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Joint with Alain Connes) a Spectral Approach to Geometry

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Joint with Alain Connes) a Spectral Approach to Geometry Spectral truncations in noncommutative geometry and operator systems Walter van Suijlekom (joint with Alain Connes) A spectral approach to geometry Can one hear the shape of a drum? (Kac, 1966) Or, more precisely, given a Riemannian manifold M, does the spectrum of wave numbers k in the Helmholtz equation 2 ∆M u = k u determine the geometry of M? The disc Wave numbers on the disc 40 30 20 10 50 100 150 200 The square Wave numbers on the square 25 20 15 10 5 50 100 150 200 Isospectral domains 2 But, there are isospectral domains in R : (Gordon, Webb, Wolpert, 1992) so the answer to Kac's question is no Weyl's estimate Nevertheless, certain information can be extracted from spectrum, such 40 as dimension25 d of M: N(Λ) = #wave numbers ≤ Λ 3020 Ωd Vol(M) ∼ Λd d(2π)d 15 20 p For the disc and square this is conrmed by the parabolic shapes ( Λ): 10 10 5 5050 100100 150150 200200 Noncommutative geometry If combined with the C ∗-algebra C(M), then the answer to Kac' question is armative. Connes' reconstruction theorem [2008]: (C(M); DM ) ! (M; g) Spectral data • This mathematical reformulation of geometry in terms of spectral data requires the knowledge of all eigenvalues of the Dirac operator. • From a physical standpoint this is not very realistic: detectors have limited energy ranges and resolution. We develop the mathematical formalism for (noncommutative) geometry with only part of the spectrum. This is in line with earlier work of [D'AndreaLizziMartinetti 2014], [GlaserStern 2019], [Berendschot 2019] and based on [arXiv:2004.14115] The usual story Given cpt Riemannian spin manifold (M; g) with spinor bundle S on M. • the C ∗-algebra C(M) • the self-adjoint Dirac operator DM • both acting on Hilbert space L2(M; S) 2 spectral triple: (C(M); L (M; S); DM ) Reconstruction of distance function [Connes 1994]: d(x; y) = sup fjf (x) − f (y)j : k[DM ; f ]k ≤ 1g f 2C(M) f x y x y Spectral triples More generally, we consider a triple (A; H; D) • a C ∗-algebra A • a self-adjoint operator D with compact resolvent and bounded commutators [D; a] for a 2 A ⊂ A • both acting (boundedly, resp. unboundedly) on Hilbert space H Generalized distance function: • States are positive linear functionals φ : A ! C of norm 1 • Pure states are extreme points of state space • Distance function on state space of A: d(φ, ) = sup fjφ(a) − (a)j : k[D; a]k ≤ 1g a2A Spectral truncations Given (A; H; D) we project onto part of the spectrum of D: • H 7! PH, projection onto closed Hilbert subspace • D 7! PDP, still a self-adjoint operator • A 7! PAP, this is not an algebra any more (unless P 2 A) Instead, PAP is an operator system: (PaP)∗ = Pa∗P. So, we turn to study (PAP; PH; PDP). We expect: • a distance formula on states of PAP. • a rich symmetry: isometries of (A; H; D) remain isometries of (PAP; PH; PDP) Operator systems Denition (Choi-Eros 1977) An operator system is a ∗-closed vector space E of bounded operators. For convenience we take E to be nite-dimensional, to contain the identity operator, and act on a xed Hilbert space H. • E is ordered: cone E+ ⊆ E of positive operators, in the sense that T 2 E+ i h ; T i ≥ 0;( 2 H): • in fact, E is completely ordered: cones Mn(E)+ ⊆ Mn(E) of positive operators on Hn for any n. Maps between operator systems E; F are complete positive maps in the sense that their extensions Mn(E) ! Mn(F ) are positive for all n. Isomorphisms are complete order isomorphisms States spaces of operator systems • The existence of a cone E+ ⊆ E of positive elements allows to speak of states on E as positive linear functionals of norm 1. • Also, the dual E d of an operator system is an operator system, with E d E d T 0 T E + = φ 2 : φ( ) ≥ ; 8 2 + and similarly for the complete order. We have E d d ∼ E as cones in E d d ∼ E. • ( )+ = + ( ) = • It follows that we have the following useful correspondence: d pure states on E ! extreme rays in (E )+ and the other way around. C ∗-envelope of an operator system Arveson introduced the notion of C ∗-envelope for operator systems in 1969, Hamana established existence and uniqueness in 1979. A C ∗-extension κ : E ! A of an operator system E is given by a complete order isomorphism onto κ(E) ⊆ A such that C ∗(κ(E)) = A. A C ∗-envelope of an operator systems is a C ∗-extension κ : E ! A with the following universal property: E κ / A O 9!