noncommutative sierpinski gasket and spectral...

$: Noncommutative Sierpinski gasket and spectral triplespi.math.cornell.edu/~fractals/6/slides/Guido.pdfNoncommutative Sierpinski gasket and spectral triples Daniele Guido Università$
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


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



Functional description: algebrasSelf-similarity and Bratteli diagramsThe algebras An for the gasketHarmonic extension, energy form and spectral triple

Outline






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}.





KΣ




Edges: E =⋃


Vertices: V =⋃









Edges: E =⋃


Vertices: V =⋃

n Vn, V0 = {x1, x2, x3},Vn = {wσ(x), |σ| = n, x ∈ V0},








Edges: E =⋃


Vertices: V =⋃





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).




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∞.




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




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.




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.




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.




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.




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.




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




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.



The C∗-algebraHarmonic structure and Dirichlet formThe spectral triple

Outline






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 ).




The level 2 diagram for the classical Sierpinski gasket

Figure: A2 ⊂M3 ⊗A1 ⊂M9 ⊗A0




The level 2 diagram for the quantum Sierpinski gasket

Figure: A2 ⊂M3 ⊗A1 ⊂M9 ⊗A0




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.




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 )}.




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.




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∞.




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∞).




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.




Thank you for your attention!


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