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introduction Let pA, Φq be a C˚-dynamical system based
- n an identity-preserving ˚-endomorphism Φ of
On the uniform convergence of Cesaro averages for C -dynamical - - PDF document
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 , q be a C -dynamical system based on an identity-preserving -endomorphism of
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uniformly converges, even for λ P σph
pppΦq, pro-
viding the formula of such a limit. To end, we also discuss some simple exam- ples arising from quantum probability, the first
- ne 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.
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(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
- f Ergodic Averages for C˚-Dynamical Sys-
tems, preprint 2019. On the uniform convergence of ergodic averages With T :“ tλ P C | |λ| “ 1u we denote the unit circle of the complex plane. It is homeomor- phic to the interval r0, 2πq by θ P r0, 2πq ÞÑ e´ıθ, after identifying the endpoints 0 and 2π. A (discrete) C˚-dynamical system is a triplet
pA, Φ, ϕq consisting of a C˚-algebra, a positive
map Φ : A Ñ A and a state ϕ P SpAq such that ϕ˝Φ “ ϕ. Consider the Gelfand-Naimark-Segal
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(GNS for short) representation
`
Hϕ, πϕ, ξϕ
˘
. If in addition ϕ
`
Φpaq˚Φpaq
˘ ď ϕpa˚aq ,
a P A , then there exists a unique contraction Vϕ,Φ P BpHϕq such that Vϕ,Φξϕ “ ξϕ and Vϕ,Φπϕpaqξϕ “ πϕpΦpaqqξϕ , a P A . The quadruple
`
Hϕ, πϕ, Vϕ,Φ, ξϕ
˘
is called the covariant GNS representation associated to the triplet pA, Φ, ϕq. If Φ is multiplicative, hence a ˚-homomorphism, then Vϕ,Φ is an isometry with final range Vϕ,ΦV ˚
ϕ,Φ,
the orthogonal projection onto the subspace πϕpAqξϕ. Now we specialise the matter to C˚-dynamical systems pA, Φ, ϕq such that A is a unital C˚- algebra with unity 1 I ” 1 IA, and Φ is multiplica- tive and unital preserving.
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Denote by AΦ :“
- a P A | Φpaq “ a
(
the fixed- point subalgebra, and σph
pppΦq :“
- 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 Aλ the relative eigenspaces. Obviously, 1 I P AΦ “ A1. Analogously, for the invariant state ϕ P SpAq, consider the pure-point peripheral spectrum σph
pppVϕ,Φq :“
- λ P T | λ is an eigenvalue of Vϕ,Φ
(
- f Vϕ,Φ. Denote with Pλ P BpHϕq the orthog-
- nal projection onto the eigenspace generated
by the eigenvectors associated to λ P T, with the convention Pλ “ 0 if λ R σph
pppVϕ,Φq.
Let SpAqΦ be the (convex, ˚-weakly compact) set of all invariant states under the action of
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the ˚-endomorphism Φ, and define the full pe- ripheral pure point spectrum as σpph,fq
pp
pΦq :“ ď
σph
pppVϕ,Φq | ϕ P SpAqΦ(
. Notice that, it is a spectral set canonically as- sociated to the C˚-dynamical system pA, Φq. We have the following Theorem: Let pA, Φq be a C˚-dynamical system. For λ P T σpph,fq
pp
pΦq, we have
lim
nÑ`8
1 n
n´1
ÿ
k“0
λ´kΦkpaq “ 0 , (1) 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 Vϕ,Φ has an atom corre- sponding to λ “ e´ıθ. But this contradicts λ R σpph,fq
pp
pΦq.
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Uniquely ergodic C˚-dynamical systems The C˚-dynamical system pA, Φq is said to be uniquely ergodic w.r.t. the fixed point subal- gebra if the ergodic average 1
n
řn´1
k“0 Φkpaq con-
verges for each fixed a P A. In such a situation, there is a unique invariant conditional expec- tation E1 : A Ñ A1 given by E1pxq :“ lim
nÑ`8
1 n
n´1
ÿ
k“0
Φkpaq, a P A. If A1 “ C, then E1pxq “ ϕpxq1 IA with ϕ P SpAq is an invariant state, which is indeed unique (i.e. SpAqΦ is the singleton tϕu). Therefore, the C˚- dynamical system pA, Φq 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
pA, Φ, ϕq by pointing out that ϕ P SpAq is the
unique invariant state.
