[PPT] - Spectral distance: Results for Moyal plane and Noncommutative Torus PowerPoint Presentation

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Spectral distance: Results for Moyal plane and Noncommutative Torus

Jean-Christophe Wallet

Laboratoire de Physique Th´ eorique, CNRS, Universit´ e Paris-Sud 11

*

coll. with:
E. Cagnache, F. d’Andrea, E. Jolibois, P. Martinetti

Workshop on Noncommutative Geometry: Topics in Mathematics and Mathematical Physics, Orsay, 24-26 November 2009

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Overview

◮ Many available exemples of Noncommutative (NC) spaces (fuzzy spaces,

almost commutatives spaces, deformations, Connes-Landi spheres, Connes-Dubois-Violette spaces,...). Relatively few works devoted to study of the spectral distance.

2

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Overview

◮ Many available exemples of Noncommutative (NC) spaces (fuzzy spaces,

almost commutatives spaces, deformations, Connes-Landi spheres, Connes-Dubois-Violette spaces,...). Relatively few works devoted to study of the spectral distance.

◮ In Connes NC geometry, there is a natural notion of distance, called spectral

distance, on the space of states of the (suitable C* completion of the) associative algebra involved in the spectral triple (ST) one starts from. [For

reviews, see e.g. Connes, Landi.].

2

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Overview

◮ Many available exemples of Noncommutative (NC) spaces (fuzzy spaces,

almost commutatives spaces, deformations, Connes-Landi spheres, Connes-Dubois-Violette spaces,...). Relatively few works devoted to study of the spectral distance.

◮ In Connes NC geometry, there is a natural notion of distance, called spectral

distance, on the space of states of the (suitable C* completion of the) associative algebra involved in the spectral triple (ST) one starts from. [For

reviews, see e.g. Connes, Landi.].

◮ On finite dimensional complete Riemann spin manifold, spectral distance

between pure states coincides with geodesic distance between corresponding

points. In NC case, actual meaning of spectral distance not clear and much

more explicit exemples are needed.

2

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Overview

◮ Many available exemples of Noncommutative (NC) spaces (fuzzy spaces,

almost commutatives spaces, deformations, Connes-Landi spheres, Connes-Dubois-Violette spaces,...). Relatively few works devoted to study of the spectral distance.

◮ In Connes NC geometry, there is a natural notion of distance, called spectral

distance, on the space of states of the (suitable C* completion of the) associative algebra involved in the spectral triple (ST) one starts from. [For

reviews, see e.g. Connes, Landi.].

◮ On finite dimensional complete Riemann spin manifold, spectral distance

between pure states coincides with geodesic distance between corresponding

points. In NC case, actual meaning of spectral distance not clear and much

more explicit exemples are needed.

◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

2

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Overview

◮ Many available exemples of Noncommutative (NC) spaces (fuzzy spaces,

almost commutatives spaces, deformations, Connes-Landi spheres, Connes-Dubois-Violette spaces,...). Relatively few works devoted to study of the spectral distance.

◮ In Connes NC geometry, there is a natural notion of distance, called spectral

distance, on the space of states of the (suitable C* completion of the) associative algebra involved in the spectral triple (ST) one starts from. [For

reviews, see e.g. Connes, Landi.].

◮ On finite dimensional complete Riemann spin manifold, spectral distance

between pure states coincides with geodesic distance between corresponding

points. In NC case, actual meaning of spectral distance not clear and much

more explicit exemples are needed.

◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

◮ Study the spectral distance on the noncommutative Moyal plane.

Corresponding ST is non compact spectral triple (NCST) proposed by [Gayral,

Gracia-Bondia, Iochum, Sch¨ ucker, Varilly, CMP 2004].

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

◮ We find explicit formula for the distance between pures states (theorem 9

given below). These are vector states generated by the elements of the matrix base.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

◮ We find explicit formula for the distance between pures states (theorem 9

given below). These are vector states generated by the elements of the matrix base.

◮ Existence of states at infinite distance so that the Moyal plane as described by

the NCST proposed recently is not a compact quantum metric space in the sense of Rieffel.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

◮ We find explicit formula for the distance between pures states (theorem 9

given below). These are vector states generated by the elements of the matrix base.

◮ Existence of states at infinite distance so that the Moyal plane as described by

the NCST proposed recently is not a compact quantum metric space in the sense of Rieffel.

◮ Exemple of “truncation” of this NCST leading to Rieffel compact quantum

metric space is given.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

◮ We find explicit formula for the distance between pures states (theorem 9

given below). These are vector states generated by the elements of the matrix base.

◮ Existence of states at infinite distance so that the Moyal plane as described by

the NCST proposed recently is not a compact quantum metric space in the sense of Rieffel.

◮ Exemple of “truncation” of this NCST leading to Rieffel compact quantum

metric space is given.

◮ Part of technical machinery can be adapted very easily to the noncommutative

torus (rational, irrational). Some partial results are presented.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Content

1

Spectral distance on Moyal plane Basic properties Moyal non compact spin geometry Spectral distance on the Moyal plane Spectral distance between pure states Discussion

2

Noncommutative Torus - preliminaries basic properties Pure states on noncommutative torus Preliminary results - Spectral distance on NC Torus

3

Conclusion

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

1

Spectral distance on Moyal plane Basic properties Moyal non compact spin geometry Spectral distance on the Moyal plane Spectral distance between pure states Discussion

2

Noncommutative Torus - preliminaries

3

Conclusion

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Basic properties

The Moyal product

◮ S(R2) ≡ S: (Frechet) space of Schwarz functions, S′(R2) ≡ S′ its topological

dual space. ||.||2, ., .: L2(R2) norm and inner product.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Basic properties

