A weak* separable C(K)* space whose unit ball is not weak* separable

A WEAK ∗ SEP ARABLE C ( K ) ∗ SP A CE WHOSE UNIT BALL IS NOT WEAK ∗ SEP ARABLE A. A VIL ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Abstract. W e provide a ZFC example o f a compact spac e K such that C ( K ) ∗ is w ∗ -separa ble but its c losed unit ball B C ( K ) ∗ is not w ∗ -separa ble. All previous exam- ples of such kind had b een constructed under CH. W e als o discuss the measurabilit y of the supremum norm on tha t C ( K ) equipp ed with its weak Baire σ - algebra . 1. Introduction Let K b e a compact s pace (all our to p ological spaces are assumed to b e Haus- dorff ) and let C ( K ) b e the Banac h space of all con tin uous real-v alued functions on K (equipp ed with the suprem um norm). One can consider the follo wing list of prop erties related to separabilit y in K and C ( K ) ∗ : (a) K is separable; (b) K carries a strictly p ositiv e measure of coun table t yp e; (c) P ( K ) (the set o f a ll R a don probabilit y measures on K ) is w ∗ -separable; (d) B C ( K ) ∗ (the closed unit ball of C ( K ) ∗ ) is w ∗ -separable; (e) C ( K ) ∗ is w ∗ -separable. It is kno wn that the follo wing implications hold (a) = ⇒ (b) = ⇒ (c) ⇐ ⇒ (d) = ⇒ (e) and cannot b e rev ersed in gene ral. Indeed, the Stone space of the measure algebra of the Leb esgue measure on [0 , 1] satisfies (b) but not (a), while T alag r a nd [18] con- structed under CH t w o examples showing that (c) ⇒ (b) and (e) ⇒ (c) do no t hold. Recen tly , D ˇ zamonja and Plebanek [2] ga v e a ZFC coun terexample to (c) ⇒ (b). In this pa p er w e pro vide a ZF C example of a compact space K w itnessing that the implication (e) ⇒ (c) do es not hold, i.e. C ( K ) ∗ is w ∗ -separable but B C ( K ) ∗ is not. The construction is giv en in Section 2 (see Theorem 2.1) and uses tec hniques w hic h are similar to those of [2]. Date : Novem ber 17, 201 8 . 2010 Mathematics S u bje ct Classific ation. 28E1 5, 4 6E15, 46E27, 54G20. Key wor ds and phr ases. W eak ∗ -separa bilit y; Baire σ -algebr a ; measurability; Kunen cardinal. A. Avil ´ es and J. Rodr ´ ıguez were supp orted b y MEC and FE DE R (Pro ject MTM2 008-0 5396) and F undac i´ on S ´ eneca (Pro ject 08848 / PI/08 ). A. Avil´ es w as s uppo r ted by R amon y Cajal co n tract (R YC-20 0 8-020 51) a nd an FP7-PE OPLE- ERG-2008 actio n. G. Plebanek was supp or ted b y MNiSW Grant N N2 01 418939 (2010–2 0 13). 1 2 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Section 3 is dev oted to discuss further prop erties of that example w hic h are relev an t within the topic of measurabilit y in Banac h space s. In eve ry Banac h space X one can consider the Bair e σ -alg ebra of the w eak top olo gy , denoted b y Ba( X , w ), whic h coincides with the σ -algebra on X generated b y X ∗ (see [3, Th eorem 2.3]). While Ba( X , w ) coincides with the Borel σ - a lgebra of the norm top ology if X is separable, b oth σ -algebras may b e differe n t if X is no nseparable. Giv en any equiv alen t norm k · k on X , its closed unit ball B X b elongs to Ba( X , w ) if and only if k · k is Ba( X , w )- measurable (as a real-v alued function on X ). It is easy to che c k that: B X ∗ is w ∗ -separable = ⇒ k · k is Ba( X , w ) -measurable = ⇒ X ∗ is w ∗ -separable. None of the reve rse implications holds in general [14]. It seem s to b e an op en problem whether t he w eak Baire measurabilit y of the supr emum norm of a C ( K ) space is equiv a len t to either C ( K ) ∗ b eing w ∗ -separable, or to B C ( K ) ∗ b eing w ∗ -separable. Our compact space K of Section 2 w ould settle o ne of the tw o questions in the negativ e, but w e ha v e b een unable to determine the degree of measurabilit y of the suprem um norm o n that C ( K ). W e shall sho w, how ev er, that at least there exists a norm dense set E ⊆ C ( K ) where the restriction of the supremum norm is relative ly Ba( C ( K ) , w )- measurable (Th eorem 3.10). This set E can be tak en to b e the