vn, V0 = {x1, x2, x3},Vn = {wσ(x), |σ| = n, x ∈ V0}.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Classical description: points, edges and cells.
Sierpinski gasket: K =⋃
i=1,2,3 wi(K )Similarities: wσ = wσn · ... · wσ1 , |σ| = n.Cells: K =
⋃|σ|=n Kσ, Kσ = wσ(K )
Edges: E =⋃
n En, E0 = {e1,e2,e3},En = {wσ(e), |σ| = n,e ∈ E0},e−,e+ end points of e.
Vertices: V =⋃
n Vn, V0 = {x1, x2, x3},Vn = {wσ(x), |σ| = n, x ∈ V0},
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Classical description: points, edges and cells.
Sierpinski gasket: K =⋃
i=1,2,3 wi(K )Similarities: wσ = wσn · ... · wσ1 , |σ| = n.Cells: K =
⋃|σ|=n Kσ, Kσ = wσ(K )
Edges: E =⋃
n En, E0 = {e1,e2,e3},En = {wσ(e), |σ| = n,e ∈ E0},e−,e+ end points of e.
Vertices: V =⋃
n Vn, V0 = {x1, x2, x3},Vn = {wσ(x), |σ| = n, x ∈ V0}.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Approximation algebras
For the Sierpinski gasket K , define An, n ≥ 0, as the algebra ofconstant functions on cells of level n + 1 which are well-definedon vertices in Vn. Observe that:• functions in An are ill-defined (or have a jump discontinuity) atmost on vertices in Vn+1 \ Vn.• Since any cell of level n + 1 contains exactly one vertex in Vn,An may be identified with C(Vn).• there is a natural embedding
An+1 ↪→ C3 ⊗An, (self-similarity)
• hence An ↪→ (C3)⊗(n+1).
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
The reconstruction of C(K )
Set (C3)⊗∞ = lim−→
(C3)⊗n, s.t. An ↪→ (C3)⊗∞, and consider
A∞ = {limn
an ∈ (C3)⊗∞,an ∈ An ⊂ (C3)⊗∞}
TheoremA∞ is a C∗-algebra, and coincides with C(K ).
NB: An is neither a sub-algebra of An+1 nor a sub-algebra ofA∞.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Examples of inclusions
Let us recall that each injective unital homomorphism offinite-dimensional C∗-algebras (e.g. finite direct sums of fullmatrix algebras) can be described up to isomorphism by abipartite graph, with vertices labeled by natural numbers.Example of an inclusion:C⊕M2 ⊕ C→ C⊕M3 ⊕M2 ⊕M3 ⊕ C
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Diagram for Sierpinski gasket
Since the algebras An are abelian, the inclusionAn+1 ⊂ C3 ⊗An can be equivalently described by the inclusionAn+1 ⊂Mr ⊗An. Consider the inclusion ψ : A1 ⊂M3 ⊗A0,where black dots have label 3, and white dots have label 1:
Figure: Sierpinski gasket diagram for the inclusion ψ.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
The inclusion ψ reconstructs the gasket
TheoremThe Sierpinski gasket K may be fully recovered by ψ.
It turns out that the algebras may be decomposed as A0 = C3,An = C3 ⊕ C3 ⊕Rn, n ≥ 1, for a suitable family Rn, and theinclusion An+1 ⊂ C3 ⊗An, n ≥ 0, splits asψ : C3 ⊕ C3 ↪→ C3 ⊗ C3, Rn+1 = C3 ⊗
(An C3).
As a consequence, R1 = 0, Rn+1 = C3 ⊗(C3 ⊕Rn
), n ≥ 1.
Therefore the An’s are reconstructed inductively, and A∞ isdefined as above.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
An inductive way of numbering cells
A cell of level n is determined by an elementw ∈ {1,2,3}n. For cells of level 1 thenumbers are assigned in a clockwise order.For a cell cw , w ∈ {1,2,3}n, its threesub-cells are numbered as in the figure.
