Hilbert C*-modules from group actions: beyond the finite orbits case

HILBER T C ∗ -MODULES FR OM GR OUP ACTIONS: BEYON D T H E FINITE ORBI T S CASE MICHAEL FRANK, VLADIMIR MANUILO V, AND EV GENIJ TR OITSKY Abstract. Con tin uous actions of top o logical groups on compact Hausdorff spaces X are inv estigated w hich induce almost per io dic functions in the corresp onding comm utative C ∗ -algebra . The unique inv ariant mean on the gro up resulting from averaging allo ws to derive a C ∗ -v alued inner pro duct and a Hilbe r t C ∗ -mo dule which serve as an environmen t to describ e c hara c teristics of the group action. F or uniformly contin uous, Lyapunov stable a ctions the derived inv ar ia nt mean M ( φ x ) is contin uous on X for an y elemen t φ ∈ C ( X ), and the induced C ∗ -v alued inner pro duct corresp o nds to a co nditional exp ectatio n from C ( X ) onto the fixed p oint algebra o f the action defined by av eraging on or bits. In the case of selfduality of the Hilb ert C ∗ -mo dule all orbits are shown to have the same cardinality . Stable actions on compact metr ic spaces give rise to C ∗ -reflexive Hilb ert C ∗ -mo dules. The same is tr ue if the car dinality of finit e orbits is uniformly b ounded and the num b er of clos ures of infinite orbits is finite. A num b er of examples illustrate typical situations a ppe a ring b eyond the class ified cases. 1. Intr oduction In v estigating contin uous group actions on top ological spaces sev eral mathematical ap- proac hes ma y b e applied. In t he presen t pap er the authors con tinu e their w ork s tarted in [5, 18 ] whic h relies on the Gel’fand dualit y of lo cally compact Hausdorff spaces and comm utativ e C ∗ -algebras. In the dual picture some w ell-kno wn results from functional analysis and no ncomm utativ e geometry can b e applied to get new insigh ts, often also for related noncomm utativ e situations of group actions on general C ∗ -algebras. Consider a contin uous action of a top olo g ical g r o up Γ on a compact Hausdorff space X . F ollo wing the Gel’fand dualit y it can be seen as a con tin uous action of Γ on the comm utativ e C ∗ -algebra C ( X ) of all con tin uous complex-v alued f unctions on X . Let us denote the subalgebra of Γ-inv arian t functions o n X by C Γ ( X ) ⊂ C ( X ). W e wish to in tro duce the s tructure of a pre-Hilb ert C ∗ -mo dule o v er C Γ ( X ) on C ( X ) whic h expresse s significan t prop erties of the action of Γ on X . One w a y to find suitable C ∗ -v alued inner pro ducts on C ( X ) is the searc h for conditional exp ectations E : C ( X ) → C Γ ( X ) whic h are a kind of mean ov er the gro up action of Γ on C ( X ) and canonically give rise to the Hilb ert C Γ ( X )-mo dule structures o n C ( X ) w e are lo oking for. W e follo w ed that approac h in [5, 18] (see [15] for a related discussion). 2000 Mathematics Subje ct Classific ation. Primary 4 6L08, Seco nda ry 43A60 , 54H20. This work is a part o f the joint DF G-RFBR pro ject (RFBR gr ant 0 7-01- 91555 / DFG pro ject ”K- Theory , C ∗ -Algebras, a nd Index Theor y ”.) The seco nd named author was a lso pa r tially supp orted by the grant HI I I-156 2.200 8 .1. The third named author was also par tially supp or ted by the RFBR grant 0 7 -01-0 0046. 1 2 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY Here we w an t to consider a more general approach closer to the top o lo gical back ground. F or elemen ts φ, ψ ∈ C ( X ) and for the deriv ed group maps (1) φ x : Γ → C , φ x ( g ) = φ ( g x ) , ( x ∈ X ) w e w an t to select a suitable norma lized in v a r ia n t mean m Γ on Γ suc h that a C Γ ( X )-v alue d inner pro duct on C ( X ) could b e defined lik e (2) h φ, ψ i ( x ) := m Γ ( φ x ψ x ) , ( x ∈ X ) . Of course, w e w ould hav e to supp ose Γ to b e amenable at this p oint to w arran t the existence of the (left) in v aria nt mean m Γ . The pro duct (2) has to satisfy at least the follo wing tw o prop erties (with Γ-in v aria nce follo wing fro m the definition): 1) The resulting functions m Γ ( φ x ψ x ) are contin uous in the argument x ∈ X . 2) The v a lue h φ, φ i ( x ) is a lwa ys p ositiv e, if φ ( g x ) 6 = 0 for some g ∈ Γ, some x ∈ X . One can observ e that the prop erty 2) w ould e.g. follo w from the following suppo sition: 2’) F or an y x ∈ X and an y non- zero φ ∈ C ( X ) the map φ x is a non-zero almost p erio dic function on Γ. The supp osition 2’) w ould allow us: • to a v oid the r estriction on Γ to b e amenable, • to o v ercome the dep ende nce on the par t icular c hoice of m Γ , b y passing f r o m (2) to (3) h φ, ψ i ( x ) := M ( φ x ψ x ) , where t he map M : Γ → C is the unique in v a rian t mean on almost p erio dic functions with resp ect to the giv e n action o f Γ, wh en 1) and 2’) a re supp osed to hold (cf. the a pp endix). The link to results in [5 , 1 8] is given b y constructing a suitable conditiona l expectation E Γ : C ( X ) → C Γ ( X ) b y the rule 1’) F or an y ϕ ∈ C ( X ) the function E Γ ( φ )( x ) := M ( φ x ) is contin uous in x . Prop erties 1’) a nd 2’) pro vide that the formu la (2) mak es C ( X ) a pre-Hilb ert C ∗ -mo dule o v er C Γ ( X ). Let us denote its completion by L Γ ( X ). W e are in terested in t w o questions here: (1) F or whic h actions the conditions 1’) and 2’) hold? (2) If they hold, what prop erties do es L Γ ( X ) ha v e? Our reference o n almost p erio dic functions is [3]. Hilbert C ∗ -mo dules w ere in tro duced in [14] and [10]. F or facts on Hilb ert C ∗ -mo dules w e refer the reader to [8, 7, 13]. Recall that for a Hilb ert C ∗ -mo dule L ov er a C ∗ -algebra A the A -dual mo dule L ′ is the mo dule of all b o unded A -linear maps from L to A . L is called self-dual (resp. C ∗ -reflexiv e) if L = L ′ (resp. if L = L ′′ ). Our pap er is o r ganized as follows : In the Section 2 w e will giv e some sufficien t conditions for conditions 1 ’) and 2’) to hold, hence, for the existence of C Γ ( X )-v alue d inner products on the C ∗ -algebra C ( X ). W e a lso show that our type of av eraging is the same as the a v eraging o v er orbits. Section 3 de als with the more restric tiv e situat io ns in whic h the resulting Hilb ert C ∗ -mo dule turns out to b e self-dual. In Section 4 we revisit the situation of resulting C ∗ -reflexiv e Hilbert C ∗ -mo dules and obtain an imp ortant restriction on X to b e supp osed. In Section 5 w e giv e some examples sho wing differen t p ossible b ehavior of MODULES FROM GROUP ACT IONS 3 a v eraging. The App endix is devoted to the pro of of the uniqueness of a measure used for a v eraging. 