linear span of the c haracteristic functions of clop en subsets of K under the set-theoretic assumption that c is a K unen cardinal ( Theorem 3.22). T erminology. W e write P ( S ) t o denote the p o w er set of any set S . The cardinality of S is denoted by | S | . The letter c stands fo r the cardinalit y of the con tin uum. A probabilit y measure ν is said to b e of countable type if the space L 1 ( ν ) is separable. F or a compact space K , we usually iden tify the dual space C ( K ) ∗ with the space of all Radon measures on K . Giv en a Bo olean alg ebra A , we write A + to denote the set of all nonzero elemen ts of A . F or t he Bo olean op erations w e use the usual sym b ols ∪ , ∩ , etc. and w e write 0 a nd 1 for the least and the greatest elemen t. The Stone space of all ultrafilters on A is den oted b y UL T( A ). Recall that the Stone isomorphism b et w een A and the algebra Clop(UL T( A )) of clop en subsets o f UL T( A ) is giv en b y A → Clop(UL T( A )) , a 7→ b a = {F ∈ UL T( A ) : a ∈ F } . Ev ery measure µ on A induces a measure b a 7→ µ ( a ) on Clop(UL T( A )) whic h can b e uniquely extended to a Radon measure on UL T( A ) (see e.g. [16, Chapter 5]); suc h Radon measure is still denoted b y the same letter µ . Giv en a set S and a family D of subsets of S , the σ - algebra on S generated b y D is denoted b y σ ( D ). If (Ω , Σ) is a n y measurable space a nd n ∈ N , we write ⊗ n Σ to denote the usual pro duct σ - algebra on Ω n , that is, ⊗ n Σ = σ  { A 1 × · · · × A n : A i ∈ Σ for all i = 1 , . . . , n }  . A cardinal κ is called a K une n c ar dinal if the equality P ( κ × κ ) = P ( κ ) ⊗ P ( κ ) A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 3 holds t rue. Of course, in this case w e ha v e P ( κ n ) = ⊗ n P ( κ ) for ev ery n ∈ N . This notion w as in v estigated by Kunen in his do ctoral dissertation [10]. It is kno wn that: (i) an y Kunen cardinal is less than or equal to c ; (ii) ω 1 is a Kunen cardinal; (iii) c is a Kune n cardinal under Martin’s axiom, while it is r elativ ely consisten t that c is not a Kunen cardinal. Kunen cardinals hav e b een considered by T alagrand [17], F remlin [7] and the au- thors [1] in connection with measurabilit y prop erties of Bana c h spaces, and also by T o dorcevic [19] in connection with univ ersalit y prop erties of ℓ ∞ /c 0 . 2. The example Fix an y cardinal κ suc h that ω 1 ≤ κ ≤ c . Let λ b e the usual pro duct probability measure on the Baire σ -algebra of 2 κ and let B b e it s measure a lgebra. Note that B has cardinalit y c since ev ery Baire subset o f 2 κ is determined b y countably man y co ordinates (see e.g. [9, 2 54M]). The letter λ will also stand for the corresp onding probabilit y measure on B . W e shall w ork in the countable simple pro duct C := B N of B , so that ev ery c ∈ C is a sequenc e c = ( c ( n )) n where c ( n ) ∈ B fo r a ll n ∈ N . On the Bo olean algebra C we consider the sequenc e of probabilit y measures { µ n : n ∈ N } defined b y µ n ( c ) := λ ( c ( n )) for all c ∈ C . Let { N b : b ∈ B + } b e a fixed indep end ent family of subsets of N , i.e. \ b ∈ s N b \ [ b ′ ∈ t N b ′ 6 = ∅ whenev er s, t ⊆ B + are finite and disjoint (see e.g. [4, p. 180 , Exercise 3.6.F ]). F or eac h b ∈ B + , define a n elemen t G b ∈ C b y declaring G b ( n ) := ( b if n ∈ N b 0 otherwise. Let A b e the subalgebra of C generated by { G b : b ∈ B + } and write K := UL T( A ). This section is dev oted to pro v e the following: Theorem 2.1. B C ( K ) ∗ is not w ∗ -sep ar able, while { µ n : n ∈ N } sep ar ates the p oints of C ( K ) , so C ( K ) ∗ is w ∗ -sep ar able. In o r der to pro v e Theorem 2.1 w e ne ed some previous w ork. F or an y finite sets s, t ⊆ B + w e consider the following elemen ts of A : J ( s ) := \ b ∈ s G b , U ( t ) := [ b ∈ t G b and W ( s, t ) := J ( s ) \ U ( t ) . The Bo olean algebra A is completely determined up to isomorphism when giv en its set of generator s { G b : b ∈ B + } and whic h elemen ts W ( s, t ) are zero. In this sense, 4 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Lemma 2.2 b elo w is in terpreted as the fact that A is isomorphic to the f r ee Bo olean algebra generated b y { G b : b ∈ B + } mo dulo the relations that W ( s, t ) = 0 if and only if s ∩ t 6 = ∅ o r T b ∈ s b = 0 . Lemma 2.2. F or two finite sets s, t ⊆ B + , we have W ( s, t ) = 0 if and only if s ∩ t 6 = ∅ or T b ∈ s b = 0 . In p articular, for a finite s ⊆ B + the fol lowing a r e e quivalent: (1) T b ∈ s b = 0 ; (2) J ( s ) = 0 ; (3) W ( s, t ) = 0 for a l l fin i te t ⊆ B + \ s ; (4) W ( s, t ) = 0 for s o me fini te t ⊆ B + \ s . Pr o of. It is clear that if s ∩ t 6 = ∅ then W ( s, t ) = 0 . On the other hand, let us observ e that for ev ery n ∈ N w e ha v e (2.1) J ( s )( n ) = \ b ∈ s G b ( n ) = ( T b ∈ s b if n ∈ T b ∈ s N b 0 otherwise. So if T b ∈ s b = 0 , the n W ( s, t ) ⊆ J ( s ) = 0 as w ell. F or the conv erse, supp o se that s ∩ t = ∅ and T b ∈ s b 6 = 0 , and pic k n 0 ∈ \ b ∈ s N b \ [ b ′ ∈ t N b ′ . Then W ( s, t )( n 0 ) = \ b ∈ s G b ( n 0 ) \ [ b ′ ∈ t G b ′ ( n 0 ) = \ b ∈ s b 6 = 0 and so W ( s, t ) 6 = 0 . The second part of the lemma, with the list of equiv alences, follo ws from the first statemen t and the argumen ts ab o v e.  W e next describ e K = UL T( A ). Let us consider the family of all subsets of B with the finite in tersection prop erty , that is FIP( B ) =  X ⊆ B + : b 1 ∩ . . . ∩ b n 6 = 0 for ev ery b 1 , . . . , b n ∈ X  . Giv en X ∈ FIP( B ), let F X b e the filter on A generated b y { W ( s , t ) : s ⊆ X finite , t ⊆ B + \ X finite } (notice that this set has the finite in tersection prop erty by Lemma 2.2). Lemma 2.3. K = {F X : X ∈ FIP( B ) } Pr o of. Ev ery filter of the form F X is an ultrafilter on A , b ecause { G b : b ∈ B + } is a set of generators of A and, for eac h b ∈ B + , w e ha v e either G b = W ( { b } , ∅ ) ∈ F X (if b ∈ X ) o r 1 \ G b = W ( ∅ , { b } ) ∈ F X (if b 6∈ X ) . Con v ersely , let F b e a n y ultrafilter on A and consider X := { b ∈ B + : G b ∈ F } . Notice that X ∈ FIP( B ) and tha t , for eac h b ∈ B + , we hav e G b ∈ F X ⇐ ⇒ G b ∈ F . A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 5 Since { G b : b ∈ B + } is a set o f generators of A , it follows that F = F X .  Let FIP 0 ( B ) be the family of all s ∈ FIP( B ) whic h are finite. The next lemma say s that the clop ens of t he form \ W ( s, t ) are a basis of the t o p ology of K , and also t hat {F s : s ∈ FIP 0 ( B ) } is dense in K . Lemma 2.4. L et a ∈ A + and F ∈ b a ⊆ K . (i) Ther e exist s ∈ F IP 0 ( B ) and a finite se t t ⊆ B + \ s such that F ∈ \ W ( s, t ) ⊆ b a . (ii) If F = F s 0 for some s 0 ∈ FIP 0 ( B ) , then w e c an cho ose s = s 0 in (i). Pr o of. (i) T ak e I ⊆ B + finite such that a b elongs to the subalgebra of A generated b y { G b : b ∈ I } . Then a can b e written as the union of finitely man y elemen ts of the form W ( s, t ), where s ∪ t = I and s ∩ t = ∅ . Since a ∈ F , there exist s and t a s b efore suc h that W ( s, t ) ∈ F , hence s ∈ FIP 0 ( B ) (b y Lemma 2.2) and F ∈ \ W ( s, t ) ⊆ b a . (ii) Since W ( s, t ) ∈ F s 0 , w e hav e G b ∈ F s 0 for a ll b ∈ s and G b 6∈ F s 0 for a ll b ∈ t . Hence s ⊆ s 0 and s 0 ∩ t = ∅ . Therefore, F s 0 ∈ \ W ( s 0 , t ) ⊆ \ W ( s, t ) ⊆ b a .  Another ingredien t to prov e Theorem 2.1 is the result isolated in Lemma 2.6 b elo w, whic h is a consequence of the follo wing c haracterization of w ∗ -separabilit y in spaces of measures, due to M¨ agerl and Namiok a [12]. F act 2.5. L e t L b e a c omp a c t sp ac e. Then the sp ac e P ( L ) is w ∗ -sep ar able if and only if ther e is a s e quenc e { ν n : n ∈ N } in P ( L ) such t hat for every nonempty op en s et V ⊆ L ther e is n ∈ N w ith ν n ( V ) > 1 / 2 . Lemma 2.6. L et A 1 and A 2 b e Bo ole an algebr as such that ther e is another Bo ole an algebr a A 3 c ontaining A 1 and A 2 as sub algebr a s and the fo l lowing hold s: ( ⋆ ) for every b ∈ A + 2 ther e is a ∈ A + 1 such that