The Moyal product

◮ S(R2) ≡ S: (Frechet) space of Schwarz functions, S′(R2) ≡ S′ its topological

dual space. ||.||2, ., .: L2(R2) norm and inner product. Proposition 1 (see e.g Gracia-Bondia, Varilly, JMP 1988) Associative bilinear Moyal ⋆-product defined as: ⋆ : S × S → S, ∀a, b ∈ S (a ⋆ b)(x) = 1 (πθ)2

d2yd2z a(x + y)b(x + t)e−i2yΘ−1t

yΘ−1t ≡ y µΘ−1

µν tν, Θµν = θ

1

−1

, θ ∈ R, θ = 0

Complex conjugation is an involution for the ⋆-product. One has: i)

d2x (a ⋆ b)(x) =
d2x (b ⋆ a)(x) =
d2x a(x)b(x)

ii) ∂µ(a ⋆ b) = ∂µa ⋆ b + a ⋆ ∂µb. iii) A ≡ (S, ⋆) is a non unital involutive Fr´ echet algebra.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Basic properties

The Moyal product

◮ S(R2) ≡ S: (Frechet) space of Schwarz functions, S′(R2) ≡ S′ its topological

dual space. ||.||2, ., .: L2(R2) norm and inner product. Proposition 1 (see e.g Gracia-Bondia, Varilly, JMP 1988) Associative bilinear Moyal ⋆-product defined as: ⋆ : S × S → S, ∀a, b ∈ S (a ⋆ b)(x) = 1 (πθ)2

d2yd2z a(x + y)b(x + t)e−i2yΘ−1t

yΘ−1t ≡ y µΘ−1

µν tν, Θµν = θ

1

−1

, θ ∈ R, θ = 0

Complex conjugation is an involution for the ⋆-product. One has: i)

d2x (a ⋆ b)(x) =
d2x (b ⋆ a)(x) =
d2x a(x)b(x)

ii) ∂µ(a ⋆ b) = ∂µa ⋆ b + a ⋆ ∂µb. iii) A ≡ (S, ⋆) is a non unital involutive Fr´ echet algebra.

◮ ◮ Set: X ⋆n ≡ X ⋆ X ⋆ ... ⋆ X, [a, b]⋆ ≡ a ⋆ b − b ⋆ a. From now on, introduce

complex coordinates ¯ z =

1 √ 2(x1 − ix2), z = 1 √ 2(x1 + ix2).

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Basic properties

The matrix base

Proposition 2 (see e.g Gracia-Bondia, Varilly, JMP 1988) The matrix base is the family of functions {fmn}m,n∈N ⊂ S ⊂ L2(R2) defined by fmn = 1 (θm+nm!n!)1/2 ¯ z⋆m ⋆ f00 ⋆ z⋆n, f00 = 2e−2H/θ, H = 1 2(x2

1 + x2 2)

i) One has the relations: fmn ⋆ fpq = δnpfmq, f ∗

mn = fnm, fmn, fkl = (2πθ)δmkδnl

(1) ii) There is a Frechet algebra isomorphism between A ≡ (S, ⋆) and the matrix algebra of decreasing sequences (amn), ∀m, n ∈ N defined by a =

m,n amnfmn,

∀a ∈ S, such that the semi-norms ρ2

k(a) ≡ m,n θ2k(m + 1 2)k(n + 1 2)k|amn|2 < ∞,

∀k ∈ N.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Basic properties

The matrix base

Proposition 2 (see e.g Gracia-Bondia, Varilly, JMP 1988) The matrix base is the family of functions {fmn}m,n∈N ⊂ S ⊂ L2(R2) defined by fmn = 1 (θm+nm!n!)1/2 ¯ z⋆m ⋆ f00 ⋆ z⋆n, f00 = 2e−2H/θ, H = 1 2(x2

1 + x2 2)

i) One has the relations: fmn ⋆ fpq = δnpfmq, f ∗

mn = fnm, fmn, fkl = (2πθ)δmkδnl

(1) ii) There is a Frechet algebra isomorphism between A ≡ (S, ⋆) and the matrix algebra of decreasing sequences (amn), ∀m, n ∈ N defined by a =

m,n amnfmn,

∀a ∈ S, such that the semi-norms ρ2

k(a) ≡ m,n θ2k(m + 1 2)k(n + 1 2)k|amn|2 < ∞,

∀k ∈ N. ⋆-product can be extended to other subspaces of S′ (use duality and continuity of ⋆ on S). Convenient: Hilbert spaces S ⊂ Gs,t ⊂ S′, s, t ∈ R, Gs,t = {a = amnfmn ∈ S′ / ||a||2

s,t = m,n θs+t(m+ 1 2)s(n+ 1 2)t|amn|2 < ∞}

Uses: ||a ⋆ b||s,r ≤ ||a||s,t||b||q,r, t + q ≥ 0 and ||a||u,v ≤ ||a||s,t if u ≤ s, v ≤ t. Then, for any a ∈ Gs,t and b ∈ Gq,r, b =

m,n bmnfmn, t + q ≥ 0, the

sequences cmn =

p ampbpn, ∀m, n ∈ N define the functions

c =

m,n cmnfmn, c ∈ Gs,r [See e.g Gracia-Bondia, Varilly, JMP 1988].

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ ◮ ◮ ◮ ◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}. ◮ B = {a ∈ C ∞(R2) / a and all derivatives bounded}. Set A1 ≡ (B, ⋆).