2. L y apuno v st ability and continuity of a veraging W e w an t to find conditions under whic h a w ell-defined a v eraging o v er the group action on orbits exists in the case of infinite orbits. F or this aim w e introduce additio na lly to the condition of uniform con tin uit y discuss ed in [5, 18 ] the condition of Lyapuno v stability . The lat t er condition ensures uniform con tin uit y , the w ell-definedness of a v eraging and the exis tence of a conditional expectation onto the fixed-p oin t algebra whic h giv es rise to a C ∗ -v alued inner pro duct and a r esulting Hilbert C ∗ -mo dule structure. In subs equen t sections w e can apply this to ol t o c haracterize those gr o up actions on compact Hausdorff spaces with infinite orbits. Definition 2.1. W e sa y that an action of a g roup G on a lo cally compact Hausdorff space X is uniformly c ontinuous if for ev ery p oint x ∈ X and every neigh b orho o d U x of x there exists a neighborho o d V x of x such that g ( V x ) ⊆ U x for ev ery g ∈ G x , where G x denotes the stabilizer o f x . Theorem 2.2. L et an action of a top olo gic al g r oup Γ o n a c omp act Hausdorff sp ac e X b e uniformly c ontinuous. If al l orbits ar e finite and if their size is u niformly b ounde d then the aver age M ( ϕ x ) is c ontinuous with r esp e ct to x ∈ X for any ϕ ∈ C ( X ) . Pr o of. If an orbit Γ x is finite then the function ϕ x on Γ is exactly p erio dic, hence M ( ϕ x ) = 1 #Γ x X g x ∈ Γ x ϕ ( g x ) , so the av erage on Γ is the same a s the av erage o v er orbits. Contin uit y of the latter is pro vided by Lemma 2.11 from [5].  Example 5.2 b elo w demonstrates that in the case of presence of infinite orbits the uniform contin uit y is not sufficien t for the contin uit y of the av e rage. No w w e g eneralize a n approac h of [18] and in tro duce a condition whic h is sufficien t to o v ercome these difficulties. Let Φ b e a uniform structure on a compact space X . Recall from [1] that, on a compact space, there is a unique uniform structure compatible with its top olog y . It consists of al l neigh b orho o ds of the diagonal in X × X [1, Ch. I I, Sect. 4, Th eorem 1]. If X is a metric space with a metric d t hen the uniform structure is the se t of the neigh b orho o ds of the diagonal ∆ ⊂ X × X of the form { ( x, y ) : x, y ∈ X , d ( x, y ) < ε } , ε ∈ (0 , ∞ ). Definition 2.3. An a ction of a group Γ on a top ological space X with a unifo rm structure Φ compatible with its top ology is called Lyapunov stable if for any U ∈ Φ and an y x ∈ X there is V ∈ Φ suc h tha t ( g x, g y ) ∈ U for any g ∈ Γ if ( x, y ) ∈ V . Note that in the case of a metric space, t his definition take s the follo wing form: Definition 2.4. An action of a gro up Γ on a metric space X is called Lyapuno v stable if for an y ε > 0 and a n y x ∈ X there exist δ > 0 suc h that ρ ( g x, g y ) < ε for an y g ∈ Γ if ρ ( x, y ) < δ. 4 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY Lemma 2.5. If an action of a discr ete gr oup Γ o n a t op olo gic al sp ac e X with a uniform structur e i s Lyapunov st able then it is uniformly c ontinuous. Pr o of. F or x ∈ X and for U ∈ Φ set U ( x ) := { y ∈ X : ( x, y ) ∈ U } . If W is a neigh b orho o d of x then there is U ∈ Φ suc h that U ( x ) ⊂ W . By stabilit y , there is V ∈ Φ suc h tha t ( g x, g y ) ∈ U for an y g ∈ Γ if ( x, y ) ∈ V . No w let g ∈ Γ x . T ak e an y y ∈ V ( x ). Then ( x, g y ) ∈ U , hence g y ∈ U ( x ) ⊂ W , i.e. g ( V ( x )) ⊂ W for an y g ∈ Γ x .  In the case when all orbits are finite, uniform con tin uit y is equiv alen t to Ly apuno v stabilit y: Prop osition 2 .6. L et a dis c r e te gr oup act unif o rmly c ontinuously on a c omp act Hausdorff sp ac e X and let al l the orbits ar e finite. The n this a ction is Lyapunov stable. Pr o of. T ake a neigh b orho o d W of the diagonal in X × X and take a p oint x ∈ X . Since its orbit is finite, we can ta k e a finite set { g 1 , . . . , g s } of elemen ts in Γ suc h that { g 1 x, . . . , g s x } is the orbit Γ x . Now find a ne ighborho o d U x of x suc h that g i ( U x ) × g i ( U x ) ⊂ W fo r each i = 1 , . . . , s . Uniform con tin uit y implies that there exists a neigh b orho o d V x of x suc h that hy ∈ U x for an y y ∈ V x and an y h ∈ Γ x . Since an y g ∈ Γ can b e written as g = g i h for some i = 1 , . . . , s and for some h ∈ Γ x , g ( V x ) = g i ( h ( V x )) ⊂ g i ( U x ). It follows from compactness of X that there is a finite nu m b er of p o ints x 1 , . . . , x r in X suc h tha t the sets V x 1 , . . . , V x r form a finite co v ering fo r X . Then W 0 = V x 1 × V x 1 ∪ . . . ∪ V x r × V x r is a neigh b orho o d of the diagonal in X × X . T ak e ( y , z ) ∈ W 0 . Then there is some 1 ≤ j ≤ r suc h that ( y , z ) ∈ V x j × V x j . Then ( g y , g z ) ∈ g i ( U x j ) × g i ( U x j ) for some i . By cons truction, g i ( U x j ) × g i ( U x j ) ⊂ W , so w e conclude that ( g y , g z ) ∈ W for an y g ∈ Γ whenev er ( y , z ) ∈ W 0 .  