a ⊆ b . If P (UL T( A 2 )) is not w ∗ -sep ar able, then P ( UL T( A 1 )) is not w ∗ -sep ar able either. Pr o of. Let { ν n : n ∈ N } b e a sequence of pr o babilit y measures on A 1 . F or eac h n ∈ N , w e can extend ν n to a probability measure ν ′ n on A 3 (see [11] or [13]) a nd we denote b y ν n the restriction of ν ′ n to A 2 . Since P (UL T( A 2 )) is not w ∗ -separable, b y F act 2.5 there is b ∈ A + 2 suc h tha t ν n ( b ) ≤ 1 / 2 for ev ery n ∈ N . Property ( ⋆ ) allows us to tak e a ∈ A + 1 suc h that a ⊆ b . Then ν n ( a ) = ν ′ n ( a ) ≤ ν ′ n ( b ) = ν n ( b ) ≤ 1 / 2 for ev ery n ∈ N . Another app eal to F act 2.5 ensures that P (UL T( A 1 )) is not w ∗ -separable.  W e are now ready to deal with Theorem 2.1 . Pr o of of The or em 2.1. According to a result of Rosen thal (see [15 , Theorem 3.1]), the space L ∞ ( ν ) ∗ is not w ∗ -separable whene v er ν is a probability measure of uncoun t- able type. This implies that P (UL T( B )) is no t w ∗ -separable (b ear in mind that C (UL T( B )) is isomorphic to L ∞ ( λ )). Let B ∗ b e the subalgebra of C consisting of all 6 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ constan t sequences. Since B ∗ is isomorphic to B , P (UL T( B ∗ )) is not w ∗ -separable. On the other hand, pro p ert y ( ⋆ ) o f Lemma 2 .6 ho lds for A 1 = A a nd A 2 = B ∗ , hence P ( K ) is not w ∗ -separable and so B C ( K ) ∗ is not w ∗ -separable either. W e no w pro v e t hat { µ n : n ∈ N } separates the p o ints of C ( K ). Fix h ∈ C ( K ) \ { 0 } . Step 1. Since {F s : s ∈ FIP 0 ( B ) } is dense in K (b y Lemma 2.4(i)), there is s ∈ FIP 0 ( B ) suc h that h ( F s ) 6 = 0. Moreo ver, w e can assume further that (2.2) h ( F s ′ ) = 0 whenev er s ′ ∈ FIP 0 ( B ) satisfies | s ′ | < | s | and that C := h ( F s ) > 0. By the con tin uit y of h and Lemma 2.4(ii), there is a finite set t ⊆ B + \ s suc h that, writing a := W ( s, t ), we hav e (2.3) h ( F ) ≥ C 2 for all F ∈ b a. Set δ := C 4 λ ( T b ∈ s b ) > 0 and define R := { r ⊆ s : r 6 = s } ⊆ FIP 0 ( B ). Step 2. Fix r ∈ R . Since h is con t inuous and h ( F r ) = 0 (by (2.2)), we can apply Lemma 2.4(ii) to find a finite set t ′ r ⊆ B + \ r suc h that | h ( F ) | ≤ δ for all F ∈ \ W ( r, t ′ r ). W riting t r := t ′ r \ s , we hav e a r := W ( r , t r ∪ ( s \ r )) ⊆ W ( r , t ′ r ) in A and therefore (2.4) | h ( F ) | ≤ δ fo r a ll F ∈ b a r . Step 3. Define t ∗ := t ∪ [ r ∈R t r ⊆ B + \ s and c ho ose n ∈ T b ∈ s N b \ S b ′ ∈ t ∗ N b ′ (whic h is nonempt y b y indep endence). Hence a ( n ) = \ b ∈ s b, a r ( n ) = \ b ∈ r b \ [ b ′ ∈ s \ r b ′ for ev ery r ∈ R , and therefore (2.5) 1 \ a ( n ) ⊆ [ r ∈R a r ( n ) in B . Step 4. Observ e t ha t µ n  K \ b a  \ [ r ∈R b a r ! = µ n  1 \ a  \ [ r ∈R a r ! = λ  1 \ a ( n )  \ [ r ∈R a r ( n ) ! (2.5) = 0 . Therefore     Z K \ b a h dµ n     ≤ Z S r ∈R c a r | h | dµ n (2.4) ≤ δ. A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 7 It follow s that µ n ( h ) = Z b a h dµ n + Z K \ b a h dµ n ≥ ≥ Z b a h dµ n − δ (2.3) ≥ C 2 µ n ( b a ) − δ = C 2 λ ( a ( n )) − δ = C 2 λ \ b ∈ s b ! − δ = C 4 λ \ b ∈ s b ! > 0 . Th us µ n ( h ) 6 = 0. This finishes the pro of.  R emark 2.7 . W e stress that Ro senthal’s theorem used in the pro of of Theorem 2.1 is a w eak ening of a result stating that L ∞ ( ν ) is not realcompact whenev er ν is a probabilit y measure of uncoun table t ype, see [6 ]. 3. Weak Baire measurability of the norm Throughout this section w e follow the notation in tro duced in Section 2. The supre- m um norm on C ( K ) is denoted by k · k . W e first s ho w that the family { µ n : n ∈ N } ⊆ P ( K ) , though separating the elemen ts of C ( K ), is not ric h enough to “measure” B C ( K ) . R emark 3.1 . The suprem um nor m on C ( K ) do es not hav e to b e measurable with resp ect to the σ -a lgebra generated by { µ n : n ∈ N } . Pr o of. W e iden tify P ( N ) and { 0 , 1 } N in the usual w ay . Let Ω ⊆ 2 N b e an indep enden t family of subs ets of N with | Ω | = c and let