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}. ◮ B = {a ∈ C ∞(R2) / a and all derivatives bounded}. Set A1 ≡ (B, ⋆). ◮ Aθ = {a ∈ S′

/ a ⋆ b ∈ L2(R2), ∀b ∈ L2(R2)}.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}. ◮ B = {a ∈ C ∞(R2) / a and all derivatives bounded}. Set A1 ≡ (B, ⋆). ◮ Aθ = {a ∈ S′

/ a ⋆ b ∈ L2(R2), ∀b ∈ L2(R2)}. Proposition 3 (Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004)

i) Aθ is a unital C*-algebra of operator of L2(R2) with the operator norm ||.||op, ||a||op = sup0=b∈L2(R2) ||a⋆b||2

||b||2

for any a ∈ Aθ, isomorphic to L(L2(R2)).

ii) A1 is a pre C* algebra. One has A ⊂ (DL2, ⋆) ⊂ A1 ⊂ Aθ.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}. ◮ B = {a ∈ C ∞(R2) / a and all derivatives bounded}. Set A1 ≡ (B, ⋆). ◮ Aθ = {a ∈ S′

/ a ⋆ b ∈ L2(R2), ∀b ∈ L2(R2)}. Proposition 3 (Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004)

i) Aθ is a unital C*-algebra of operator of L2(R2) with the operator norm ||.||op, ||a||op = sup0=b∈L2(R2) ||a⋆b||2

||b||2

for any a ∈ Aθ, isomorphic to L(L2(R2)).

ii) A1 is a pre C* algebra. One has A ⊂ (DL2, ⋆) ⊂ A1 ⊂ Aθ.

◮ ◮ Consider a non compact spectral triple [Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004] as noncommutative generalisation of non compact

Riemannian spin manifold: (A, A1, H, D; J, χ) (2)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}. ◮ B = {a ∈ C ∞(R2) / a and all derivatives bounded}. Set A1 ≡ (B, ⋆). ◮ Aθ = {a ∈ S′

/ a ⋆ b ∈ L2(R2), ∀b ∈ L2(R2)}. Proposition 3 (Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004)

i) Aθ is a unital C*-algebra of operator of L2(R2) with the operator norm ||.||op, ||a||op = sup0=b∈L2(R2) ||a⋆b||2

||b||2

for any a ∈ Aθ, isomorphic to L(L2(R2)).

ii) A1 is a pre C* algebra. One has A ⊂ (DL2, ⋆) ⊂ A1 ⊂ Aθ.

◮ ◮ Consider a non compact spectral triple [Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004] as noncommutative generalisation of non compact

Riemannian spin manifold: (A, A1, H, D; J, χ) (2)

◮ The antiunitary operator J and involution χ will not be relevant here.

A1 ⊃ A is called a prefered unitization of A .

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple”

◮ DL2 = {a ∈ L2(R2) ∩ C ∞(R2) / a(n) ∈ L2(R2), ∀n ∈ N}. ◮ B = {a ∈ C ∞(R2) / a and all derivatives bounded}. Set A1 ≡ (B, ⋆). ◮ Aθ = {a ∈ S′

/ a ⋆ b ∈ L2(R2), ∀b ∈ L2(R2)}. Proposition 3 (Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004)

i) Aθ is a unital C*-algebra of operator of L2(R2) with the operator norm ||.||op, ||a||op = sup0=b∈L2(R2) ||a⋆b||2

||b||2

for any a ∈ Aθ, isomorphic to L(L2(R2)).

ii) A1 is a pre C* algebra. One has A ⊂ (DL2, ⋆) ⊂ A1 ⊂ Aθ.

◮ ◮ Consider a non compact spectral triple [Gayral, Gracia-Bondia, Iochum, Sch¨

ucker, Varilly, CMP 2004] as noncommutative generalisation of non compact

Riemannian spin manifold: (A, A1, H, D; J, χ) (2)

◮ The antiunitary operator J and involution χ will not be relevant here.

A1 ⊃ A is called a prefered unitization of A .

◮ H = L2(R2) ⊗ C2, i.e Hilbert space of integrable square sections of trivial

spinor bundle S = R2 ⊗ C2 with standard Hilbert product ψ, φ =

d2x(ψ∗

1φ1 + ψ∗ 2φ2) ∀ψ, φ ∈ H, ψ = (ψ1, ψ2), φ = (φ1, φ2).

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple“

◮ Define now ∂ = 1 √ 2(∂1 − i∂2), ¯

∂ =

1 √ 2(∂1 + i∂2).

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple“

◮ Define now ∂ = 1 √ 2(∂1 − i∂2), ¯

∂ =

1 √ 2(∂1 + i∂2). ◮ Unbounded Euclidean self-adjoint Dirac operator D = −iσµ∂µ (densely

defined on Dom(D) = (DL2 ⊗ C2) ⊂ H). One has σµσν + σνσµ = 2δµν, ∀µ, ν = 1, 2 and σ1 = 1 1

, σ2 =

i −i

, D = −i

√ 2

¯

∂ ∂

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Moyal non compact spin geometry

Relevant elements of the spectral “triple“

◮ Define now ∂ = 1 √ 2(∂1 − i∂2), ¯

∂ =

1 √ 2(∂1 + i∂2). ◮ Unbounded Euclidean self-adjoint Dirac operator D = −iσµ∂µ (densely

defined on Dom(D) = (DL2 ⊗ C2) ⊂ H). One has σµσν + σνσµ = 2δµν, ∀µ, ν = 1, 2 and σ1 = 1 1

, σ2 =

i −i

, D = −i

√ 2

¯

∂ ∂

◮ A can be represented faithfully on space of bounded operators on H:

π(a) = L(a) ⊗ I2, π(a)ψ = (a ⋆ ψ1, a ⋆ ψ2), ∀ψ = (ψ1, ψ2) ∈ H, ∀a ∈ A L(a): left multiplication operator by any a ∈ A. π(a) and [D, π(a)] are bounded operators on H for any a ∈ A.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

◮ Spectral distance is related naturally to spectral triple [see e.g Connes, Landi].