Prop osition 2.7. L et a discr ete gr oup Γ a ct Ly apuno v stably on a c omp act Hausdorff sp ac e X an d let ϕ : X → C b e a c ontinuous function. Then, for any x ∈ X , the function ϕ x : Γ → C , ϕ x ( g ) := ϕ ( g x ) , is a l m ost p erio dic. Pr o of. T ake ε > 0. The n compactness of X implies existence of some U ∈ Φ suc h that (4) | ϕ ( x ) − ϕ ( y ) | < ε if ( x, y ) ∈ U , since ϕ is uniformly con tin uous and Φ consists of all neigh bo rho o ds of the diagonal. Stabilit y implies tha t there is V ∈ Φ suc h t ha t ( g x, g y ) ∈ U for an y g if ( x, y ) ∈ V . Denote the closure of the orbit Γ x b y Y ⊂ X . It is a compact subset. Compactness of Y implies that o ne can find in Y a finite num ber of p oin ts of the form g i x , i = 1 , . . . , s , suc h that for an y p ∈ Y there is some i , for whic h ( g i x, p ) ∈ V and, therefore, for any g , h ∈ Γ, (5) ( hg i x, hg x ) ∈ U . Then t he functions L g i ϕ x , i = 1 , . . . , s , form an ε -net for the set { L g ϕ x : g ∈ Γ } , with resp ect to the uniform norm. Indeed, for any g ∈ Γ, there is an index i suc h that (5) holds. Then, b y (4), w e hav e sup h ∈ Γ | ( L g ϕ x )( h ) − ( L g i ϕ x )( h ) | = sup h ∈ Γ | ϕ ( hg x ) − ϕ ( hg i x ) | < ε.  MODULES FROM GROUP ACT IONS 5 So, under the conditions of Prop osition 2.7 the in v ariant mean M ( ϕ x ) is w ell-defined on C ( X ). Theorem 2.8. L et a d iscr ete gr oup Γ act on a c omp act Hausdorff s p ac e X . If the action is Lyapunov stable, then the c onditional exp e ctation E Γ : C ( X ) → C Γ ( X ) define d by E Γ ( φ )( x ) = M ( φ x ) is wel l-define d, i.e. the c onditions 1’) and 2’) hold . Pr o of. By Prop osition 2.7 w e only need to ve rify the con tin uit y of the mean M ( ϕ x ) with resp ect to x ∈ X . Let x ∈ X and ε > 0 be arbitrary . Let us remind (cf. [6, pp. 250–251]) that w e can c hoose h 1 , . . . , h p ∈ Γ in suc h a w ay that the uniform distance on Γ × Γ b et w een the function 1 p p X j =1 D h j ϕ x : Γ × Γ → C (where ( D h ψ )( g 1 , g 2 ) := ψ ( g 1 hg 2 )) and some constan t is les s then ε . In this case the uniform distance satisfies the inequality      M ( ϕ x ) − 1 p p X j =1 D h j ϕ x      u < 2 ε. Let us ch o o se a neigh b orho od V ∈ Φ suc h that | ϕ ( g y ) − ϕ ( g x ) | < ε , for an y g ∈ Γ , ( x, y ) ∈ V . This neigh b orho o d can b e found as in the pro of of Prop osition 2.7: first we can find a neigh b orho o d U suc h that | ϕ ( y ) − ϕ ( z ) | < ε whenev er ( y , z ) ∈ U (using compactness of X ). Then, b y stabilit y of the action, we can find for this U another neigh b orho o d V ∈ Φ suc h that ( g y , g x ) ∈ U for any g ∈ Γ whe nev er ( y , x ) ∈ V . Then for an y y ∈ U ( x ) o ne has 1 p p X j =1 D h j ϕ y ! ( g 1 , g 2 ) = = 1 p p X j =1 D h j ϕ x ! ( g 1 , g 2 ) + 1 p p X j =1 ( ϕ y ( g 1 h j g 2 ) − ϕ x ( g 1 h j g 2 )) = 1 p p X j =1 D h j ϕ x ! ( g 1 , g 2 ) + 1 p p X j =1 ( ϕ ( g 1 h j g 2 y ) − ϕ ( g 1 h j g 2 x )) . Eac h term of the second summand is less then ε . Hence, the second summand is less then ε . Th us,      M ( ϕ x ) − 1 p p X j =1 D h j ϕ y      u < 3 ε for an y y ∈ U ( x ). Therefore, considering M ( ϕ x ) as an a rbitrary constan t, w e ha v e      M ( ϕ y ) − 1 p p X j =1 D h j ϕ y      u < 6 ε, and finally , | M ( ϕ y ) − M ( ϕ x ) | < 9 ε 6 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY for an y y ∈ U ( x ). Consequen tly , E Γ ( φ ) is Γ-inv ariant and con tin uous on X .  F or x ∈ X let us denote its orbit Γ x b y γ and the closure of the o r bit γ in X b y γ . Theorem 2.9. L et a disc r e te gr oup Γ act on a c omp act Hausdorff sp ac e X . 1) F or a Lyapunov stable action and for the unique invariant me an M : Γ → C we have the e qu ality (6) M ( ϕ x ) = Z γ ϕ | γ dµ γ , wher e x ∈ γ , µ γ is a (unique) invariant me asur e on γ of total mass 1. 2) If γ is finite, then M ( ϕ x ) , x ∈ γ , c an b e taken as the standar d aver age, as it was c onsider e d in [5, 18] . Pr o of. Eviden tly , 2) follows from 1). Let us sho w that for a ϕ ∈ C ( X ) the left-hand side of (6 ) d o es no t dep end on x ∈ γ . First, eviden tly , it does not dep end on t he choice of x inside the same orbit. Hence, it is sufficie n t to v erify tha t the v alue is the same for g x 2 sufficien tly close to an y x 1 for x 1 , x 2 ∈ γ to demons trate the in v arian tness with resp ect to the action of Γ. By the Ly apuno v stabilit y prop ert y , for an y ε > 0 w e can find an elemen t g ε ∈ Γ suc h that g ε x 2 is so close to x 1 that | ϕ ( g x 1 ) − ϕ ( g g ε x 2 ) | < ε fo r an y g ∈ Γ. Then | M ( ϕ x 1 ) − M ( ϕ x 2 ) | = | M ( ϕ x 1 ) − M ( ϕ g ε x 2 ) | = | M ( ϕ x 1 − ϕ g ε x 2 ) | ≤ sup g ∈ Γ | ϕ ( g x 1 ) − ϕ ( g g ε x 2 ) | < ε. Since ε is arbitrary small, M ( ϕ x 1 ) = M ( ϕ x 2 ) and t he v alue is constan t on the closure o f orbits. A similar estimation implies the contin uit y of t his (w ell-defined by the ab o v e arg umen t) functional m : C ( γ ) → C , m ( φ ) = M ( φ x ) for x ∈ γ , with resp ect to the v ariation of closures of orbits. By the Riesz-Mark o v-Kakutani theorem [4, Theorem 3, Sect. IV.6], m has the form m ( f ) = Z γ f dµ, where µ is some regular coun tably additiv e complex measure on γ . Eviden tly , µ is inv ari- an t. It remains to explain wh y µ ia unique. In fact, this follo ws from [2, Ch. VI I, § 1, Problem 14]. W e giv e details in the App endix.  3. Self-duality After a characterization of the inner structure of Hilb ert C ∗ -mo dules that arise from Ly apuno v stable actions w e are going to describe the inte rrelation b etw ee n certain prop- erties of the action and self-daulit y of the resulting Hilb ert C ∗ -mo dule. Lemma 3.1. L et a discr e te gr oup Γ act on a c omp act Hausdorff sp ac e X . I f the action is Lyapunov stable then any two orbits ar e either sep ar ate d fr om e ach other, or have the same clo s ur e. Thus, closur es of orbits ar e sep ar ate d sets in X/ Γ . The Gelfand sp e ctrum of C Γ ( X ) is the set of clo sur es of orbits. MODULES FROM GROUP ACT IONS 7 Pr o of. Suppo se, t w o orbits γ = Γ x and γ ′ = Γ y are not separated but ha v e distinct closures. This means (after a shift, if ne cessary) that y ∈ γ , but g 0 y 6∈ γ for some g 0 ∈ Γ. Then there exists some U of the uniform structure Φ suc h that there a r e no p oints of the form ( g 0 y , g 1 x ) in U . Let us tak e a neigh b orho o d V ∈ Φ corresponding to U by the definition of Ly apuno v stabilit y . T a k e ( y , g 2 x ) ∈ V . Then ( g 0 y , g 0 g 2 x ) ∈ U . T ak e g 1 = g 0 g 2 . A contradiction. Th us, the quotient space of clos ures of orbits is Hausdorff and, hence, it coincide s with the Gelfand sp ectrum of C Γ ( X ).  