Σ denote the trace of Borel(2 N ) on Ω. Then | Σ | = c and so w e can find A ⊆ Ω suc h that A / ∈ Σ and | A | = | Ω \ A | = c . No w we can c ho ose an en umeration Ω = { N b : b ∈ B + } suc h that (3.1) λ ( b ) = 1 / 2 ⇐ ⇒ N b ∈ A. Define a f unction f : Ω → C ( K ) b y f ( N b ) := 1 2 λ ( b ) 1 c G b . W e c laim that f is measurable with respect to Σ and the σ - algebra on C ( K ) generated b y { µ n : n ∈ N } . Indeed, for fixed n ∈ N , we hav e ( µ n ◦ f )( N b ) = 1 2 λ ( b ) µ n ( G b ) = 1 2 λ ( b ) λ ( G b ( n )) = ( 1 / 2 if n ∈ N b 0 otherwise. It f ollo ws that µ n ◦ f = (1 / 2) π n , where π n : Ω ⊆ 2 N → { 0 , 1 } denotes the n -th co ordinate pro jection, hence µ n ◦ f is Σ-measurable. On the o t her ha nd, t he comp osition k f ( · ) k : Ω → R is not Σ-measurable b ecause k f ( N b ) k = 1 (3.1) ⇐ ⇒ N b ∈ A and A / ∈ Σ. This implies that k · k is not measurable with resp ect to the σ - algebra on C ( K ) generated b y { µ n : n ∈ N } .  8 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ 3.1. Measurabilit y on a norm dense set. Let S ( K ) denote the dense subspace of C ( K ) consisting of all linear com binations of c haracteristic functions of clopen subsets of K . The next simple lemm a provides a useful represen tat io n of the elemen ts of S ( K ). Lemma 3.2. L et g ∈ S ( K ) . T hen ther e exis t s ⊆ B + finite an d a c o l le ction of r e a l numb ers { y r : r ⊆ s } such that g = X r ⊆ s y r 1 d J ( r ) . Pr o of. W e can write g as g = X r ′ ⊆ s z r ′ 1 \ W ( r ′ ,s \ r ′ ) for some s ⊆ B + finite a nd certain collection of real num b ers { z r ′ : r ′ ⊆ s } . Note that there is a (unique) collection of real n um b ers { y r : r ⊆ s } suc h that (3.2) z r ′ = X r ⊆ r ′ y r for ev ery r ′ ⊆ s. Since J ( r ) = [ r ⊆ r ′ ⊆ s W ( r ′ , s \ r ′ ) for ev ery r ⊆ s w e conclude that g = X r ′ ⊆ s z r ′ 1 \ W ( r ′ ,s \ r ′ ) (3.2) = X r ′ ⊆ s X r ⊆ r ′ y r 1 \ W ( r ′ ,s \ r ′ ) = X r ⊆ s y r X r ⊆ r ′ ⊆ s 1 \ W ( r ′ ,s \ r ′ ) = X r ⊆ s y r 1 d J ( r ) and the pro of is ov er.  W e denote b y S ′ ( K ) the set o f all g ∈ S ( K ) whic h can b e written as g = X r ⊆ s y r 1 d J ( r ) for some finite set s ⊆ B + and some collection o f nonzer o real n um b ers { y r : r ⊆ s } . It is easy to c hec k (via Lemma 3.2) that S ′ ( K ) is norm dense in S ( K ) and so, S ′ ( K ) is nor m dense in C ( K ). In Theorem 3.10 we shall pro v e that the restriction of the suprem um norm to S ′ ( K ) is measurable with respect to the trace of Ba( C ( K ) , w ) on S ′ ( K ). The pro of requires some w ork and the first step is to find a substan tially larger family of measures to deal with. Lemma 3.3. F or e ach T ⊆ B + and e ach k ∈ N ther e is a pr ob ability me asur e µ k T on A such that, for ev ery fi n ite s et r ⊆ B + , we have µ k T ( J ( r )) = ( λ  T b ∈ r b  k if r ⊆ T , 0 otherwise . A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 9 Mor e over, for every finite disjoint sets r , s ⊆ B + , we have µ 1 T ( W ( r, s )) = ( λ  T b ∈ r b \ S b ′ ∈ s ∩ T b ′  if r ⊆ T , 0 otherwise . Sometimes we shal l write µ T := µ 1 T . Pr o of. Let B k := B ⊗ · · · ⊗ B denote the free pro duct of k man y copies of B and let λ k denote the pro duct measure on B k (see e.g. [8, 2.25]). Consider the function ϕ k T : B + → B k defined b y ϕ k T ( b ) := ( b ⊗ · · · ⊗ b if b ∈ T , 0 otherwise . Then ϕ k T preserv es disjoin tness, so there is a Bo olean homomorphism ˜ ϕ k T : A → B k suc h t ha t ˜ ϕ k T ( G b ) = ϕ k T ( b ) for all b ∈ B + (b ear in mind that A is isomorphic to t he Bo olean algebra freely g enerated by the G b ’s mo dulo the relations that W ( s, t ) = 0 if and only if s ∩ t 6 = ∅ or T b ∈ s b = 0 ; see Lemma 2 .2 and the preceding commen ts). No w, it is not difficult to c hec k that the form ula µ k T ( a ) := λ k ( ˜ ϕ k T ( a )) defines a probabilit y measure µ k T on A satisfying the required prop erty .  