On the uniform convergence of Cesaro averages for C -dynamical - PDF document

On the uniform convergence of Cesaro averages for C ˚ -dynamical systems Francesco Fidaleo University of Tor Vergata August 5, 2019

introduction Let p A , Φ q be a C ˚ -dynamical system based on an identity-preserving ˚ -endomorphism Φ of the unital C*-algebra A . We study the uniform convergence of Cesaro averages n ´ 1 M a,λ p n q : “ 1 λ ´ k Φ k p a q , ÿ a P A , n k “ 0 uniformly for values λ in the unit circle. For such a purpose, we define a spectral set σ p ph , f q p Φ q Ă T canonically associated to the pp given dynamical system, and show that n Ñ`8 M a,λ p n q “ 0 lim whenever λ P T � σ p ph , f q p Φ q . pp If in addition, if p A , Φ q is uniquely ergodic w.r.t. the fixed-point algebra, then we can provide ` ˘ conditions for which the sequence M a,λ p n q n

uniformly converges, even for λ P σ ph pp p Φ q , pro- viding the formula of such a limit. To end, we also discuss some simple exam- ples arising from quantum probability, the first one not enjoying the property to be uniquely ergodic w.r.t. the fixed point subalgebra, and the second one satisfying such a strong ergodic property, to which our results apply. Other more involved examples coming from noncom- mutative geometry ( i.e. the noncommutative 2-torus) can be exhibited. The present talk is based on the papers: (i) F. Fidaleo Uniform Convergence of Cesaro Averages for Uniquely Ergodic C ˚ -Dynamical Systems , Entropy 20 (2018), 987.

(ii) S. Del Vecchio, F. Fidaleo, L. Giorgetti, S. Rossi The Anzai skew-product for the noncommutative torus , preprint 2019. (iii) F. Fidaleo On the Uniform Convergence of Ergodic Averages for C ˚ -Dynamical Sys- tems , preprint 2019. On the uniform convergence of ergodic averages With T : “ t λ P C | | λ | “ 1 u we denote the unit circle of the complex plane. It is homeomor- phic to the interval r 0 , 2 π q by θ P r 0 , 2 π q ÞÑ e ´ ıθ , after identifying the endpoints 0 and 2 π . A (discrete) C ˚ -dynamical system is a triplet p A , Φ , ϕ q consisting of a C ˚ -algebra, a positive map Φ : A Ñ A and a state ϕ P S p A q such that ϕ ˝ Φ “ ϕ . Consider the Gelfand-Naimark-Segal

` ˘ (GNS for short) representation . If H ϕ , π ϕ , ξ ϕ in addition Φ p a q ˚ Φ p a q ď ϕ p a ˚ a q , ` ˘ a P A , ϕ then there exists a unique contraction V ϕ, Φ P B p H ϕ q such that V ϕ, Φ ξ ϕ “ ξ ϕ and V ϕ, Φ π ϕ p a q ξ ϕ “ π ϕ p Φ p a qq ξ ϕ , a P A . ` ˘ The quadruple is called the H ϕ , π ϕ , V ϕ, Φ , ξ ϕ covariant GNS representation associated to the triplet p A , Φ , ϕ q . If Φ is multiplicative, hence a ˚ -homomorphism, then V ϕ, Φ is an isometry with final range V ϕ, Φ V ˚ ϕ, Φ , the orthogonal projection onto the subspace π ϕ p A q ξ ϕ . Now we specialise the matter to C ˚ -dynamical systems p A , Φ , ϕ q such that A is a unital C ˚ - algebra with unity 1 I ” 1 I A , and Φ is multiplica- tive and unital preserving.

Denote by A Φ : “ � ( a P A | Φ p a q “ a the fixed- point subalgebra, and σ ph � ( pp p Φ q : “ λ P T | λ is an eigenvalue of Φ the set of the peripheral eigenvalues of Φ ( i.e. the peripheral pure-point spectrum ), with A λ I P A Φ “ the relative eigenspaces. Obviously, 1 A 1 . Analogously, for the invariant state ϕ P S p A q , consider the pure-point peripheral spectrum σ ph � ( pp p V ϕ, Φ q : “ λ P T | λ is an eigenvalue of V ϕ, Φ of V ϕ, Φ . Denote with P λ P B p H ϕ q the orthog- onal projection onto the eigenspace generated by the eigenvectors associated to λ P T , with the convention P λ “ 0 if λ R σ ph pp p V ϕ, Φ q . Let S p A q Φ be the (convex, ˚ -weakly compact) set of all invariant states under the action of

the ˚ -endomorphism Φ, and define the full pe- ripheral pure point spectrum as σ p ph , f q σ ph pp p V ϕ, Φ q | ϕ P S p A q Φ ( ď � p Φ q : “ . pp Notice that, it is a spectral set canonically as- sociated to the C ˚ -dynamical system p A , Φ q . We have the following Theorem : Let p A , Φ q be a C ˚ -dynamical system. For λ P T � σ p ph , f q p Φ q , we have pp n ´ 1 1 λ ´ k Φ k p a q “ 0 , ÿ lim (1) n Ñ`8 n k “ 0 uniformly, for each a P A . Proof (sketch): If (1) does not hold, then there exists an invariant state ϕ , for which the spectral measure of V ϕ, Φ has an atom corre- sponding to λ “ e ´ ıθ . But this contradicts λ R σ p ph , f q p Φ q . pp