Theorem 3.2. L et a discr et e gr oup Γ a c t on a c omp ac t Hausdorff sp ac e X . In the c ase of a L yapunov stable action the mo dule L Γ ( X ) has the fol lowing description: it c onsists of al l function s ψ : X → C such that 1) ψ | γ ∈ L 2 ( γ , µ γ ) , wher e µ γ is a unique normalize d invarian t me asur e on γ for any orbit γ , 2) for any ϕ ∈ C ( X ) the function h ψ , ϕ i L is c ontinuous. In p a rticular, the ave r a ge h ψ , 1 i L of such a function ψ is c ontinuous on X/ Γ . Pr o of. W e should prov e the following t w o facts: a) the set of con tin uous functions on X satisfies these conditions, and b) it is dense in this set with resp ect to t he C ∗ -v alued inner pro duct on the mo dule L Γ ( X ). Condition 1) of the assertion ab ov e should b e inte rpreted via the equalit y (7) Z γ ψ | γ dµ γ = M ( ψ x ) , ψ ∈ C ( X ) , x ∈ γ . This equality follows from t w o facts: i) the left-hand side dep ends only on ψ | γ and defines an in v aria nt mean on almost p erio dic functions on Γ, ii) suc h a mean is unique. ( see Theorem 2.9) Th us, b y the res ults of Prop osition 2.7 and Theorem 2.8 of Section 2, condition a) is fulfilled. No w ta ke an arbitrary function ψ ( x ) satisfying t he conditions 1) and 2) of the assertion ab ov e, an arbitrary function ϕ ∈ C ( X ) with k ϕ k L ≤ 1, and an a rbitrary small ε > 0. Consider the closure of an orbit γ . Cho ose a con tin uous function f γ : γ → C suc h that (8) Z γ | ψ | γ − f γ | 2 dµ < ε 2 . By normalit y o f X , f γ can b e extended to a contin uous function b f γ : X → C . There exists a neighborho o d U γ of γ in the Gelfand sp ectrum g X/ Γ of C Γ ( X ), suc h that (9) Z γ ′′ ( ψ | γ ′′ − b f γ | γ ′′ ) ϕ γ ′′ dµ < 2 ε , γ ′′ ∈ U γ . This follo ws fro m (8) and 2). Cho ose a finite sub cov ering U γ i of g X/ Γ and a subo rdinated partition o f unity ω i , i = 1 , . . . , I . This can b e done by Lemma 3.1. Then sup |h ψ − f , ϕ i| ≤ 2 ε , where f = I X i =1 ω i b f γ i . Th us f ∈ C ( X ) is 2 ε -close to ψ .  8 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY Theorem 3.3. L et a disc r e te gr oup Γ act on a c omp act Hausdorff sp ac e X . 1) Supp ose, the mo dule L Γ ( X ) is self-dual and the Gelfand sp e ctrum g X/ Γ of the algebr a of c ontinuous invarian t functions C Γ ( X ) has n o isolate d p oints. Then ther e ar e only finitely many γ with infini te γ an d al l fi n ite orb i ts ha v e the same c ar dinality. 2) If ther e a r e on l y finitely many γ with infi nite γ and al l finite orbits have the same c ar dinality, then the mo dule L Γ ( X ) is se lf-dual. Pr o of. 1) By [12], the restriction on the Gelfand sp ectrum im plies that L Γ ( X ) is finitely generated pro jectiv e. Let N b e the cardinality of some of its generator systems. Th us, the n um ber of p oints of eac h finite orbit is ≤ N . This follows from the epimorphit y of the restriction map L Γ ( X ) → L Γ ( Y ), where Y ⊂ X is a closed Γ-in v a rian t s et. Indeed, Y is a closed set in a normal space, hence con tin uous functions on it are extendable b y the Tietze theorem. In this situatio n of uniform b o undness of the cardinalit y o f finite or bits, the subset X f , formed by all finite orbits, is a closed (inv arian t) subset of X . Indeed, suppo se, an infinite orbit γ is in the closure of X f . Cho ose a co v er of X b y (a finite n um ber of ) open sets U i , suc h tha t no one of them is co v ered by the others, and γ is cov ered b y more than N of these U i ’s. Let U = ∪ i ( U i × U i ) b e an elemen t of the uniform structure on X . Then there exists another neigh b orho o d V of the diagonal in X × X suc h that Γ( V ) ⊂ U under the diagonal action. Cho osing a finite orbit (of cardinalit y ≤ N ) V -close to γ w e obtain a con tradiction to prop erties of U . Th us, as ab ov e, L Γ ( X f ) is finitely generated. Moreov e r, it is pro jectiv e, b ecause the pro- jection asso ciated with a canonical isometric embedding of the finitely generated pro jectiv e C ( g X/ Γ)-mo dule L Γ ( X ) into a s tandard finitely generated C ( g X/ Γ)-mo dule C ( g X/ Γ) N , sa y π : C ( g X/ Γ) N → L Γ ( X ), restricted to X f giv es an epimorphic idempotent mapping π ′ : C ( ^ X f / Γ) N → L Γ ( X f ) , defined by the restriction of matrix elemen t s of the pro j ection π . Epimorphity f ollo ws from the ab ov e argumen t whic h relied on the Tietze theorem. W e arriv e to the case considered in [5 ] and [18]. As it is explained in Theorem 2 .9, the a v erage o v er finite orbits is the same as in these pap ers, a nd the inner pro duct is the same. Th us, L Γ ( X f ) = C ( X f ). By the results of [5] and [1 8], under our assumptions this mo dule is finitely generated pro jectiv e if and only if all (finite) or bits ha v e the same cardinalit y . No w w e pass to pro ving the statemen t ab out infinite orbits. Supp ose there exists an infinite n um ber o f closures of infinite orbits: γ i , i = 1 , 2 , . . . . W e need to construct a C Γ ( X )-functional on L Γ ( X ), whic h is not an elemen t of L Γ ( X ). P assing to a subseque nce, if necessary , w e can a ssume that for each p oin t z i ∈ g X/ Γ represen ting γ i , w e can choose an op en neigh b o rho o d U i , suc h tha t U i ∩ U j = ∅ if i 6 = j . Indeed, supp ose the opp osite. This implies that for one of the p oin ts, say z 1 , and an y its neigh b orho o d only finitely man y p oin ts fr o m the set { z 1 , z 2 , . . . } are o ff t his neigh b orho o d (i.e., z 2 , z 3 , · · · → z 0 ). W e c ho ose a neigh b orho od U ′ 1 ∋ z 0 , suc h that there is z i 1 6∈ U ′ 1 , a nd (b y normality ) a neigh b orho o d U ′′ 1 of z 0 , suc h t ha t U ′′ 1 ⊂ U ′ 1 and there is a neighborho od U 1 of z i 1 , such tha t U 1 ∩ U ′′ 1 = ∅ . T ake U ′ 2 ⊂ U ′′ 1 , such tha t there exists z i 2 ∈ U ′′ 1 , z i 2 6∈ U ′ 2 . T ak e, by normality , U ′′ 2 ∋ z i 0 , such that U ′′ 2 ⊂ U ′′ 2 ⊂ U ′ 2 and there exists a neigh borho o d