10

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

◮ Spectral distance is related naturally to spectral triple [see e.g Connes, Landi]. ◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

10

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

◮ Spectral distance is related naturally to spectral triple [see e.g Connes, Landi]. ◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

◮ The spectral distance can be defined as follows

Definition 4 (see e.g Connes, Landi) The spectral distance between any two states ω1 and ω2 of ¯ A is defined by d(ω1, ω2) = sup

a∈A

|ω1(a) − ω2(a)|; ||[D, π(a)]||op ≤ 1
(3)

where ||.||op is the operator norm for the representation of A in B(H).

10

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

◮ Spectral distance is related naturally to spectral triple [see e.g Connes, Landi]. ◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

◮ The spectral distance can be defined as follows

Definition 4 (see e.g Connes, Landi) The spectral distance between any two states ω1 and ω2 of ¯ A is defined by d(ω1, ω2) = sup

a∈A

|ω1(a) − ω2(a)|; ||[D, π(a)]||op ≤ 1
(3)

where ||.||op is the operator norm for the representation of A in B(H).

◮ Spectral distance between pure states: noncommutative analog of geodesic

distance between two points. Recall: spectral distance for the spectral triple encoding

the geometry of compact Riemann spin manifold equals geodesic distance.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

◮ Spectral distance is related naturally to spectral triple [see e.g Connes, Landi]. ◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

◮ The spectral distance can be defined as follows

Definition 4 (see e.g Connes, Landi) The spectral distance between any two states ω1 and ω2 of ¯ A is defined by d(ω1, ω2) = sup

a∈A

|ω1(a) − ω2(a)|; ||[D, π(a)]||op ≤ 1
(3)

where ||.||op is the operator norm for the representation of A in B(H).

◮ Spectral distance between pure states: noncommutative analog of geodesic

distance between two points. Recall: spectral distance for the spectral triple encoding

the geometry of compact Riemann spin manifold equals geodesic distance.

◮ (3) extends the notion of distance to non-pure states, i.e objects that are not

analog to points. determination of spectral distance between 2 pure states not enough to

exhaust the full metric information involved in (3) [Rieffel].

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

◮ Spectral distance is related naturally to spectral triple [see e.g Connes, Landi]. ◮ A few past studies [lattice(Dimakis, M¨

uller-Hoissen; Bimonte, Lizzi, Sparano), finite spaces (Iochum, Krajewski, Martinetti), inspired by physics (Martinetti), quantum metric spaces (Rieffel) ]

◮ The spectral distance can be defined as follows

Definition 4 (see e.g Connes, Landi) The spectral distance between any two states ω1 and ω2 of ¯ A is defined by d(ω1, ω2) = sup

a∈A

|ω1(a) − ω2(a)|; ||[D, π(a)]||op ≤ 1
(3)

where ||.||op is the operator norm for the representation of A in B(H).

◮ Spectral distance between pure states: noncommutative analog of geodesic

distance between two points. Recall: spectral distance for the spectral triple encoding

the geometry of compact Riemann spin manifold equals geodesic distance.

◮ (3) extends the notion of distance to non-pure states, i.e objects that are not

analog to points. determination of spectral distance between 2 pure states not enough to

exhaust the full metric information involved in (3) [Rieffel].

◮ Relationship with the Wasserstein distance of order 1 between probability

distributions on a metric space and the spectral distance [d’Andrea, Martinetti ].

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

Pure states

◮ Convenient to use the matrix base (Proposition 2). Start from the very simple

bservation that any vector state defined by any fmn depends only on the first

indice m ∈ N, thanks to (1).

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

Pure states

◮ ◮ ◮ Convenient to use the matrix base (Proposition 2). Start from the very simple

bservation that any vector state defined by any fmn depends only on the first

indice m ∈ N, thanks to (1). Proposition 5 The pure states of ¯ A are the vector states ωψ : ¯ A → C defined by any unit vector ψ ∈ L2(R2) of the form ψ =

m∈N ψmfm0, m∈N |ψm|2 = 1 2πθ and one has

ωψ(a) ≡

(ψ, 0), π(a)(ψ, 0)
= 2πθ
m,n∈N

ψ∗

mψnamn

(4)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

Pure states

◮ ◮ ◮ Convenient to use the matrix base (Proposition 2). Start from the very simple

bservation that any vector state defined by any fmn depends only on the first

indice m ∈ N, thanks to (1). Proposition 5 The pure states of ¯ A are the vector states ωψ : ¯ A → C defined by any unit vector ψ ∈ L2(R2) of the form ψ =

m∈N ψmfm0, m∈N |ψm|2 = 1 2πθ and one has

ωψ(a) ≡

(ψ, 0), π(a)(ψ, 0)
= 2πθ
m,n∈N

ψ∗

mψnamn

(4) Proof.