Uniquely ergodic C ˚ -dynamical systems The C ˚ -dynamical system p A , Φ q is said to be uniquely ergodic w.r.t. the fixed point subal- ř n ´ 1 gebra if the ergodic average 1 k “ 0 Φ k p a q con- n verges for each fixed a P A . In such a situation, there is a unique invariant conditional expec- tation E 1 : A Ñ A 1 given by n ´ 1 1 ÿ Φ k p a q , E 1 p x q : “ lim a P A . n Ñ`8 n k “ 0 If A 1 “ C , then E 1 p x q “ ϕ p x q 1 I A with ϕ P S p A q is an invariant state, which is indeed unique ( i.e. S p A q Φ is the singleton t ϕ u ). Therefore, the C ˚ - dynamical system p A , Φ q is said to be uniquely ergodic if there exists only one invariant state ϕ for the dynamics induced by Φ. For a uniquely ergodic C ˚ -dynamical system, we simply write p A , Φ , ϕ q by pointing out that ϕ P S p A q is the unique invariant state.

Here, there are some standard results relative to uniquely ergodic C ˚ -dynamical systems. Proposition : Let the C ˚ -dynamical system p A , Φ , ϕ q be uniquely ergodic. Then σ ph pp p Φ q is a subgroup of T , and the corresponding eigenspaces A λ , λ P σ ph pp p Φ q are generated by a single unitary u λ . We have the following immediate corollary of the above result: Corollary : Let the C ˚ -dynamical system p A , Φ , ϕ q be uniquely ergodic. Then σ ph pp p Φ q Ă σ ph pp p V ϕ, Φ q . The main result involving the uniquely ergodic C ˚ -dynamical systems is the following

Theorem : Let p A , Φ , ϕ q be a uniquely ergodic C ˚ -dynamical pp p Φ q Ť σ ph system. Fix λ P σ ph pp p V ϕ, Φ q c . Then for each a P A , n ´ 1 1 Φ k p a q λ ´ k “ ϕ p u ˚ ÿ λ a q u λ , lim n n k “ 0 uniformly for n Ñ `8 , where u λ P A λ is any unitary eigenvalue corresponding to λ P σ ph pp p Φ q (with the convention that if λ P σ ph pp p V ϕ, Φ q c , then u λ “ 0). Proof : First consider the case λ P σ ph pp p Φ q , and take a unitary eigenvector u λ P A λ , unique up to a phase-factor. Since Φ is multiplicative, we have n ´ 1 ˆ 1 ˙ ϕ p u ˚ Φ k p u ˚ ÿ λ a q u λ “ u λ lim λ a q n n k “ 0 n ´ 1 ˆ 1 ˙ Φ k p a q λ ´ k ÿ “ lim . n n k “ 0

The case λ R σ ph pp p V ϕ, Φ q follows by the result in the previous section because σ p ph , f q p Φ q “ σ ph pp p V ϕ, Φ q . pp � We can exhibit simple examples based on the tensor product construction, for which σ ph pp p Φ q � σ ph pp p V ϕ, Φ q and for some a P A and λ P σ ph pp p V ϕ, Φ q � σ ph pp p Φ q such that n ´ 1 1 λ ´ k Φ k p a q ÿ lim n n k “ 0 fails to exist, even in the weak topology. More complicated examples of this phenomenon are constructed by using the cross-product con- struction ( i.e. a ”genuine” noncommutative framework) coming from the noncommutative 2-torus. Concerning the C ˚ -dynamical systems p A , Φ q which are uniquely ergodic w.r.t. the fixed

I A , for λ P σ ph point subalgebra A 1 � C 1 pp p Φ q we can provide conditions on A λ for which there exists a norm one projection E λ : A Ñ A λ such that n ´ 1 1 λ ´ k Φ k p a q “ E λ p a q , ÿ lim a P A . (2) n n k “ 0 More precisely, suppose that u P A λ is an isom- etry or a co-isometry. We can prove that λ P σ ph � pp p Φ q | A λ contains an isometry Ă σ p ph , f q ( p Φ q . or a co-isometry pp In addition, (i) A Q x ÞÑ E λ p x q : “ E 1 p xu ˚ q u P A λ (isometry case), (ii) A Q x ÞÑ E λ p x q : “ uE 1 p u ˚ x q P A λ (co-isometry- case),

and (2) holds true. Notice that, for uniquely ergodic C ˚ -dynamical systems p A , Φ , ϕ q ( i.e. when A 1 “ C 1 I A ), this is always the case because E λ p a q “ ϕ p u ˚ λ x q u λ “ ϕ p xu ˚ λ q u λ , where u λ is the unique unitary (up to a phase- factor) generating A λ . Examples We are listing simple examples coming from quantum probability for which the obtained re- sults apply. More complicated examples can be obtained by considering skew-products on the noncommutative 2-torus.

the monotone case We consider the C ˚ -dynamical system p m , s q where m is the concrete C ˚ -algebra generated by the identity I “ 1 I m and the monotone cre- ators t m : n | n P Z u acting on the monotone Fock space Γ mon p ℓ 2 p Z qq on ℓ 2 p Z q . It has the structure m “ a ` C I where I R a , and thus the state at infinity ω 8 is meaningful. The one-step shift s is defined on generators as s p m : j q “ m : j ` 1 , j P Z . The main properties of p m , s q are summarised as follows: – for the fixed-point subalgebra, m s “ C I , – the set of all invariant states S p m q s “ � ( p 1 ´ t q ω o ` tω 8 | t P r 0 , 1 s