MODULES FROM GROUP ACT IONS 9 U 2 of z i 2 suc h that U 2 ⊂ U ′′ 1 and U 2 ∩ U ′′ 2 = ∅ . And so on. Finally , collec t the points { z i n } whic h hav e the required prop ert y . Let us define f i : γ i → { 0 , √ i } to b e the indicator functions of subsets of γ i with µ i (supp f i ) = 1 i (where µ i is the in v ariant measure on γ i of total mass 1). Thus f i ∈ L 2 ( γ i , µ i ), h f i , f i i γ i = R γ i | f i | 2 dµ i = 1 and R γ i | f i | dµ i = 1 / √ i . Cho ose α i ∈ C ( X ) ( i = 1 , 2 , . . . ) suc h that 1) supp α i ⊂ p − 1 ( U i ), where p : X → g X/ Γ is the canonical pro jection; 2) α i ( X ) ⊂ h 0 , 1 √ i i , α i ( γ i ) = h 0 , 1 √ i i ; 3) k α i | γ i − f i k L 2 < 1 2 i , k α i | γ i k 2 L 2 − k f i k 2 L 2 < 1 2 i ; 4) R γ | α i | γ | dµ γ ≤ 1 √ i + 1 2 i − 1 for any γ ; 5) R γ | α i | γ | 2 dµ γ ≤ 1 + 1 2 i − 1 for an y γ . T o construct these functions w e first appro ximate f i b y an appropriate contin uous func- tion, t hen extend b y the Tietze theorem, a nd finally m ultiply by an appropriate parti- tion of unit y function. More precise ly , w e first choose a con tin uous function α ′ i : γ i →  0 , 1 / √ i  , restricted to satisfy prop erties 2 and 3. Then we extend b y the Tietze theorem α ′ i to a con tin uous function α ′′ i : X →  0 , 1 / √ i  . By Theorem 3.2 the functions h α ′′ i , 1 i L : g X/ Γ → [0 , + ∞ ) , h α ′′ i , α ′′ i i L : g X/ Γ → [0 , + ∞ ) , are con tin uous and h α ′′ i , 1 i L ( z i ) ∈  1 √ i − 1 2 i , 1 √ i + 1 2 i  , h α ′′ i , α ′′ i i L ( z i ) ∈  1 − 1 2 i , 1 + 1 2 i  . Cho ose a neigh b orho o d U ′ i ⊂ U i of z i suc h that h α ′′ i , 1 i L ( z i ) ∈  1 √ i − 1 2 i − 1 , 1 √ i + 1 2 i − 1  , h α ′′ i , α ′′ i i L ( z i ) ∈  1 − 1 2 i − 1 , 1 + 1 2 i − 1  . Let ω i : g X/ Γ → [0 , 1] be a con tin uous function with ω i ( z i ) = 1 and supp ω i ∈ U ′ i . Put b ω i := p ∗ ω i : X → [0 , 1] and α i := b ω i α ′′ i . They are the required ones. Define a function h : X → [0 , + ∞ ) to b e equal to α i on p − 1 ( U i ) ( i = 1 , 2 , . . . ) and 0 otherwise. F ir st, w e wish to show that h 6∈ L Γ ( X ). Indeed, h h, h i L is greater than 1 − 1 2 i > 1 / 2 a t eac h z i and v a nishes at any accum ula tion po in t of { z i } . No w let us sho w that h ∈ L Γ ( X ) ′ . L et ϕ be a con tin uous function on Y suc h that kh ϕ, ϕ ik L ≤ 1. Then for an y γ in some p − 1 ( U i ) w e hav e (using prop erty 5) |h h, ϕ i | γ | = |h α i | γ , ϕ | γ i| ≤ h α i | γ , α i | γ i 1 / 2 · h ϕ, ϕ i 1 / 2 ≤ 2 . F or the remaining γ ’s this pro duct v anishes. It remains to sho w that h h, ϕ i is a con tin uous (in v a r ia n t) function, i.e. that fo r an y ε > 0 and an y p oin t γ 0 from the closure of ∪ i p − 1 ( U i ) there is an in v ariant neigh b o rho o d W of γ 0 suc h that Z γ h γ · ϕ | γ dµ γ < ε for an y γ ∈ W . Cho o se W not in tersecting with p − 1 ( U i ) for i = 1 , . . . , k , where k > max  2 ,  2 sup x ∈ X | ϕ ( x ) | ε  2  . Then (b ey ond the trivial cases) γ ∈ p − 1 ( U i ), i > k . Let us 10 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY estimate using pro p ert y 4):     Z γ h γ · ϕ | γ dµ γ     ≤ sup x ∈ X | ϕ ( x ) | · Z γ | α i | γ | dµ γ = sup x ∈ X | ϕ ( x ) | ·  1 √ i − 1 2 i − 1 , 1 √ i + 1 2 i − 1  < sup x ∈ X | ϕ ( x ) | · 2 √ i < ε for i > k . Hence, the mo dule is not self-dual. 2) As it is explained in the first part of the pro o f, in this case X = X f ⊔ γ 1 ⊔ · · · ⊔ γ n , L Γ ( X ) = L Γ ( X f ) ⊕ L 2 ( γ 1 , µ γ 1 ) ⊕ L 2 ( γ n , µ γ n ) , ( L Γ ( X )) ′ C Γ ( X ) = ( L Γ ( X f )) ′ C Γ ( X f ) ⊕ ( L 2 ( γ 1 , µ γ 1 )) ′ C Γ ( γ 1 ) ⊕ ( L 2 ( γ n , µ γ n )) ′ C Γ ( γ n ) = ( L Γ ( X f )) ′ C Γ ( X f ) ⊕ ( l 2 ( C )) ′ C ⊕ · · · ⊕ ( l 2 ( C )) ′ C = ( L Γ ( X f )) ′ C Γ ( X f ) ⊕ L 2 ( γ 1 , µ γ 1 ) ⊕ L 2 ( γ n , µ γ n ) . As it was explained a b o v e L Γ ( X f ) = C ( X f ) in this case, and ( C ( X f )) ′ C Γ ( X f ) = C ( X f ).  Example 3.4. Let Y be the cone giv en b y the equation x 2 + y 2 = z 2 , Z ⊂ Y b e the subset of all points with z ∈ J = { 0 , 1 , 1 / 2 , 1 / 3 , . . . } . The n Z is an infinite collection of circles with one limit point (0 , 0 , 0) added. Let X b e a union o f three distinct copies of Z . T o describe an action of Z on Z num ber the circles in t he double-cone consecutiv ely b y n um b ers of Z where the nu m b er zero is fixed to the p oin t ( 0 , 0 , 0). Consider the discrete group Γ = Z ⊕ Z 3 , where Z acts on eac h circle b y an irratio na l rotatio n b y an angle α i ( i = 1 , 2 , . . . ), where α i → 0, and where Z 3 transp oses the cones. Then the mo dule L Γ ( X ) is not self-dual since the orbits are all infinite except for t he fixed-po in t. 4. C ∗ -Reflexivity 4.1. The metric case In this section w e w ould like to understand, in whic h situations the Hilb ert C ∗ -mo dule L Γ ( X ) is C ∗ -reflexiv e o v er C Γ ( X ). Our pre vious pa r t ia l results [5, 18] made us believ e that the Hilb ert C ∗ -mo dule L Γ ( X ) is C ∗ -reflexiv e in muc h mor e general situations b eyond the finite orbit case. It turns out that any coun tably generated mo dule o v er a wide class of comm utativ e C ∗ -algebras is C ∗ -reflexiv e. Theorem 4.1. L et X b e a c om p act metric sp ac e. Then any c ountably gener ate d mo dule over C ( X ) is C ∗ -r eflexive. Pr o of. The first version of a pro of app eared in [9]. Then T rofimo v [17] realized that the form ulation in [9] was to o general and pro vided a proo f for an y compact X with a certain prop ert y L. While preparing this pap er, we understo o d tha t t he prop ert y L of T rofimo v for X is the same as the prop ert y of X to b e a compact Baire space. So, an y comp act Hausdorff space has this prop erty L, and C ∗ -reflexivit y w ould hav e place f o r any countably generated mo dule ov er an y unital commutativ e C ∗ -algebra, which is ob viously not true, e.g. fo r v on Neumann alg ebras [11]. Nev ertheless , the main part of T rofimo v’s proof is correct. But it was o v erlooked that implicitly the pro of used that, for a ny sub set E ⊂ X and for an y po in t t 0 in the closure of E , there exists a sequenc e of p oin ts t n ∈ E , whic h MODULES FROM GROUP ACT IONS 11 con v erges to t 0 . In other w ords, the top ology on X is su pp osed to p ossess a coun table base of neigh borho o ds at an y p oint of X . This is not true in general, but if we restrict ourselv es t o the case of compact metric spaces then this is obviously true. Under this additional assumption, T r o fimo v’s pro of is correct.  