C*-algebras, composition operators and dynamics

Transcription

1 University of Florida SEAM 23, March 9, 2007

2 The setting: D = fjzj < 1g C T = fjzj = 1g dm=normalized Lebesgue measure on T ' : D! D holomorphic (' 6= const.) H 2 = Hardy space, P : L 2! H 2 Riesz proj. Some operators on H 2 : Toeplitz operators: For f 2 L 1 (T), g 2 H 2, T f g := P(fg ) Composition operators: For g 2 H 2, C'g = g '

3 General question: What can be said about the C*-algebra generated by a set of composition operators? (or composition and Toeplitz operators)? We seek C*-algebraic relations that obtain between Toeplitz and composition operators (and their adjoints).

4 One easy relation: Lemma For any ' 2 Hol(D) and f 2 H 1, C ' T f = T f 'C ' Proof: From denitions for any h 2 H 2, (C ' T f )h = C ' (fh) = (f ')(h ') = (T f 'C ' )h

5 Polar decomposition of automorphisms Covariance algebras The extension of the crossed product Theorem (Bourdon-MacCluer '07) Let ' be inner. Then is Toeplitz, with symbol C ' C ' F (z) = 1 j'(0)j2 jz '(0)j 2 Proof: Recall the Brown-Halmos criterion: T is Toeplitz if and only if S TS = T. Now S C ' C 'S = C ' T ' T 'C ' = C ' C ' To nd symbol, apply to C ' C ' to scalars...

6 Polar decomposition of automorphisms Covariance algebras The extension of the crossed product Let now be a group of automorphisms of D. What can we say about C = C fc ' : ' 2 g E.g., how is it related to C*-algebras associated to? We will need the notion of a covariance algebra: Consider: X a compact Hausdor space G a group of homeomorphisms of X In the category of C*-algebras, these correspond to C(X ), a commutative unital C*-algebra C (u(g)), where u : G! U(H) is a unitary representation

7 Polar decomposition of automorphisms Covariance algebras The extension of the crossed product Denition Given: X = compact Hausdor space G = group of homeomorphisms of X G acts on C(X ) via (g f )(x) = f (g 1 x). A covariant representation of X ; G is a triple (; u; H) with satisfying : C(X )! B(H) a *-homomorphism u : G! U(H) a unitary representation u(g) (f )u(g) = (g f ) A covariance algebra is a C*-subalgebra of B(H) generated by a covariant representation.

8 Polar decomposition of automorphisms Covariance algebras The extension of the crossed product Consider now ' 2 Aut(D). Dene U ' = C ' (C ' C ') 1=2 Suppose also that is discrete and non-elementary (any orbit accumulates at at least three points of T). Using the two relations proved so far, it can be shown that U ' U = U ' + compact U ' T f U ' = T f ' 1 S 2 C fc ' : ' 2 g + compact for all '; 2 and all f 2 C(T). It follows that C fc ' : ' 2 g = C fs; U ' : ' 2 g

9 Polar decomposition of automorphisms Covariance algebras The extension of the crossed product Theorem (J., to appear) Let be a non-elementary Fuchsian group Aut(D). Let C = C fc ' : ' 2 g There is an exact sequence of C*-algebras 0! K! C! C(T)! 0 The quotient C =K is generated by the images in the Calkin algebra of of T f and U ' = C ' (C C ' ') 1=2. By previous slide, the quotient is thus a covariance algebra for (T; ). With more work it can be shown that it is isomorphic to the full crossed product C*-algebra C(T) which is \universal" among covariance algebras (any other is a quotient of this).

10 What are the \next" examples? Analogy between dynamics of groups and rational functions on the Riemann sphere (\Sullivan's dictionary") suggests replacing with = Fuchsian group = group leaving T invariant f' n g = iterates of rational function leaving T invariant That is: replace C with the C*-algebra generated by C ' with ' a nite Blaschke product. Then look for 0! K! C (C ' )!??! 0 where?? is some kind of \covariance algebra" for the dynamical system (T; ').

11 Consider X = compact Hausdor space ' : X! X continuous, surjective Suppose L : C(X )! C(X ) is a positive linear map satisfying L(f 1 (f 2 ')) = L(f 1 ) f 2 for all f 1 ; f 2 2 C(X ) (examples later). The Exel crossed product for the triple (X ; '; L) is the universal C*-algebra generated by C(X ) and an operator V satisfying (f ')V = Vf V fv = L(f ) \redundancies"... Thinking f = T f ; V = C ' (mod K) leads us to consider the operators C ' T f C '

12 Theorem Suppose ' : D! D is inner and f 2 L 1 (T). Then is a Toeplitz operator. Proof: Brown-Halmos again: C ' T f C ' S C T ' f C ' S = C T T ' ' f T ' C ' = C T ' f j'j 2C ' (' analytic) = C T ' f C ' (' inner) Which Toeplitz operator is it?

13 Given ' : D! D, 2 T, + '(z) Re '(z) = 1 j'(z)j2 j '(z)j 2 = Z 1 jzj 2 j zj 2 d () The measures are the for '. Denition (The Aleksandrov operator) Let f be a Borel function on T. Dene A ' (f )() = Z f () d ()

14 Theorem (Aleksandrov '87) The Aleksandrov operator A ' (f )() = Z f d extends to a well-dened, bounded operator on C(T) L p (T), 1 p 1; also H p if '(0) = 0 BMO, VMO, Besov spaces...