A vertex v ∈ V0, v ⊂ cw , |w | = n + 1 is described by theprojection w ∈ An, where w = wn . . .w1 = ewnwn ⊗ · · · ⊗ ew1w1 .A vertex v ∈ Vn \ V0, v ⊂ cw ∩ cw ′ , |w | = n + 1 is described bythe sum of projection w + w ′ ∈ An.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
The algebras An
Let us denote by 1,2,3 the matrix units e11,e22,e33 inC3 ⊂M3(C) and set α1 = (11 + 33), α2 = (21 + 13), α3 =(31 + 23) ∈ (C3)⊗2 ⊂M3(C)⊗2:
Proposition
An is the sub-algebra of (C3)⊗(n+1) generated by{xk ⊗ αj2n−k−1 j = 1,2,3, k = 0 · · · n − 1, xk ∈ (C3)⊗k ,
j2n j = 1,2,3.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Harmonic extension
The harmonic extension ϕ0 : Ao → A1 can be written, ongenerators, as
ϕ0(1) = 12 +25α1 +
25α2 +
15α3
ϕ0(2) = 22 +15α1 +
25α2 +
25α3
ϕ0(3) = 32 +25α1 +
15α2 +
25α3
Set ϕn :M⊗n3 ⊗ C3 →M⊗n
3 ⊗ C3 ⊗ C3, ϕn = idn ⊗ ϕ0, andobserve that An ⊂M⊗n
3 ⊗ C3. It turns out that ϕn(An) ⊂ An+1,and is the harmonic extension.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
Energy form
Since An ⊂ (C3)⊗n ⊗ C3, any a ∈ An may be uniquelydecomposed as a =
∑i=1,2,3 ai ⊗ ei , with ai ∈ (C3)⊗n. It turns
out that the combinatorial energy on An is given bya ∈ An → En[a] =
∑i 6=j
tr(|ai − aj |2
), tr the non-normalized trace.
As is known, a relation between harmonic extension andcombinatorial energy holds: En+1[ϕn(a)] = 3/5 En[a], a ∈ An,and, ∀f ∈ A∞, (5/3)n En[ρn(f )] is increasing, where ρn denotesthe restriction map from K to Vn
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
A discrete spectral triple for C(K )
The spectral triple studied in [D.G.-T.Isola 2003], [D.G.-T.Isola2017, JNCG] may be described as follows:• Hn = (C3)⊗n ⊗ E , E = {a ∈M3 : aii = 0, i = 1,2,3}.• Fn : Hn → Hn,Fn(h1 ⊗ h2) = h1 ⊗ hT
2 , Dn = 2nFn.• πn : A∞ → B(Hn), πn(a)h = ρn(a)h.• H = ⊕nHn, F = ⊕nFn, D = ⊕nDn, π = ⊕nπn.
Theorem (D.G.-T.Isola 2017, JNCG)
(H,D, π) is a spectral triple on A∞. It recovers the Hausdorffdimension and measure of K , the geodesic distance, and theDirichlet energy (up to a constant) via the formula
ress=δtr(|[D,a]|2|D|−s), δ = 2− log 5/3
log 2.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Outline
1 A functional approach to the Sierpinski gasketFunctional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple
2 The noncommutative Sierpinski gasketThe C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Quantization procedure: the NC-gasket
Let K the Sierpinski gasket, ψ : C3 ⊗ C3 →M3 ⊗ C3 as above.Set Rq
1 = {0} and construct inductively Rqn such that
Rqn+1 =M3 ⊗
(C3 ⊕Rq
n), n ≥ 1. Finally set Aq
0 = A0,Aq
n = C3 ⊕ C3 ⊕Rqn,n ≥ 1.
LemmaBy construction, the map ψ determines an inclusionAq
n+1 ⊂M3 ⊗Aqn. Hence Aq
n ⊂ (M3)⊗(n+1) ⊂ UHF (3∞).
Theorem
The space Aq∞ = {lim
nan ∈ UHF (3∞),an ∈ Aq
n} is a C∗-algebra(functions on the quantized K ) containing C(K ).