Corollary 4.2. L et X b e a c omp act metric sp ac e, and let an action o f Γ on X b e Lyapuno v stable. Then the mo dule L Γ ( X ) is C ∗ -r eflexive. Pr o of. Since X is metric, the module L Γ ( X ) is c oun tably generated and the C ∗ -algebra C Γ ( X ) is separable, hence its Gelfand sp ectrum is metrizable.  Example 4.3. Let D = Q ∞ k − 1 D k , where eac h D k is the tw o-po in ts space with the distance b et w een the t w o points equal to 2 − k , a nd let X = J × D . Let G = ⊕ ∞ k =1 Z 2 , G n = ⊕ n k =1 Z 2 and π n : G → G n , i n : G n → G b e the standard pro jection and inclusion homomorphisms. Denote their comp osition by p n = i n ◦ π n : G → G . Let α denote the standard action of G on D . Define an action β of G on X b y the formula β g  1 n , d  =  1 n , α p n ( g ) ( d )  , n ∈ N \ { 0 } , and β g (0 , d ) = (0 , α g ( d )) , where d ∈ D . It is easy to see that the following prop erties hold for t his action: • The orbit of a ny p oin t of the form ( 1 n , d ) is finite and consists of 2 n elemen ts. • The orbit of a ny p oin t of the form (0 , d ) is infinite. • The action β is con tin uous. • The action β is Ly apuno v stable. It follo ws from Corollary 4.2 that the mo dule L Γ ( X ) is C ∗ -reflexiv e in this example. 4.2. The non-me tric case After we ha v e clarified, how C ∗ -reflexivit y arises in the metric case, let us pass to the case, when X is non-metric. T o begin with, we give an example of a non- C ∗ -reflexiv e mo dule L Γ ( X ). Example 4.4. Let K b e a (non-metrizable) compact space suc h that l 2 ( A ) is not C ∗ - reflexiv e, where A = C ( K ). That is the case for A b eing a v o n Neumann a lg ebra, and one of the most important cases is that of K = β N , the Stone– ˇ Cec h compactification of in tegers. Consider the compact space X = K × S 1 equipped with the action of Z b y irrational rotat io n in the second argumen t: m ( y , s ) = ( y , e απ m s ) , m ∈ Z , α ∈ R \ Q , s ∈ S 1 ⊂ C . This is an isometric a ctio n on S 1 and a trivial one on K , hence it is Ly apuno v stable. Eviden tly , the alg ebra of con tin uous in v a r ian t functions C Γ ( X ) is A = C ( K ). By Theorem 3.2, the mo dule L Γ ( X ) is the set of all functions ψ : X → C s uc h that 1) ψ | γ ∈ L 2 ( γ , µ γ ) for eac h orbit γ , i.e. ψ y ( s ) = ψ ( y , s ) ∈ L 2 ( S 1 ); 2) for an y ϕ ∈ C ( X ) the function h ψ , ϕ i L is con tin uous. Let { e j } be a c ountable system of orthonormal func tions forming an orthonormal basis of L 2 ( S 1 ) (e.g. exp onents). Then { 1 A · e j } is an o r t honormal system in L Γ ( X ): h 1 A · e j , 1 A · e k i L ( y ) = Z S 1 1 A ( y ) e j ( s ) e k ( s )1 A ( y ) ds = δ j k . 12 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY Let us s ho w that the C ( K ) -linear span of { 1 A · e j } is de nse in C ( X ) (hence, in L Γ ( X )) with resp ect to the Hilb ert mo dule distance. L et ϕ ∈ C ( X ). Then for a n y ε > 0 w e can c ho ose a division ∆ 1 , . . . , ∆ d of S 1 suc h that sup ∆ i ( ϕ − f i ) < ε d , where f i is indep enden t on s ∈ S 1 , i.e. actually f i ∈ A , and sup ∆ i | f i | ≤ 2 sup X | ϕ | . Let χ i b e the indicator function of ∆ i , i = 1 , . . . , d . T ak e ˆ χ i to b e a C - linear com bination of { e j } suc h that k χ i − ˆ χ i k L 2 ( S 1 ) < ε d , i = 1 , . . . , d. Then ˆ ϕ ( y , s ) := d X i =1 f i ( y ) · ˆ χ i ( s ) ∈ span C ( K ) { 1 a · e j } . Let ψ ∈ C ( X ), k ψ k L ≤ 1. The n kh ϕ − ˆ ϕ, ψ ik L = sup y ∈ K     Z S 1 ( ϕ ( y , s ) − ˆ ϕ ( y , s )) ψ ( y , s ) d s     ≤ sup y ∈ K      Z S 1 ϕ ( y , s ) − d X j =1 f j ( y ) χ j ( s ) ! ψ ( y , s ) ds      + sup y ∈ K      d X j =1 Z S 1 f j ( y )( χ j ( s ) − ˆ χ j ( s )) ψ ( y , s ) ds      ≤ sup y ∈ K sup s ∈ S 1      ϕ ( y , s ) − d X j =1 f j ( y ) χ j ( s )      !  Z S 1 | ψ ( y , s ) | 2 ds  1 / 2 + sup y ∈ K d · sup j =1 ,...,d | f j ( y ) | ·  Z S 1 | χ j ( s ) − ˆ χ j ( s ) | 2 ds  1 / 2 ·  Z S 1 | ψ ( y , s ) | 2 ds  1 / 2 ≤ sup i =1 ,...,d sup K × ∆ i | ϕ ( y , s ) − f i ( y ) | · k ψ k L + (2 sup x ∈ X k ϕ ( x ) k ) · d · ε d · k ψ k L < ε  1 + 2 sup x ∈ X k ϕ ( x ) k  . Th us, L Γ ( X ) = l 2 ( A ) and is not C ∗ -reflexiv e. Although we are far f r o m obtaining a criterium f o r C ∗ -reflexivit y , w e can give a sufficien t condition ev en in the non-metric case. Theorem 4.5. Consider a Lyapunov stable action of Γ on a c omp a c t Hausdorff sp ac e X , wher e X is n ot ne c essarily m e trizable. Supp ose, the c ar dinali ty of finite orbits is uniform ly b ounde d a n d the numb er of closur es of infinite orbits is fi n ite. Then L Γ ( X ) is C ∗ -r eflexive. Pr o of. By the argument in the proo f of the first part of Theorem 3.3 (se e p age 8 ) finite orbits form a closed in v ariant subset X f ⊂ X . The Gelfand spectrum consists of a closed subspace X f / Γ and a finite n um b er of isolated po in ts corresp onding to the closures of infinite orbits. Arguing as in the second part of Theorem 3.3 (see page 10), w e reduce this case to t he case of pure finite orbits [16].  