F rom the tec hnical p o in t of view, the following subsets of S ( K ) will play a relev ant role in our pro of of Theorem 3.10. Definition 3.4. L et D b e a fin i te p artition of B + . We de note by S D ( K ) (r e s p . S ′ D ( K ) ) the set of al l g ∈ S ( K ) which c an b e written a s g = X r ⊆ s y r 1 d J ( r ) for some finite s e t s ⊆ B + such that | T ∩ s | ≤ 1 for e v e ry T ∈ D and some c ol le ction of r e al numb ers (r esp. nonzer o r e al numb ers) { y r : r ⊆ s } . Definition 3.5. L et D b e a fi n ite p artition of B + and C ⊆ D . We define: (i) T C := S T ∈ C T ⊆ B + ; (ii) a signe d me asur e ν k C on A (for k = 1 , 2 ) by ν k C := X B ⊆ C ( − 1) | C \ B | µ k T B ; (iii) a function θ C : C ( K ) → R by θ C ( g ) := ( ν 1 C ( g )) 2 ν 2 C ( g ) if ν 2 C ( g ) 6 = 0 and θ C ( g ) := 0 otherwise; 10 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ (iv) a function η C : C ( K ) → R by η C ( g ) := ν 1 C ( g ) θ C ( g ) if θ C ( g ) 6 = 0 and η C ( g ) := 0 otherwise. Clearly , the functions θ C and η C defined ab ov e are Ba( C ( K ) , w )- measurable. The follo wing lemma collects sev eral useful prop erties of suc h functions. Lemma 3.6. L et D b e a fin ite p a rtition of B + and le t g ∈ S D ( K ) b e as in Defini- tion 3.4. Fix r ⊆ s and k ∈ { 1 , 2 } . Writing C r := { T ∈ D : T ∩ r 6 = 0 } , the fol lowing statements hold: (i) If C ⊆ C r then ν k C ( J ( r )) = µ k T C ( J ( r )) . (ii) If C r ( C ⊆ D then ν k C ( J ( r )) = 0 . (iii) ν k C r ( g ) = y r ( λ ( T b ∈ r b )) k . (iv) I f J ( r ) 6 = 0 then θ C r ( g ) = y r . (v) I f J ( r ) 6 = 0 and y r 6 = 0 , then η C r ( g ) = λ ( T b ∈ r b ) . Pr o of. (i) F or eac h B ( C w e hav e r 6⊆ T B (b ecause C ⊆ C r ) and so µ k T B ( J ( r )) = 0. Hence ν k C ( J ( r )) = X B ⊆ C ( − 1) | C \ B | µ k T B ( J ( r )) = µ k T C ( J ( r )) . (ii) F or eac h B ⊆ C with C r 6⊆ B we hav e r 6⊆ T B and so µ k T B ( J ( r )) = 0. On the other hand, g iv en any C r ⊆ B ⊆ C w e ha ve r ⊆ T C r ⊆ T B and therefore µ k T B ( J ( r )) = ( λ ( T b ∈ r b )) k . Hence ν k C ( J ( r )) = X B ⊆ C ( − 1) | C \ B | µ k T B ( J ( r )) = X C r ⊆ B ⊆ C ( − 1) | C \ B | µ k T B ( J ( r )) = = λ \ b ∈ r b !! k · X C r ⊆ B ⊆ C ( − 1) | C \ B | . The assumption C 6 = C r implies X C r ⊆ B ⊆ C ( − 1) | C \ B | = X A ⊆ C \ C r ( − 1) | A | = 0 , as can b e easily c hec ke d by induction on | C \ C r | . Therefore, ν k C ( J ( r )) = 0. (iii) T ak e an y r ′ ⊆ s . Note first that if r ′ 6⊆ r then r ′ 6⊆ T B for ev ery B ⊆ C r , hence ν k C r ( J ( r ′ )) = X B ⊆ C r ( − 1) | C r \ B | µ k T B ( J ( r ′ )) = 0 . A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 11 On the other hand, if r ′ ( r , then C r ′ ( C r and (ii) implies ν k C r ( J ( r ′ )) = 0. Observ e also that ν k C r ( J ( r )) = ( λ ( T b ∈ r b )) k (b y (i)). It follows that ν k C r ( g ) = X r ′ ⊆ s y r ′ ν k C r ( J ( r ′ )) = y r ν k C r ( J ( r )) = y r λ \ b ∈ r b !! k . (iv) Bearing in mind (iii) and the fact tha t J ( r ) 6 = 0 , w e hav e y r = 0 if a nd only if ν 2 C r ( g ) = 0. Th us, b y the v ery definition of θ C r , t he eq uality θ C r ( g ) = y r holds whenev er y r = 0. On the other ha nd, if y r 6 = 0 then θ C r ( g ) = ( ν 1 C r ( g )) 2 ν 2 C r ( g ) (iii) = y r and η C r ( g ) = ν 1 C r ( g ) θ C r ( g ) (iii) = λ \ b ∈ r b ! , whic h pro v es (v).  Our nex t step is to pro v e tha t , for an y Z ⊆ B + and p ∈ N , the mapping g 7→ µ Z ( g p ) is measurable with respect to the trace of Ba ( C ( K ) , w ) on S ′ ( K ) (see Lemma 3.9 b elo w). W e b egin by ch ec king the measurabilit y on subsets of t he fo rm S ′ D ( K ). Lemma 3.7. L et Z ⊆ B + b e a set, D a finite p artition of B + finer than { Z , B + \ Z } and p ∈ N . Then the mapping φ Z,p : S ′ D ( K ) → R , φ Z,p ( g ) := µ Z ( g p ) , is me asur able with r esp e c t to the tr ac e of Ba( C ( K ) , w ) on S ′ D ( K ) . Pr o of. W rite D ( Z ) := { T ∈ D : T ⊆ Z } and let Λ b e the set o f all β = ( β C ) C ⊆ D ( Z ) suc h that β C ∈ N ∪ { 0 } for all C ⊆ D ( Z ) and P C ⊆ D ( Z ) β C = p . W e write  p β  := p ! Q C ⊆ D ( Z ) β C ! and C ( β ) := [ C ⊆ D ( Z ) β C > 0 C . T o pro v e the measurabilit y of φ Z,p with resp ect to the trace of Ba( C ( K ) , w ) on S ′ D ( K ) it is sufficien t to c hec k that, for eac h g ∈ S ′ D ( K ), the following equalit y holds: (3.3) φ Z,p ( g ) = X β ∈ Λ  p β      Y C ⊆ D ( Z ) β C > 0 θ β C C ( g )     η C ( β ) ( g ) . Step 1. W rite g = X r ⊆ s y r 1 d J ( r ) , 12 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ where s ⊆ B + is finite, | T ∩ s | ≤ 1 for ev ery T ∈ D and y r ∈ R \ { 0 } for all r ⊆ s . Let ∆ b e the set o f all δ = ( δ r ) r ⊆ s suc h that δ r ∈ N ∪ { 0 } fo r all r ⊆ s and P r ⊆ s δ r = p . W riting  p δ  := p ! Q r ⊆ s δ r ! and r ( δ ) := [ r ⊆ s δ r > 0 r , w e ha v e g p = X δ ∈ ∆  p δ     Y r ⊆ s δ r > 0 y δ r r    1 \ J ( r ( δ )) and so (3.4) φ Z,p ( g ) = µ Z ( g p ) = X δ ∈ ∆  p δ     Y r ⊆ s δ r > 0 y δ r r    µ Z  J ( r ( δ ))  . Step 2. Let δ ∈ ∆ suc h tha t µ Z ( J ( r ( δ ))) 6 = 0. F or an y r ⊆ s with δ r > 0 we hav e J ( r ) ⊇ J ( r ( δ )), hence µ Z ( J ( r )) 6 = 0 (in particular, J ( r ) 6 = 0 ) and so r ⊆ Z , whic h implies that C r = { T ∈ D : T ∩ r 6 = ∅} ⊆ D ( Z ). Set β = ( β C ) C ⊆ D ( Z ) b y declaring (3.5) β C := ( δ r if C = C r for some r ⊆ s with δ r > 0, 0 otherwise . Then P C ⊆ D ( Z ) β C = P r ⊆ s δ r = p , so that β ∈ Λ. Moreo v er, w e hav e C ( β ) = [ C ⊆ D ( Z ) β C > 0 C = [ r ⊆ s δ r > 0 C r = C r ( δ ) . Since µ Z ( J ( r ( δ ))) 6 = 0, w e hav e J ( r ( δ )) 6 = 0 and µ Z ( J ( r ( δ ))) = λ ( T b ∈ r ( δ ) b ). Th us, from Lemma 3.6(v) it follows that η C ( β ) ( g ) = η C r ( δ ) ( g ) = µ Z ( J ( r ( δ ))). On the other hand, for eac h r ⊆ s with δ r > 0 we hav e y r = θ C r ( g ) by Lemma 3.6(iv), hence Y C ⊆ D ( Z ) β C > 0 θ β C C ( g ) = Y r ⊆ s δ r > 0 θ δ r C r ( g ) = Y r ⊆ s δ r > 0 y δ r r . Therefore  p β      Y C ⊆ D ( Z ) β C > 0 θ β C C ( g )     η C ( β ) ( g ) =  p δ     Y r ⊆ s δ r > 0 y δ r r    µ Z  J ( r ( δ ))  . This show s that each nonzero summand of (3.4) can b e w ritten as a summand of (3.3). Note also that if δ ′ ∈ ∆ satisfies µ Z ( J ( r ( δ ′ ))) 6 = 0 and w e define β ′ = ( β ′ C ) C ⊆ D ( Z ) ∈ Λ as in (3.5) (with δ replaced b y δ ′ ), t hen β 6 = β ′ whenev er δ 6 = δ ′ . A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 13 Step 3. Let β ∈ Λ such that (3.6)     Y C ⊆ D ( Z ) β C > 0 θ β C C ( g )     η C ( β ) ( g ) 6 = 0 . Fix C ⊆ D ( Z ) with β C > 0. W e claim that r C := T C ∩ s satisfies C r C = C . Indeed, the inclusion C r C ⊆ C is clear. T o pro v e the rev erse inclusion, w e argue by con tradiction. Supp ose C r C ( C . By Lemma 3.6(ii) , w e hav e ν 1 C ( J ( r ′ )) = 0 whenev er r ′ ⊆ T C . Since w e also ha v e ν 1 C ( J ( r ′ )) = 0 for eve ry r ′ ⊆ s with r ′ 6⊆ T C (b y the very definition o f ν 1 C ), it follows that ν 1 C ( g ) = P r ′ ⊆ s y r ′ ν 1 C ( J ( r ′ )) = 0, hence θ C ( g ) = 0, whic h contradicts (3.6). This show s that C r C = C , as claimed. No w Lemma 3.6(iii) ensures that ν 1 C ( g ) = ν 1 C r C ( g ) = y r C λ \ b ∈ r C b ! . Since ν 1 C ( g ) 6 = 0, the previous equality implies that J ( r C ) 6 = 0 . F rom Lemma 3.6(iv) it follow s that y r C = θ C r C ( g ) = θ C ( g ). Set δ = ( δ r ) r ⊆ s b y declaring (3.7) δ r := ( β C if r = r C for some C ⊆ D ( Z ) with β C > 0 , 0 otherwise . Then P r ⊆ s δ r = P C ⊆ D ( Z ) β C = p , hence δ ∈ ∆. F rom our previous considerations w e deduce that Y C ⊆ D ( Z ) β C > 0 θ β C C ( g ) = Y C ⊆ D ( Z ) β C > 0 y β C r C = Y r ⊆ s δ r > 0 y δ r r . Moreo v er, w e claim that η C ( β ) ( g ) = µ Z ( J ( r ( δ ))). Indeed, since C ( β ) = [ C ⊆ D ( Z ) β C > 0 C = [ C ⊆ D ( Z ) β C > 0 C r C = [ r ⊆ s δ r > 0 C r = C r ( δ ) , w e ha v e η C ( β ) ( g ) = η C r ( δ ) ( g ) and so (3.6) implies that ν 1 C r ( δ ) ( g ) 6 = 0. Bearing in mind Lemma 3.6(iii) and the f a ct that y r ( δ ) 6 = 0, w e infer that J ( r ( δ )) 6 = 0 . An app eal to Lemma 3.6(v) no w yields η C ( β ) ( g ) = λ ( T b ∈ r ( δ ) b ). On the other ha nd, t he fa ct that r ( δ ) = [ r ⊆ s δ r > 0 r = [ C ⊆ D ( Z ) β C > 0 r C = [ C ⊆ D ( Z ) β C > 0 T C ∩ s ⊆ Z ensures that µ Z ( J ( r ( δ ))) = λ ( T b ∈ r ( δ ) b ). It follo ws that η C ( β ) ( g ) = µ Z ( J ( r ( δ ))). 