Let H0 be the Hilbert space spanned by the familly (fm0)m∈N. For any a = P

m,n amnfmn ∈ A,

ne has P

p ||L(a)fp0||2 2 = P p,m |apm|2 = ||a||2 2 < ∞ so that L(a) is Hilbert-Schmidt on H0 and

therefore compact on H0. Let π0 be this representation of A on H0 and ¯ π0(A) be the completion

f π0(A). One has

¯ π0(A) ⊆ K(H0). π0(A) involves all finite rank operators. Then ¯ π0(A) ⊇ K(H0) and so ¯ π0(A) = K(H0).This latter has a unique irreducible representation (up to unitary equivalence) and the corresponding pure states are exactly given by vectors states defined by any unit vector ψ = P

m∈N ψmfm0 ∈ H0

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

On the ”unit ball“

◮ ◮ ◮ Algebraic relations:

Proposition 6 i) The fmn’s and their derivatives fullfill: ∂fmn = n θ fm,n−1−

m + 1

θ fm+1,n; ¯ ∂fmn = m θ fm−1,n−

n + 1

θ fm,n+1, ∀m, n ∈ N (5) ii) a =

m,n amnfmn ∈ A, define ∂a ≡ m,n αmnfmn, ¯

∂a ≡

m,n βmnfmn.

a) The following relations hold: αm+1,n =

n + 1

θ am+1,n+1−

m + 1

θ am,n, α0,n =

n + 1

θ a0,n+1, ∀m, n ∈ N (6) βm,n+1 =

m + 1

θ am+1,n+1 −

n + 1

θ am,n, βm,0 =

m + 1

θ am+1,0, ∀m, n ∈ N (7) b) One has the inversion formula ap,q = δp,qa0,0 + √ θ

min(p,q)

k=0

αp−k,q−k−1 + βp−k−1,q−k √p − k + √q − k , ∀p, q ∈ N, p + q > 0 (8)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

On the ”unit ball“

◮ The condition defining the unit ball ||[D, π(a)]||op ≤ 1 can be translated into

constraints on the coefficients the expansion of ∂a and ¯ ∂a in the matrix base.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

On the ”unit ball“

◮ ◮ The condition defining the unit ball ||[D, π(a)]||op ≤ 1 can be translated into

constraints on the coefficients the expansion of ∂a and ¯ ∂a in the matrix base. Lemma 7 We set ∂a =

m,n αmnfmn and ¯

∂a =

m,n βmnfmn, for any a ∈ A and any unit

vector ϕ =

m,n ϕmnfmn ∈ L2(R2). Assume that ||[D, π(a)]||op ≤ 1.

i) The following property (P) holds: (P)

p

|αmp||ϕpn| ≤ 1 2 √ πθ and

p

|βmp||ϕpn| ≤ 1 2 √ πθ , ∀ϕ ∈ H0, ||ϕ||2 = 1, ∀m, n (9) ii) If (P) holds, then |αmn| ≤

1 √ 2 and |βmn| ≤ 1 √ 2, ∀m, n ∈ N

iii) For any radial function a ∈ A (i.e amn = 0 if m = n), ||[D, π(a)]||op ≤ 1 is equivalent to |αmn| ≤

1 √ 2 and |βmn| ≤ 1 √ 2, ∀m, n ∈ N.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance on the Moyal plane

On the ”unit ball“

Proof.

If ||[D, π(a)]||op ≤ 1, then ||∂a||op ≤

1 √ 2 and ||¯

∂a||op ≤

1 √ 2 . Use matrix base: for any ϕ ∈ H0,

||∂a ⋆ ϕ||2

2 = 2πθ P m,n | P p αmpϕpn|2. Then (def.of ||∂a||op), P m,n | P p αmpϕpn|2 ≤ 1 4πθ for

any ϕ ∈ H0 with P

m,n |ϕmn|2 = 1 2πθ . This implies

| X

p

αmpϕpn| ≤ 1 2 √ πθ , ∀ϕ ∈ H0, ||ϕ||2 = 1, ∀m, n ∈ N (10) Now, | P

p αmpϕpn| ≤ 1 2 √ πθ true for any ϕ ∈ H0 with ||ϕ||2 = 1 and one can construct ˜

ϕ with || ˜ ϕ||2 = ||ϕ||2 via αmp ˜ ϕpn = |αmp||ϕpn|. Therefore X

p

|αmp||ϕpn| ≤ 1 2 √ πθ , ∀ϕ ∈ H0, ||ϕ||2 = 1, ∀m, n ∈ N (11) Note: (11) implies (10). Property i) shown. Property ii): direct consequence of the property P. To prove iii), show that any radial function a such that |αmn| ≤

1 √ 2 and |βmn| ≤ 1 √ 2 , ∀m, n ∈ N

is in the unit ball. One first observe that if a is radial, one has αmn = 0 if m = n + 1 thanks to (6). Then, for any unit vector ψ ∈ H0 ||∂a ⋆ ψ||2

2 = 2πθ

X

p,q

| X

r

αprψrq|2 = 2πθ X

p,q

|αp,p−1ψp−1,q|2 ≤ πθ X

p,q∈N

|ψpq|2 (12) so that ||∂a||2

p ≤ 1

2 and a is in the unit ball. Similar considerations for βmn.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance between pure states

Definition 8 We denote by ωm the pure state generated by the unit vector

1 √ 2πθfm0, ∀m ∈ N.

For any a =

m,n amnfmn ∈ A, one has ωm(a) = amm.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance between pure states

Definition 8 We denote by ωm the pure state generated by the unit vector

1 √ 2πθfm0, ∀m ∈ N.

For any a =

m,n amnfmn ∈ A, one has ωm(a) = amm.

Theorem 9 The spectral distance between any two pure states ωm and ωn is d(ωm, ωn) =

θ

2

m

k=n+1

1 √ k , ∀m, n ∈ N, n < m

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Spectral distance between pure states

Proof.

From Proposition 6, αn+1,n = r n + 1 θ (an+1,n+1 − an,n) = r n + 1 θ (ωn+1,n+1(a) − ωnn(a)), ∀n ∈ N Then, use ii) of Lemma 7 implies that, for any a in the unit ball, one has |ωn+1,n+1(a) − ωnn(a))| ≤ q

θ 2 1 √n+1 , ∀n ∈ N. Then, a reapeted use of the triangular inequality

for the distance gives rises to d(ωm, ωn) ≤ q

θ 2

Pm

k=n+1 1 √ k , ∀m, n ∈ N, n < m.