MODULES FROM GROUP ACT IONS 13 5. Fur ther examples W e wan t to show by examples that there are other situations b ey ond the described ab o v e in whic h a we ll-defined av e raging can b e found leading to admissible C ∗ -v alued inner pro ducts and deriv ed Hilb ert C ∗ -mo dule structures o n the corresp onding comm utativ e C ∗ -algebras. The follo wing example sho ws that w e can hav e a non- L y apuno v stable action with a go o d av erage. Example 5.1. Let Γ = Z . Let X b e the direct pro duct X = J × S 1 of the subset J = { 0 , 1 , 1 / 2 , 1 / 3 , . . . } ⊂ R and the unit circle. Let α i → α b e a sequence of irrational n um bers, suc h that α is irrational and α /α i is irrat ional for ev ery i . L et the generator of Z rotate { 1 /i } × S 1 b y α i , and the limit circle { 0 } × S 1 b y α . Clearly w e ha v e 1’) and 2 ’) in this case. The next example demonstrates that in the case of presence of infinite orbits, uniform con tin uit y is not sufficien t f o r con tin uit y of the av erag e. Example 5.2. [18, Example 25] Let X ⊂ R 3 consist of tw o circles S ± :    x = cos 2 π t y = sin 2 π t z = ± 1 , t ∈ ( −∞ , + ∞ ) , and of a non- uniform spiral Σ :    x = cos 2 π τ y = sin 2 π τ z = 2 π · arctan τ , ( −∞ , + ∞ ) . Let the generator g of Γ = Z act on all three comp onents b y t 7→ t + α, τ 7→ τ + α, where α is a p ositive irra tional num ber. Then the isotrop y group of eac h point of X is trivial. Hence, the condition of uniform con tin uit y holds automatically . Let ϕ : X → R b e the restriction of the function R 3 ∋ ( x, y , z ) 7→ z on to X , then the function ϕ x on Z has the follo wing f orm: if x ∈ S ± then ϕ x = ± 1; if x ∈ Σ t hen ϕ x is a function o n Z suc h that ϕ x ( n ) ∈ [ − 1 , 1] for an y n ∈ Z and lim n →±∞ ϕ x ( n ) = ± 1. So, ϕ x is in general not almost p erio dic a nd w e cannot a v erage it using our definition. Nev ertheles s, w e can a v erage it using the amenabilit y of the group Z . In this case w e get E Γ ( ϕ x ) =  ± 1 fo r x ∈ S ± 0 for x ∈ Σ . Th us w e see that E Γ ( ϕ x ) is not con tin uous with resp ect to x ∈ X . Example 5.3. In the previous example let us iden tify the tw o circles , S + and S − . Then X w ould consist of the spiral Σ and of the circle S . Still, the function ϕ x on Γ = Z need not b e almost perio dic, but there is an almost perio dic function ρ on Z such that for an y ε > 0 there is finite subset F ⊂ Z such t ha t k ϕ x − ρ k < ε on Z \ F . This make s it p ossible to define an a v erage E Γ ( ϕ ) b y E Γ ( ϕ x ) = M ( ρ ). And it is easy to see that, this time, E Γ ( ϕ x ) is contin uous with resp ect to x ∈ X . 14 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY Example 5.4. Our next example is a mo dification of Example 4.4. Let Y = N × S 1 , X = β Y its Stone– ˇ Cec h compactification. Let Γ = Z act on Y b y rotating eac h circle b y the irrational a ng le α . This action canonically extends to an action on X . Let s ∈ S 1 . Then the inclusion N → N × S 1 , n 7→ ( n, s ), canonically extends to a map s ∗ : β N → X and m ( s ∗ ( x )) = ( s · e imα ) ∗ ( x ) for any x ∈ β N and any m ∈ Γ = Z . Let ϕ ∈ C ( X ). Since C ( X ) = C b ( Y ) (con tin uous functions on X are canonically iden tified with b ounded con tin uous f unctions on Y ), ϕ can b e iden tified with a uniformly b ounded sequence ( ϕ (1) , ϕ (2) , . . . ) of con tin uous functions on S 1 . Let x ∈ β N . Then ϕ s ∗ ( x ) ( m ) = ϕ (( s · e imα ) ∗ ( x )) , where m ∈ Z . Let U x b e an ultrafilter on N , whic h corresp o nds to the p o in t x ∈ β N . Then ϕ s ∗ ( x ) = lim U x ( ϕ ( n ) ( s )) ∞ n =1 , where the limit of the sequence ( ϕ ( n ) ( s )) ∞ n =1 is tak en o v er U x , hence ϕ s ∗ ( x ) ( m ) = lim U x ( ϕ ( n ) (( s · e imα ) ∗ ( x ))) ∞ n =1 . T ak e ϕ ( n ) ( s ) = e ins . Then ϕ ∈ C b ( Y ) = C ( X ). Then ϕ 1 ∗ ( x ) ( m ) = lim U x ( e inmα ) ∞ n =1 . Let U b e an ultrafilter o n N suc h that lim U ( e inλ ) ∞ n =1 = 0 fo r any λ ∈ ( 0 , 2 π ), and let x 0 ∈ β N b e the p oin t that corresp onds to U . The n ϕ 1 ∗ ( x 0 ) ( m ) =  1 , if m = 0 , 0 , if m 6 = 0 . Th us, for the point y = 1 ∗ ( x 0 ) ∈ X and for t he function ϕ ∈ C ( X ) w e see that the function ϕ y is not almost p erio dic on Z . Nev ertheles s, there is a ‘go o d’ av eraging in this example. Since any contin uous function ϕ on X is a uniformly b ounded sequence of functions ϕ ( n ) , n ∈ N , on S 1 , it is easy to see that C Γ ( X ) ∼ = C b ( N ), and one can define E Γ ( ϕ ) b y the form ula ( E Γ ( ϕ )) n = R S 1 ϕ ( n ) ( s ) ds . These examples show that a go o d av erag ing ( a nd an inner pro duct with v alues in C Γ ( X )) can be defined in a wider class than Lyapuno v stable actions. On the other hand, as the la st tw o examples sho w, a go o d av erag ing, when exists, ma y give rise to a degenerate inner pro duct. 6. Appe ndix In this section w e will prov e the fo llo wing assertion on the uniqueness of inv ariant regular measures: Lemma 6.1. Supp ose, a discr ete gr oup Γ acts on a c omp act Hausdorff sp ac e X in such a way that the orbit γ of an element a ∈ X is dens e in X . If the action is Lyapunov stable, then X c arries not mor e than one inva ria nt r e g ular me asur e. MODULES FROM GROUP ACT IONS 15 In fact the assertion follo ws from [2, Ch. VI I, § 1, Problem 14]. More precisely , denote the closure γ of an or bit γ by T . W e can simplify t he idea of the argument of [2, Ch. VI I, § 1, Problem 14] b ecause T is a compact Hausdorff space. F or an y subset A ⊂ T and an y B ⊂ T with interior p oin ts denote b y ( A : B ) the minimal cardinality of co v ers of A formed b y sets g B , g ∈ Γ. Densit y o f γ in T and Ly apuno v stabilit y imply the finiteness of this n um b er, or, more precisely , that suc h a cov e r ex ists. T o pro v e this, we will sho w that for an y op en set B w e ha v e Γ( B ) = T . Supp ose the opp osite: x 6∈ Γ( B ). Then Γ x is not dense in X . A con tradiction to L emma 3.1. In the uniform structure of T there exists a fundamen tal sub-system S formed by in v ar ia n t sets in T × T (under the dia g onal action of Γ). Not e, if Γ( V ) ⊂ U b y Ly apuno v stabilit y , then ∪ g ∈ Γ g ( V ) is an in v arian t neighborho o d of the diag onal, c ontaining V and con tained in U . If C ⊂ T is a third relativ ely compact set with interior p o in ts, then ( A : C ) ≤ ( A : B ) · ( B : C ) . Recall the follo wing notation. If V and W are subs ets of T × T , the n VW ⊂ T × T is formed by all pairs ( x, y ) suc h that t here exists an elemen t z ∈ T with ( x, z ) ∈ V and ( z , y ) ∈ W . If K ⊂ T is an arbitrary set, then V ( K ) := { y ∈ T | ( x, y ) ∈ V for some x ∈ K } . If K = { a } , w e write V ( a ). Supp ose, K ⊂ T is a compact subset, L ⊃ K is a o p en set, and V ∈ S is an in v ar ian t symmetric op en neigh b orho o d of the diagonal, suc h that V ( K ) ⊂ L , and W ∈ S is a closed in v arian t symmetric neighborho o d of the diagonal suc h that W ⊂ V . Let U b e a symmetric inv arian t set con taining the diagonal suc h tha t UW ⊂ V and WU ⊂ V . Then for an y inv arian t (regula r) p ositive measure ν one has ( W ( a ) : U ( a )) · ν ( K ) ≤ ( L : U ( a )) · ν ( V ( a )) , (10) ( K : U ( a )) · ν ( W ( a )) ≤ ( V ( a ) : U ( a )) · ν ( L ) . (11) Indeed, let L = S ( L : U ( a )) i =1 g i U ( a ), g i ∈ Γ. Then eac h x ∈ K b elongs at least to ( W ( a ) : U ( a )) sets from the collection { g i V ( a ) } . Indeed, W ( x ) is co v ered by g i U ( a ). Hence, its n um ber (n um b er of those of them, whic h really in tersect W ( x )) is greater-equal to ( W ( x ) : U ( a )) = ( W ( a ) : U ( a )). The last equalit y follows from the dens ity of the orbit, Ly apuno v prop ert y and the inv ariance of W : W ( g x ) = g W ( x ). It r emains to show that if g i U ( a ) ∩ W ( x ) 6 = ∅ , then x ∈ g i V ( a ). In this case let g i s ∈ g i U ( a ) ∩ W ( x ) for some s ∈ T , then ( g i a, g i s ) ∈ U , ( x, g i s ) ∈ W . Since the sets are symmetric and WU ⊂ V , w e ha v e ( x, g i a ) ∈ V . Then ( a, g − 1 i x ) ∈ V , g − 1 i x ∈ V ( a ), x ∈ g i V ( a ). Thus , ν ( K ) ≤ 1 ( W ( a ) : U ( a )) ( L : U ( a )) X i =1 ν ( g i U ( a )) . Since ν is in v a rian t, w e obtain (10). T o obtain (11) in a similar w a y , w e will s how that eac h y ∈ L b elongs to not more than ( V ( a ) : U ( a )) sets of the form g i W ( a ). Indeed, if y ∈ g i W ( a ), then y = g i s and ( a, s ) ∈ W for some s ∈ T , as w ell as ( g i a, g i s ). Let ( a, t ) ∈ U . Since WU ⊂ V , ( s, t ) ∈ V , t ∈ V ( s ), U ( a ) ⊂ V ( s ), g i U ( a ) ⊂ g i V ( s ) = V ( y ). So, g i U ( a ) ( i = 1 , . . . , ( L : U ( a )) form a minimal cov e r of L while a pa r t of this co v er is inside V ( y ) ⊂ L . Thus the cardinality of this part is low er-equal to ( V ( y ) : U ( a )) = 16 MICHAEL FRANK, V LADIMIR MAN UILOV, A ND EVGENIJ TROITSKY ( V ( a ) : U ( a )). Hence, ν ( L ) ≥ 1 ( V ( a ) : U ( a )) · ( L : U ( a )) X i =1 ν ( g i W ( a )) = ( L : U ( a )) ν ( W ( a )) ( V ( a ) : U ( a )) ≥ ( K : U ( a )) ν ( W ( a )) ( V ( a ) : U ( a )) and w e obtain (11). Supp ose, A, A 0 ⊂ T a re non-empt y o p en set, K ⊂ T is a compact set. Cho ose a fundamen tal system of op en inv arian t symmetric neighborho o ds U α of the diagonal ∆ indexed b y a net A . Put λ α ( A ) := ( A : U α ( a )) ( A 0 : U α ( a )) , λ ( A ) := lim A λ α ( A ) and λ ′ ( K ) := inf λ ( B ) , where B runs ov er the set of all r elativ ely compact op en neigh b orho o ds o f K . Then from (10) and (1 1) w e obtain (12) λ ′ ( W ( a )) · ν ( K ) ≤ λ ( L ) · ν ( W ( a )) , λ ′ ( K ) · ν ( W ( a )) ≤ λ ′ ( W ( a )) · ν ( L ) for W sufficien tly close to the diagonal to hav e W ( K ) ⊂ L . Indeed, c ho ose op en symmetric in v ar ia n t neigh b orho o ds of W inside its sufficien tly small U α -neigh b orho od: W α and V α (i.e. W α ⊂ U α W , V α ⊂ U α W ) s uc h that W α ⊂ V α and still V α ( K ) ⊂ L . Then choose β 0 = β 0 ( α ) suc h that U β W α ⊂ V α , for all β > β 0 . T ak e in (10 ) W = W α , V = V α , U ( a ) := U β ( a ), and divide b oth parts by ( A 0 : U β ( a )): ( W α ( a ) : U β ( a )) ( A 0 : U β ( a )) ν ( K ) ≤ ( L : U β ( a )) ( A 0 : U β ( a )) ν ( V α ( a )) , ( W α ( a ) : U β ( a )) ( A 0 : U β ( a )) ν ( K ) ≤ ( L : U β ( a )) ( A 0 : U β ( a )) ν ( V α ( a )) . P assing to the limit o v er β ∈ A we obtain λ ( W α ( a )) · ν ( K ) ≤ λ ( L ) · ν ( V α ( a )) for an y α ∈ A . P assing to the limit ov e r α ∈ A we get λ ′ ( W ( a )) · ν ( K ) ≤ λ ( L ) · ν ( W ( a )) . T o obtain the second inequalit y in (12) c hoose a sufficien tly small op en neighborho o d K α of K to ha v e K α ⊂ L and add to the ab ov e restrictions the following one: V α ( K α ) ⊂ L . Then from (11) w e ha v e ( K α : U β ( a )) ( A 0 : U β ( a )) ν ( W ( a )) ≤ ( V α ( a ) : U β ( a )) ( A 0 : U β ( a )) ν ( L ) , ( K α : U β ( a )) ( A 0 : U β ( a )) ν ( W ( a )) ≤ ( V α ( a ) : U β ( a )) ( A 0 : U β ( a )) ν ( L ) . P assing to the limit o v er β ∈ A the inequalit y λ ( K α ) · ν ( W ( a )) ≤ λ ( V α ( a )) · ν ( L ) holds for any α ∈ A . No w passing to the limit o v er α ∈ A we get λ ′ ( K ) · ν ( W ( a )) ≤ λ ′ ( W ( a )) · ν ( L ) . MODULES FROM GROUP ACT IONS 17 In part icular, if K 1 ⊂ T and K 2 ⊂ T are compact subsets, and L 1 ⊃ K 1 and L 2 ⊃ K 2 are relativ ely compact op en sets, t hen (12) implies ν ( K 2 ) λ ( L 2 ) ≤ ν ( W ( a ) λ ′ ( W ( a )) ≤ ν ( L 1 ) λ ′ ( K 1 ) , λ ′ ( K 1 ) · ν ( K 2 ) ≤ λ ( L 2 ) · ν ( L 1 ) . P assing to t he infim um ov e r L 2 ⊃ K 2 (b y the definition o f λ ′ ) a nd ov er L 1 ⊃ K 1 (b y the regularity of ν ) w e obtain λ ′ ( K 1 ) · ν ( K 2 ) ≤ λ ′ ( K 2 ) · ν ( K 1 ). T ransp osing K 1 and K 2 w e hav e λ ′ ( K 2 ) · ν ( K 1 ) ≤ λ ′ ( K 1 ) · ν ( K 2 ). Hence, λ ′ ( K 1 ) · ν ( K 2 ) = λ ′ ( K 2 ) · ν ( K 1 ). This uniquely determines ν (up to a constan t multiplier). Reference s 1. N. Bourba k i, Gener al T op ol o gy , Springer-V erla g, Berlin, 1 989. MR0979 294 (90a:54 001a), MR09792 95 (90a:54001b) 2. N. Bourbak i, Inte gr atio n. II. 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MR1990 630 (2004d:46069) HTWK Leipzig, FB IMN , Postf ach 301166, D-04251 Leipzig, Germany E-mail addr ess : mfrank@i mn.htw k-leipzig.de URL : h ttp:// www.im n.htwk-leipzig.de/~mfrank Dept. o f Mech. and Ma th., Moscow St a te Univ ersity, 119991 GSP -1 Moscow, Russia, . and Harbin Institute of Technology, Harbin, P. R. China E-mail addr ess : manuilov @mech. math.msu.su URL : h ttp:// mech.m ath.msu.su/~manuilov Dept. of Mech. and Ma th., Mosco w St a te U niversity, 119991 GSP-1 Mosco w, R ussia E-mail addr ess : troitsky @mech. math.msu.su URL : h ttp:// mech.m ath.msu.su/~troitsky