ρ λ B Shilov boundaries There is a useful description of C ∗-envelopes using Shilov ideals. Denition Let κ : E ! A be a C ∗-extension of an operator system. A boundary ideal is given by a closed 2-sided ideal I ⊆ A such that the quotient map q : A ! A=I is completely isometric on κ(E) ⊆ A. The Shilov ideal is the largest of such boundary ideals. Proposition Let κ : E ! A be a C ∗-extension. Then there exists a Shilov boundary ∗ ideal J andC env(E) =∼ A=J. As an example consider the operator system of continuous harmonic functions Charm(D) on the closed disc. Then by the maximum modulus principle the Shilov boundary is S 1. Accordingly, its C ∗-envelope is C(S 1). Propagation number of an operator system One lets E ◦n be the norm closure of the linear span of products of ≤ n elements of E. Denition The propagation number prop(E) of E is dened as the smallest integer ◦n ∗ ∗ n such that κ(E) ⊆ Cenv(E) is a C -algebra. Returning to harmonic functions in the disk we have prop(Charm(D)) = 1. Proposition The propagation number is invariant under complete order isomorphisms, as well as under stable equivalence: prop(E) = prop(E ⊗min K) Operator system spectral triples Denition An operator system spectral triple is a triple (E; H; D) where E is an operator system in B(H), H is a Hilbert space and D is a self-adjoint operator in H with compact resolvent and such that [D; T ] is a bounded operator for all T 2 E ⊂ E. It gives a distance function for states φ, on E using the same formula: d(φ, ) = sup fjφ(T ) − (T )j : k[D; T ]k ≤ 1g T 2E We will illustrate this with spectral truncations of the circle. Spectral truncation of the circle Consider the circle (C(S 1); L2(S 1); D = −id=dx) ikx • Eigenvectors of D are Fourier modes ek (x) = e for k 2 Z • Orthogonal projection P Pn onto span e1 e2 en = Cf ; ;:::; g • The space C(S 1)(n) := PC(S 1)P is an operator system • Any T = PfP in C(S 1)(n) can be written as a Toeplitz matrix t t t t 0 0 −1 ··· −n+2 −n+1 1 t1 t0 t−1 t−n+2 B . .. C . t t . PfP ∼ tk−l = B 1 0 C kl B . C @ .. .. A tn−2 t−1 tn−1 tn−2 ··· t1 t0 ∗ 1 (n) ∼ 1 (n) We have: Cenv(C(S ) ) = Mn(C) and prop(C(S ) ) = 2 (for any n) 1 (n) n and C(S ) ; C ; D = diagf1; 2;:::; ng is an operator system spectral triple. Dual operator system: FejérRiesz We introduce the FejérRiesz operator system C ∗ : (Z)(n) • functions on S 1 with a nite number of non-zero Fourier coecients: a = (:::; 0; a−n+1; a−n+2;:::; a−1; a0; a1;:::; an−2; an−1; 0;:::) P ikx 1 • an element a is positive i k ak e is a positive function on S . Proposition 1. The Shilov boundary of the operator system C ∗ is S 1. (Z)(n) 2. The C ∗-envelope of C ∗ is given by C ∗ . (Z)(n) (Z) 3. The propagation number is innite. Lemma (Fejér, Riesz) Let I ⊆ [−m; m] be an interval of length m + 1. Suppose that p z Pm p zk is a Laurent polynomial such that p 0 for all ( ) = k=−m k (ζ) ≥ ζ 2 C for which jζj = 1. Then there exists a Laurent polynomial q z P q zk so that p q 2 for all S 1 . ( ) = k2I k (ζ) = j (ζ)j ζ 2 ⊂ C Proposition 1. The extreme rays in C ∗ are given by the elements a a ( (Z)(n))+ = ( k ) P k 1 for which the Laurent series k ak z has all its zeroes on S . 2. The pure states of C ∗ are given by a P a k ( S 1). (Z)(n) 7! k k λ λ 2 Pure states on the Toeplitz matrices The duality between C S 1 (n) and C ∗ is given by ( ) (Z)(n) C S 1 (n) C ∗ ( ) × (Z)(n) ! C X (T = (tk−l )k;l ; a = (ak )) 7! ak t−k k Proposition 1. The extreme rays in C S 1 (n) are f f for any S 1. ( )+ γ(λ) = j λih λj λ 2 2. The pure states of C(S 1)(n+1) are given by functionals T 7! hξ; T ξi n+1 where the vector ξ = (ξ0; : : : ; ξn) 2 C is such that the P n−k 1 polynomial z 7! k ξk z has all its zeroes on S . 1 (n+1) ∼ n 3. The pure state space P(C(S ) ) = T =Sn is the quotient of the n-torus by the symmetric group on n objects.
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