To show that the upper bound is actually reached, consider the radial element ˆ a = X

p,q∈N

ˆ apqfpq, ˆ apq = δpq r θ 2

m0

X

k=p

1 √ k + 1 , m0 ∈ N fixed (13) as a linear combination of m0 + 1 − p elements of the matrix base, ˆ a ∈ A. By using ii) of Proposition 6 and iii) of Lemma 7, one easily shows that ˆ αp+1,p = − 1

√ 2 , 0 ≤ p ≤ m0 (the other

ˆ α′s vanish) and ˆ a is in the unit ball (for the ˆ βm,n, the proof is similar).

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Discussion

◮ ◮ ◮ ◮ ◮ ◮ ◮ ◮ Observe d(ωm, ωn) = d(ωm, ωp) + d(ωp, ωn) for any m < p < n ∈ N.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Discussion

◮ Observe d(ωm, ωn) = d(ωm, ωp) + d(ωp, ωn) for any m < p < n ∈ N. ◮ There exist states at infinite distance. Use radial element ˆ

a (13) to get lower bound on spectral distance between 2 pure states Prop 5. Start from ωψ′(a) − ωψ(a) = (2πθ)2

m,n,p,q

(da)mn,pqψ′∗

mψ∗ pψ′ nψq

(14) (da)mn,pq = (amnδpq − apqδmn) Using (14) yields d(ωψ′, ωψ) ≥ (2πθ)2

θ

2|

p≤k≤q

1 √ k (|ψpψ′

q|2 − |ψqψ′ p|2)|

(15) Choose now ψ0 =

1 √ 2πθf00, ψ′ 0 = 1 √ 2πθ

m
ζ(s)

(m+1)s fm0 where ζ(s) is the

Riemann zeta function, s > 1. Then d(ωψ′, ωψ) ≥ ζ(s)

θ

2

1≤k≤m

1 (m + 1)s√ k (16) The right hand side is divergent for s ≤ 3

2.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Discussion

Definition 10 (Rieffel, Contemp. Math. 2004) A Compact Quantum Metric Space (CQMS) is a order unit space A equipped with a seminorm l such that l(1) = 0 and the distance defined by d(ω1, ω2) = sup

|ω1 − ω2(a)|, / l(a) ≤ 1
(17)

induced the weak* topology on the state space of A.

◮ Then, a (unital) ST with l(a) = ||[D, π(a)]||op as seminorm is CQMS

whenever the spectral distance induces weak* topology on the state space.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Discussion

Definition 10 (Rieffel, Contemp. Math. 2004) A Compact Quantum Metric Space (CQMS) is a order unit space A equipped with a seminorm l such that l(1) = 0 and the distance defined by d(ω1, ω2) = sup

|ω1 − ω2(a)|, / l(a) ≤ 1
(17)

induced the weak* topology on the state space of A.

◮ Then, a (unital) ST with l(a) = ||[D, π(a)]||op as seminorm is CQMS

whenever the spectral distance induces weak* topology on the state space.

◮ In the Moyal case, existence of states at infinite spectral distance implies that

the NCST proposed recently is not a CQMS in the sense of Rieffel.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Spectral distance on Moyal plane

Discussion

A truncation of the Moyal ST

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

1

Spectral distance on Moyal plane

2

Noncommutative Torus - preliminaries basic properties Pure states on noncommutative torus Preliminary results - Spectral distance on NC Torus

3

Conclusion

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

basic properties

The noncommutative torus

Definition 11 (For reviews see, e.g Landi, Gracia-Bondia, Varilly) A2

θ universal C*-algebra generated by u1, u2 with u1u2 = ei2πθu2u1. Algebra of the

noncommutative torus T2

θ is the dense (unital) pre-C* subalgebra of A2 θ defined by

T2

θ = {a = i,j∈Z aijui 1uj 2 / supi,j∈Z(1 + i2 + j2)k|aij|2 < ∞}. ◮ Weyl generators defined by UM ≡ e−iπm1θm2um1 1 um2 2 , ∀M = (m1, m2) ∈ Z2.

For any a ∈ T2

θ, a = m∈Z2 aMUM. Let δ1 and δ2: canonical derivations

δa(ub) = i2πuaδab, ∀a, b ∈ {1, 2}. One has δb(a∗) = (δb(a))∗, ∀b = 1, 2. Proposition 12 One has for any M, N ∈ Z2, (UM)∗ = U−M, UMUN = σ(M, N)UM+N where the commutation factor σ : Z2 × Z2 → C satisfies σ(M +N, P) = σ(M, P)σ(N, P), σ(M, N +P) = σ(M, N)σ(M, P), ∀M, N, P ∈ Z2 σ(M, ±M) = 1, ∀M ∈ Z2 δa(UM) = i2πmaUM, ∀a = 1, 2, ∀M ∈ Z2

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

basic properties

The noncommutative torus

◮ Let τ be tracial state:

For any a =

M∈Z2 aMUM ∈ T2 θ, τ : T2 θ → C, τ(a) = a0,0.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

basic properties

The noncommutative torus

◮ Let τ be tracial state:

For any a =

M∈Z2 aMUM ∈ T2 θ, τ : T2 θ → C, τ(a) = a0,0. ◮ Hτ: GNS Hilbert space (completion of T2 θ in the Hilbert norm induced by

< a, b >≡ τ(a∗b)). One has τ(δb(a)) = 0, ∀b = 1, 2.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

basic properties

The noncommutative torus

◮ Let τ be tracial state:

For any a =

M∈Z2 aMUM ∈ T2 θ, τ : T2 θ → C, τ(a) = a0,0. ◮ Hτ: GNS Hilbert space (completion of T2 θ in the Hilbert norm induced by

< a, b >≡ τ(a∗b)). One has τ(δb(a)) = 0, ∀b = 1, 2.