15 The Cauchy transform of : Assume now '(0) = 0. Then for jj < 1; jj = 1, A ' (k )() = Z 1 1 d () = 1 1 '() = C '(k )() Since A ' is bounded on H 2, this proves Theorem (inner case: Lotto-McCarthy '93; general case{?) If '(0) = 0, then C ' = A ' Application: we can now compute the symbol of C T ' f C '...

16 Theorem (J., in progress) Let ' : D! D be inner and f 2 L 1 (T). Then is Toeplitz, with symbol A ' (f ). C ' T f C ' There is an \asymptotic" version for general ': Denition Say A 2 B(H 2 ) is (weakly) asymptotically Toeplitz if lim n!1 S n AS n exists (WOT). If so, limit is a Toeplitz operator T g ; g is called the asymptotic symbol of A.

17 Theorem (J., in progress) If ' : D! D, f 2 L 1, then C ' T f C ' is asymptotically Toeplitz, with asymptotic symbol A s '(f )() := Z f d (recall = h m + ). Moreover if E = f : j'()j = 1g and f = 0 a.e. on E c, then C ' T f C ' is Toeplitz. (Converse holds if f 0.)

18 Returning to the case of nite Blaschke products, it is not hard to show that the Clark measures are given by = X '()= 1 j' 0 ()j So A ' (f )() is a certain weighted average of f over the preimages of. For general inner ', the set f 2 T : '() = g is a carrier for but has atoms only at points where ' has an angular derivative.

19 Denition (the transfer operator) ' : X! X continuous, surjective, nitely valent g : X! R, continuous Dene for f 2 C(X ) L g (f )(x) = L g X '(y )=x exp(g(y))f (y) is called the transfer operator or Perron-Frobenius-Ruelle operator; it is bounded and completely positive on C(X ). For all f 1 ; f 2 2 C(X ) we have L g (f 1 (f 2 ')) = L g (f 1 ) f 2 In particular if L g (1) 1 then L g is a left inverse for C '.

20 Proto-"Perron-Frobenius"-theorem: (unital version, for simplicity) Suppose L g (1) 1. With suitable hypotheses on ('; g), L n g (f )! c 1 uniformly, where c is a scalar. The assignment f! c determines a probability measure satisfying L g = Under good conditions such is unique; it describes the asymptotic distribution of the (weighted) backward orbits of '.

21 Recall that for a rational map of the Riemann sphere, its Julia set J is the complement of the maximal open set on which the iterates of are a normal family. (For a nite Blaschke product J T; either J = T or J is a Cantor set.) Theorem (Denker-Urbanski '91 (simplied version)) Given: rational with Julia set J g : J! R Holder continuous, L g (1) 1 Then there is a unique measure with support equal to J such that 8f 2 C(J), L n g (f )! uniformly. Z J f d 1

22 Now x a nite Blaschke product ' and a Holder continuous function g : X! R. We have a transfer operator L g (f )(z) = X '()=z exp(g())f () dened on C(T). For simplicity we assume L g (1) 1; we call such g an admissible weight. Theorem Let g be an admissible weight. Then there exists h 2 A(D) such that V := T h C ' satises for all f 2 C(T). V T f V = T Lg (f )

23 By suitably modifying Przytycki's proof of the Denker-Urbanski result, we obtain: Theorem Given: ' a nite Blaschke product g : T! R an admissible weight Then for all f 2 C(T), L n g (f )! Z f d g 1 uniformly, where g is the unique exp(g) conformal measure supported on the Julia set of '.

24 For a xed admissible weight g, we earlier obtained a weighted composition operator V := T h C ' such that V T f V = T Lg (f ) for all f 2 C(T). Using the Perron-Frobenius result for L g we obtain Theorem Fix an admissible weight g and V as above. For all f 2 C(T), Z V n T f V n! f d g I in norm.

25 In general some weight is needed to obtain convergence; for example let 3z '(z) = 3 + z The orbit of 0 under ' tends radially to 1; using this one can show kc n ' C n 'k! 1:

26 Since the measure g always has support equal to the Julia set of ', the composition operator C ' can \see" the Julia set via iteration of the completely positive map: Corollary Let ' be a nite Blaschke product, g an admissible weight and V = T h C ' as above. Let f 2 C(T); f 0. Then f vanishes on the Julia set J T of ' if and only if in norm. V n T f V n! 0

27 Let ' be inner, with Clark measures f g. Consider the normalized Aleksandrov operator fa ' (f )() = Z f d k k In the rational case this corresponds to the admissible weight g() = log j' 0 ()j k '() k

28 Conjecture: For any inner function ' and all f 2 C(T), the sequence fa ' n (f ) converges uniformly to a scalar c. If so, c = R f d and describes the (weighted) \asymptotic distribution" of the backwards orbits of ' on T. This would be particularly interesting when ' has an attracting xed point on T; presumably supp() ( T (a \Julia set" for ' on T). Where does Holder continuity come in? (Possibly via Matheson's smoothness theorem; which implies that A ' is bounded on the Holder classes.)

29 Theorem Let ' be a nite Blaschke product (deg ' 2), with xed point in D g : T! R Holder continuous C(T) o ';L N the Exel crossed product Then there is an exact sequence of C*-algebras 0! K! C (S; C ' )! C(T) o ';L N! 0