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Here, there are some standard results relative to uniquely ergodic C˚-dynamical systems. Proposition: Let the C˚-dynamical system pA, Φ, ϕq be uniquely
- ergodic. Then σph
pppΦq is a subgroup of T, and
the corresponding eigenspaces Aλ, λ P σph
pppΦq
are generated by a single unitary uλ. We have the following immediate corollary of the above result: Corollary: Let the C˚-dynamical system pA, Φ, ϕq be uniquely
- ergodic. Then σph
pppΦq Ă σph pppVϕ,Φq.
The main result involving the uniquely ergodic C˚-dynamical systems is the following
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Theorem: Let pA, Φ, ϕq be a uniquely ergodic C˚-dynamical
- system. Fix λ P σph
pppΦq Ť σph pppVϕ,Φqc. Then for
each a P A, lim
n
1 n
n´1
ÿ
k“0
Φkpaqλ´k “ ϕpu˚
λaquλ ,
uniformly for n Ñ `8, where uλ P Aλ is any unitary eigenvalue corresponding to λ P σph
pppΦq
(with the convention that if λ P σph
pppVϕ,Φqc,
then uλ “ 0). Proof: First consider the case λ P σph
pppΦq, and take
a unitary eigenvector uλ P Aλ, unique up to a phase-factor. Since Φ is multiplicative, we have ϕpu˚
λaquλ “uλ lim n
ˆ1
n
n´1
ÿ
k“0
Φkpu˚
λaq
˙ “ lim
n
ˆ1
n
n´1
ÿ
k“0
Φkpaqλ´k
˙
.
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The case λ R σph
pppVϕ,Φq follows by the result in
the previous section because σpph,fq
pp
pΦq “ σph
pppVϕ,Φq .
- We can exhibit simple examples based on the
tensor product construction, for which σph
pppΦq
σph
pppVϕ,Φq and for some a P A and λ P σph pppVϕ,Φq
σph
pppΦq such that
lim
n
1 n
n´1
ÿ
k“0
λ´kΦkpaq 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 pA, Φq which are uniquely ergodic w.r.t. the fixed
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point subalgebra A1 C1 IA, for λ P σph
pppΦq we
can provide conditions on Aλ for which there exists a norm one projection Eλ : A Ñ Aλ such that lim
n
1 n
n´1
ÿ
k“0
λ´kΦkpaq “ Eλpaq, a P A. (2) More precisely, suppose that u P Aλ is an isom- etry or a co-isometry. We can prove that
- λ P σph
pppΦq | Aλ contains an isometry
- r a co-isometry
( Ă σpph,fq
pp
pΦq .
In addition, (i) A Q x ÞÑ Eλpxq :“ E1pxu˚qu P Aλ (isometry case), (ii) A Q x ÞÑ Eλpxq :“ uE1pu˚xq P Aλ (co-isometry- case),
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and (2) holds true. Notice that, for uniquely ergodic C˚-dynamical systems pA, Φ, ϕq (i.e. when A1 “ C1 IA), this is always the case because Eλpaq “ ϕpu˚
λxquλ “ ϕpxu˚ λquλ,
where uλ is the unique unitary (up to a phase- factor) generating Aλ. Examples We are listing simple examples coming from quantum probability for which the obtained re- sults apply. More complicated examples can be
- btained by considering skew-products on the
noncommutative 2-torus.
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the monotone case We consider the C˚-dynamical system pm, sq where m is the concrete C˚-algebra generated by the identity I “ 1 Im and the monotone cre- ators tm:
n | n P Zu acting on the monotone Fock
space Γmonpℓ2pZqq on ℓ2pZq. It has the structure m “ a ` CI where I R a, and thus the state at infinity ω8 is meaningful. The one-step shift s is defined on generators as spm:
jq “ m: j`1, j P Z.