◮ The even real spectral triple:

(T2

θ, H, D; J, Γ)

H = Hτ ⊗ C2. One has δ†

b = −δb, ∀b = 1, 2, in view of

< δb(a), c >= τ((δb(a)∗c) = τ(δb(a∗)c) = −τ(a∗δb(c)) = − < a, δb(c) > for any b = 1, 2 and δb(a∗) = (δb(a))∗.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

basic properties

The noncommutative torus

◮ Let τ be tracial state:

For any a =

M∈Z2 aMUM ∈ T2 θ, τ : T2 θ → C, τ(a) = a0,0. ◮ Hτ: GNS Hilbert space (completion of T2 θ in the Hilbert norm induced by

< a, b >≡ τ(a∗b)). One has τ(δb(a)) = 0, ∀b = 1, 2.

◮ The even real spectral triple:

(T2

θ, H, D; J, Γ)

H = Hτ ⊗ C2. One has δ†

b = −δb, ∀b = 1, 2, in view of

< δb(a), c >= τ((δb(a)∗c) = τ(δb(a∗)c) = −τ(a∗δb(c)) = − < a, δb(c) > for any b = 1, 2 and δb(a∗) = (δb(a))∗.

◮ Define δ = δ1 + iδ2 and ¯

δ = δ1 − iδ2. D: unbounded self-adjoint Dirac

perator D = −i 2

b=1 δb ⊗ σb, densely defined on

Dom(D) = (T2

θ ⊗ C2) ⊂ H.

D = −i δ ¯ δ

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

basic properties

The noncommutative torus

◮ Let τ be tracial state:

For any a =

M∈Z2 aMUM ∈ T2 θ, τ : T2 θ → C, τ(a) = a0,0. ◮ Hτ: GNS Hilbert space (completion of T2 θ in the Hilbert norm induced by

< a, b >≡ τ(a∗b)). One has τ(δb(a)) = 0, ∀b = 1, 2.

◮ The even real spectral triple:

(T2

θ, H, D; J, Γ)

H = Hτ ⊗ C2. One has δ†

b = −δb, ∀b = 1, 2, in view of

< δb(a), c >= τ((δb(a)∗c) = τ(δb(a∗)c) = −τ(a∗δb(c)) = − < a, δb(c) > for any b = 1, 2 and δb(a∗) = (δb(a))∗.

◮ Define δ = δ1 + iδ2 and ¯

δ = δ1 − iδ2. D: unbounded self-adjoint Dirac

perator D = −i 2

b=1 δb ⊗ σb, densely defined on

Dom(D) = (T2

θ ⊗ C2) ⊂ H.

D = −i δ ¯ δ

◮ Faithfull representation π : T2

θ → B(H) : π(a) = L(a) ⊗ I2,

π(a)ψ = (aψ1, aψ2), ψ = (ψ1, ψ2) ∈ H, ∀a ∈ T2

θ. L(a): left multiplication

perator by any a ∈ T2

θ. π(a) and [D, π(a)] bounded on H for any T2 θ.

[D, π(a)]ψ = −i(L(δb(a)) ⊗ σb)ψ = −i L(δ(a)) L(¯ δ(a)) ψ2 ψ1

(18)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Pure states on noncommutative torus

◮ Classification of the pure states in the irrational case is lacking.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Pure states on noncommutative torus

◮ Classification of the pure states in the irrational case is lacking. ◮ Consider rational case: θ = p q, p < q, p and q relatively prime, q = 1. Set

T2

p q ≡ Tp/q [see e.g Connes, Landi, Rieffel]. Unitary equivalence classes of irreps. Tp/q

classified by a torus parametrized by (α, β). Irreps. given by πα,β : Tp/q → Cq, α, β ∈ C unitaries and πα,β(u1), πα,β(u2) ∈ Mq(C) are the usual clock and shift matrices in the basis defined by ˘ ek = β−k/quk

2 e0

¯ , ∀k ∈ {0, 1, ..., q − 1} and u1e0 = α1/qe0.

Proposition 13 The set of pure states of the rational noncommutative torus is exactly the set of vector states ωψ

α,β : Tp/q → C

ωψ

α,β(a) = (ψ, πα,β(a)ψ), ∀ψ ∈ Cq, ||ψ|| = 1

(19) where ψ is given up to an overall phase. The pure states are then classified by a bundle over a commutative torus parametrized by (α, β) with fiber P(Cq). Proof.

By standard results on C*-algebras, any irrep. (πα,β, Cq) is unitarily equivalent to the GNS representation (ωψ, πα,β) for any ψ ∈ Cq. Then, the ωψ are pure states. Write now ωψ

α,β(a) = (ψ, πα,β(a)ψ) for any a ∈ Tp/q.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Preliminary results - Spectral distance on NC Torus

Lemma 14 Set δ(a) =

N∈Z2 αNUN. One has αN = i2π(n1 + in2)aN, ∀N = (n1, n2) ∈ Z2.

i) For any a in the unit ball, ||[D, π(a)||op ≤ 1 implies |αN| ≤ 1, ∀N ∈ Z2. Similar results hold for ¯ δ(a). ii) The elements ˆ aM ≡

UM 2π(m1+im2) verify ||[D, π(ˆ

aM)||op = 1, ∀M = (m1, m2) ∈ Z2, M = (0, 0)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Preliminary results - Spectral distance on NC Torus

Lemma 14 Set δ(a) =

N∈Z2 αNUN. One has αN = i2π(n1 + in2)aN, ∀N = (n1, n2) ∈ Z2.

i) For any a in the unit ball, ||[D, π(a)||op ≤ 1 implies |αN| ≤ 1, ∀N ∈ Z2. Similar results hold for ¯ δ(a). ii) The elements ˆ aM ≡

UM 2π(m1+im2) verify ||[D, π(ˆ

aM)||op = 1, ∀M = (m1, m2) ∈ Z2, M = (0, 0) Proof.