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
The level 2 diagram for the classical Sierpinski gasket
Figure: A2 ⊂M3 ⊗A1 ⊂M9 ⊗A0
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
The level 2 diagram for the quantum Sierpinski gasket
Figure: A2 ⊂M3 ⊗A1 ⊂M9 ⊗A0
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
The algebras Aqn for the gasket
As above, 1,2,3 denote the matrix units e11,e22,e33 inM3(C)and α1 = (11 + 33), α2 = (21 + 13), α3 = (31 + 23) ∈ (C3)⊗2 ⊂M3(C)⊗2. Then
Proposition
Aqn is the sub-algebra ofM3(C)⊗(n+1) generated by{xk ⊗ αj2n−k−1 j = 1,2,3, k = 0 · · · n − 1, xk ∈M⊗k
3 ,j2n j = 1,2,3.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
The restriction maps
PropositionThe restriction maps ρn extend to unital ∗-homomorphisms (stilldenoted by ρn) from Aq
∞ to Aqn. We have:
• a = limn ρn(a),a ∈ Aq∞,
• ker(ρn) = (M3)⊗n ⊗ ker(ρ0),
• ker(ρ0) ∩ C(K ) = C0(K ),• Aq
∞ is generated by the elements{f , x ⊗ g : f ∈ C(K ), x ∈ (M3)
⊗n,g ∈ C0(K )}.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Representations of Aq∞
Since Aqn ⊂M
⊗(n+1)3 , the ρn can be thought as finite
dimensional representations of Aq∞ on (C3)⊗(n+1).
Theorem
Let π be a representation of Aq∞ which is disjoint from ρn,
n ≥ 0. Then π extends to UHF (3∞), andπ(UHF (3∞))′′ = π(Aq
∞)′′. In particular, π is faithful.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
The harmonic extension
Let ϕ0 : Ao → A1 as above, setϕn :M⊗n
3 ⊗ C3 →M⊗n3 ⊗ C3 ⊗ C3, ϕn = idn ⊗ ϕ0, and observe
that Aqn ⊂M⊗n
3 ⊗ C3. Then
Proposition
ϕn(Aqn) ⊂ A
qn+1, and it is a completely positive contraction, still
called harmonic extension.
Composing the ϕn’s, we get ϕn,n+k : Aqn → A
qn+k .
Proposition
∀a ∈ Aqn,∃ϕn,∞(a) := limk ϕn,n+k (a) ∈ Aq
∞. It turns out thatρn(ϕn,∞(a)) = a, a ∈ Aq
n, limn ϕn,∞(ρn(a)) = a,a ∈ Aq∞.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
The Dirichlet form
Since Aqn ⊂M⊗n
3 ⊗ C3, any a ∈ Aqn may be uniquely
decomposed as a =∑
i=1,2,3 ai ⊗ ei , with ai ∈M⊗n3 . One sets
Eqn [a] =
∑i 6=j
tr(|ai − aj |2
), tr the non-normalized trace.
Theorem
If a ∈ Aqn, 5
3Eqn+1[ϕn(a)] = Eq
n [a]. If a ∈ Aq∞, (5/3)nEq
n [ρn(a)] isincreasing. The set B ⊂ Aq
∞ for which the limit Eq∞[a] is finite is
the domain of a closed quadratic form on Aq∞. Eq
∞ is a Dirichletform on L2(Aq
∞, τ) with domain B, where τ is the restriction ofthe standard trace on UHF (3∞).
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
A spectral triple on Aq∞
For the spectral triple on Aq∞ we keep H,F and D as in the
classical case, set πqn : Aq
∞ → B(Hn), πqn (a)h = ρn(a)h,
πq = ⊕nπqn .
Theorem
(H,D, πq) is a spectral triple on Aq∞. The abscissa of
convergence d of tr(|D|−s) equals log 3
log 2 (Hausdorff dimension),the formula a→ ress=d tr(a|D|−s) gives, up to a constant, therestriction to Aq
∞ of the finite trace on UHF (3∞) (self-similarmeasure), the Dirichlet energy (up to a constant) is obtainedwith the same formula as in the classical case, with the same δ.
Daniele Guido Noncommutative Sierpinski gasket and spectral triples
A functional approach to the Sierpinski gasketThe noncommutative Sierpinski gasket
The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple
Thank you for your attention!
Daniele Guido Noncommutative Sierpinski gasket and spectral triples