The main properties of pm, sq are summarised as follows: – for the fixed-point subalgebra, ms “ CI, – the set of all invariant states Spmqs “
- p1 ´ tqωo ` tω8 | t P r0, 1s
(
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is the convex combination of the vacuum state ωo and the state at infinity ω8. Therefore, pm, sq cannot be uniquely ergodic w.r.t. the fixed-point subalgebra. Indeed, it can be viewed by direct inspection because 1 n
n´1
ÿ
k“0
skpmlm:
lq “ 1
n
n´1
ÿ
k“0
ml`km:
l`k Ó Peo ,
the self-adjoint projection onto the subspace generated by the vacuum vector eo. But such a convergence in the strong operator topology, cannot be uniform. We can check that σpppαq “ t1u “ σpfq
pppαq, ∗
and thus for each x P m and λ P T t1u, lim
nÑ`8
1 n
n´1
ÿ
k“0
λ´kskpxq “ 0 ,
∗For ˚-automorphisms, all spectra are included in the
unit circle T, and therefore we omit the suffix ”ph”.
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uniformly. the boolean case We consider the C˚-dynamical system pb, sq, where b is the concrete C˚-algebra generated by the identity and the boolean creators tb:
n |
n P Zu acting on the boolean Fock space Γboolepℓ2pZqq
- n ℓ2pZq, and (with an abuse of notation) s
is the one-step shift acting on generators as spb:
jq “ b: j`1, j P Z.
It was shown that b is nothing but the C˚- algebra K
`
ℓ2ptou\Z
˘ `CI generated by all com-
pact operators acting on Γboolepℓ2pZqq “ ℓ2`
tou\
Z
˘
and the identity I :“ 1 Iℓ2ptou\Zq. The shift is therefore generated by the adjoint action adV , with V defined on the canonical basis
teou \ tej | j P Zu of ℓ2` tou \ Z ˘
by V eo “ eo , V ej “ ej`1 , j P Z .
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Furthermore, we have: – for the fixed point subalgebra, b1 ” bs “ CPeo
À CP K
eo;
– the set of all invariant states Spbqs “
- p1 ´ tqωo ` tω8 | t P r0, 1s
(
is the convex combination of the vacuum state ωo and the state at infinity ω8. – with a P K
`
ℓ2ptou \ Z
˘
, b Q A`bI ÞÑ E1pa`bIq :“
` xAeo, eoy`b ˘
Peo`bP K
eo P b1
is a conditional expectation, invariant un- der the shift s; – the C˚-dynamical system pb, sq is uniquely mixing (and therefore uniquely ergodic) w.r.t.
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the fixed-point subalgebra with conditional expectation E1. Notice that the set Spbqs of the boolean invari- ant states has the same structure as that Spmqs
- f the monotone invariant ones. Furthermore,
σpppsq “ t1u “ σpfq
pppsq as for the monotone case.
Differently to pm, sq, the C˚-dynamical system
pb, sq is uniquely ergodic w.r.t. the fixed-point
- subalgebra. Therefore, for the convergence of
ergodic averages we have for x P b and λ P T, lim
nÑ`8
1 n
n´1
ÿ
k“0
λ´kskpxq “
" E1pxq
if λ “ 1 , if λ ‰ 1 . In order to provide an example for which the involved spectra are non trivial, we consider the tensor product construction of the previ-
- us boolean C˚-dynamical system with the ir-
rational rotations on the unit circle. For the irrational number θ P p0, 1q, denote by Rθ the rotation on T of the angle 2πθ. Let
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pA, αq be the tensor product C˚-dynamical sys-
tem, where A “ CpTq b b “ C
`
T; b
˘
, αpfqpzq :“ s
`
fpe2πıθzq
˘
, z P T , f P C
`
T; b
˘
, Finally, for each f P C
`
T; b
˘
define E1pfq :“
ˆż b E1 ˙ pfq “ ¿
E1
`
fpzq
˘ dz
2πız . Notice that, with 1 P CpTq the constant func- tion identically equal to 1, E1 is projecting onto the fixed-point subalgebra A1 “ 1 b b „ b. The main results involving such a dynamical system, whose spectral sets are non-trivial by construction, are: Proposition: The C˚-dynamical system pA, αq is uniquely er- godic w.r.t. the fixed-point subalgebra with expectation E1.
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In addition, σpppαq “
- e2πılθ | l P Z