The relation involving αN obvious. Then, ||[D, π(a)||op ≤ 1 is equivalent to ||δ(a)||op ≤ 1 and ||¯ δ(a)||op ≤ 1 in view of (18). For any a ∈ A2

θ and any unit ψ = P N∈Z2 ψNUN ∈ Hτ, one has

||δ(a)ψ||2 = P

N∈Z2 | P P∈Z2 αPψN−Pσ(P, N)|2. Then ||δ(a)||op ≤ 1 implies

| P

P∈Z2 αPψN−Pσ(P, N)| ≤ 1, for any N ∈ Z2 and any unit ψ ∈ Hτ. By a straighforward

adaptation of the proof carried out for ii) of Lemma 7, this implies |αM| ≤ 1, ∀M ∈ Z2. This proves ii). Finally, iii) stems simply from an elementary calculation.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Preliminary results - Spectral distance on NC Torus

Proposition 15 Let the familly of unit vectors ΦM = ( 1+UM

√ 2 , 0) ∈ H, ∀M ∈ Z2, M = (0, 0)

generating the family of vector states of T2

θ

ωΦM : T2

θ → C, ωΦM(a) ≡ (ΦM, π(a)ΦM)H = 1

2 < (1 + UM), (a + aUM) > (20) The spectral distance between any state ωΦM and the tracial state is d(ωΦM, τ) = 1 2π|m1 + im2|, ∀M = (m1, m2) ∈ Z2, M = (0, 0) (21)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Preliminary results - Spectral distance on NC Torus

Proposition 15 Let the familly of unit vectors ΦM = ( 1+UM

√ 2 , 0) ∈ H, ∀M ∈ Z2, M = (0, 0)

generating the family of vector states of T2

θ

ωΦM : T2

θ → C, ωΦM(a) ≡ (ΦM, π(a)ΦM)H = 1

2 < (1 + UM), (a + aUM) > (20) The spectral distance between any state ωΦM and the tracial state is d(ωΦM, τ) = 1 2π|m1 + im2|, ∀M = (m1, m2) ∈ Z2, M = (0, 0) (21) Proof.

Set a = P

N∈Z2 aNUN. Using Proposition 12 yields ωΦM (a) = τ(a) + 1 2 (aM + a−M). This,

combined with Lemma 14 yields d(ωΦM , τ) ≤

1 2π|m1+im2|. Upper bound obviousley saturated by

the element ˆ aM of iii) of Lemma 14 which belongs to the unit ball.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Preliminary results - Spectral distance on NC Torus

SLIDE 63

Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Noncommutative Torus - preliminaries

Preliminary results - Spectral distance on NC Torus

Proposition 16 For any two pure states in the familly

ωek

α,β

, one has

d(ωek

α,β, ωel α′,β′)

≥ sup(m1,m2)=(0,0)

1 4π|m1+im2q|

sin 2π
m1
θ k−l

2

− φ−φ′

2q

− m2

ψ−ψ′ 2

where

k, l ∈

0, 1, ..., q − 1
and α = e2iπφ, β = e2iπψ, α′ = e2iπφ′ and β′ = e2iπψ′.

◮

Proof.

Set a = P

m1,m2∈Z am1,m2um1 1 um2 2 . One first obtains by standard calculation

ωek

α,β(a) =

X

M∈Z2

a(m1,m2q)α

m1 q βm2e−2iπθm1k =

X

M∈Z2

a(m1,m2q) exp „ −2iπ „ θm1k − m1 q φ − m2ψ «« For ˆ aM defined in ii) of Lemma 14 such that M = (m1, m2q) = (0, 0), we have |ωek

α,β(ˆ

aM) − ωel

α′,β′(ˆ

aM)| = 1 4π|m1 + im2q| ˛ ˛ ˛ ˛sin 2π „ m1 „ θ k − l 2 − φ − φ′ 2q « − m2 ψ − ψ′ 2 «˛ ˛ ˛ ˛ (22) d(ωek

α,β, ωel α′,β′) then larger than supremum of these quantities for (m1, m2) = (0, 0) and φ, ψ,

φ′, ψ′ mod 1.

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Conclusion

1

Spectral distance on Moyal plane

2

Noncommutative Torus - preliminaries

3

Conclusion

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Conclusion

◮ Determination of distance between arbitrary pure states for Moyal plane

difficult. In progress

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Conclusion

◮ Determination of distance between arbitrary pure states for Moyal plane

difficult. In progress

◮ Noncommutative torus has been undertaken (ways ”inspirated by the Moyal

case“)

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Spectral Distance, Workshop on Noncommutative Geometry, Orsay, 24 - 26 November 2009 Jean-Christophe Wallet, LPT-Orsay

Conclusion

◮ Determination of distance between arbitrary pure states for Moyal plane

difficult. In progress

◮ Noncommutative torus has been undertaken (ways ”inspirated by the Moyal

case“)

◮ Other exemples of noncommutative spaces : SU(2)q, Connes-Landi

28