14 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Therefore  p β      Y C ⊆ D ( Z ) β C > 0 θ β C C ( g )     η C ( β ) ( g ) =  p δ     Y r ⊆ s δ r > 0 y δ r r    µ Z ( J ( r ( δ ))) . This show s that each nonzero summand of (3.3) can b e w ritten as a summand of (3.4). Note t ha t if β ′ ∈ Λ satisfies (3.6) (with β replaced by β ′ ) and w e define δ ′ = ( δ ′ r ) r ⊆ s ∈ ∆ as in (3.7) (with β replaced b y β ′ ), then δ 6 = δ ′ whenev er β 6 = β ′ . Th us, equalit y (3.3 ) ho lds true and the pro of is o v er.  The follow ing folklore fa ct will allow us to prov e Lemma 3.9 as an easy consequence of Lemma 3.7 a b o v e. R emark 3.8 . Le t (Ω , Σ) b e a measurable space. W rite Ω = S j ∈ N Ω j where Ω j ⊆ Ω j +1 for all j ∈ N . Let A ⊆ Ω b e a set suc h that, f o r each j ∈ N , the in tersection A ∩ Ω j b elongs to the trace of Σ on Ω j . Then A ∈ Σ. Pr o of. F or each j ∈ N we hav e A ∩ Ω j = E j ∩ Ω j for some E j ∈ Σ. W e claim that A = S k ∈ N T j ≥ k E j . Indeed, w e ha v e A = [ k ∈ N A ∩ Ω k = [ k ∈ N \ j ≥ k A ∩ Ω j = [ k ∈ N \ j ≥ k E j ∩ Ω j ⊆ [ k ∈ N \ j ≥ k E j . On the other hand, for eac h k ∈ N , w e hav e \ j ≥ k E j = [ n ≥ k \ j ≥ k E j ∩ Ω n ⊆ [ n ≥ k E n ∩ Ω n = [ n ≥ k A ∩ Ω n = A. It follow s that A = S k ∈ N T j ≥ k E j ∈ Σ.  Lemma 3.9. L et Z ⊆ B + b e a se t an d p ∈ N . Then the mapp ing φ Z,p : S ′ ( K ) → R , φ Z,p ( g ) := µ Z ( g p ) , is me asur able with r esp e c t to the tr ac e of Ba( C ( K ) , w ) on S ′ ( K ) . Pr o of. Since | B + | = c , there is a sequence D (1) , D (2) , . . . of finite partitions of B + , eac h one b eing finer than { Z , B + \ Z } , suc h that: • D ( j + 1) is finer than D ( j ) for all j ∈ N ; • for ev ery s ⊆ B + finite there is j ∈ N suc h that | T ∩ s | ≤ 1 for all T ∈ D ( j ). Indeed, let ξ : { 0 , 1 } N → B + b e any bijection and, for eac h j ∈ N and σ ∈ { 0 , 1 } j , set E σ j := { x ∈ { 0 , 1 } N : x ( i ) = σ ( i ) f o r every i = 1 , . . . , j } . Then the partitions D ( j ) :=  ξ ( E σ j ) ∩ Z : σ ∈ { 0 , 1 } j  ∪  ξ ( E σ j ) \ Z : σ ∈ { 0 , 1 } j  , j ∈ N , fulfill the required prop erties. A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 15 Clearly , S ′ ( K ) = S j ∈ N S ′ D ( j ) ( K ) and S ′ D ( j ) ( K ) ⊆ S ′ D ( j +1) ( K ) fo r all j ∈ N . The measurabilit y of φ Z,p with resp ect to the trace of Ba ( C ( K ) , w ) on S ′ ( K ) now follo ws from Lemma 3.7 and Remark 3.8.  W e hav e a lr eady gathered all the to ols needed to pro v e the main result of this subsection: Theorem 3.10. The r estriction of the supr e m um norm to S ′ ( K ) is me asur able w ith r esp e ct to the tr ac e of Ba( C ( K ) , w ) on S ′ ( K ) . Pr o of. W e fix a coun table a lgebra Z on B + whic h separates the p oin ts of B + (the algebra o f clop en subsets of { 0 , 1 } N can be t ransferred to B + via any bijection betw een { 0 , 1 } N and B + ). W e claim that (3.8) k g k = sup Z ∈Z lim sup p →∞  µ Z ( g 2 p )  1 2 p for ev ery g ∈ C ( K ) . Indeed, the ine qualit y “ ≥ ” is obv ious (eac h µ Z is a probability measure). T o verify the rev erse inequality , fix g ∈ C ( K ) and tak e ε > 0. By Lemma 2.4(i) there exist finite disjoin t sets r, s ⊆ B + suc h that | g ( F ) | ≥ k g k − ε for every F ∈ \ W ( r, s ) 6 = ∅ . Since Z separates t he p oin ts of B + , we can find Z ∈ Z suc h that r ⊆ Z and s ∩ Z = ∅ , hence µ Z ( W ( r, s )) = λ ( T b ∈ r b ) > 0. Since  µ Z ( g 2 p )  1 2 p ≥ ( k g k − ε )  µ Z ( W ( r, s ))  1 2 p for ev ery p ∈ N , w e ha v e lim sup p →∞  µ Z ( g 2 p )  1 2 p ≥ ( k g k − ε ) lim p →∞  µ Z ( W ( r, s ))  1 2 p = k g k − ε. As ε > 0 is arbitr a ry , equalit y (3.8) holds true. Once w e kno w that k · k is expressed b y the formula (3.8), the assertion follo ws from Lemma 3.9.  3.2. Measurabilit y on the set of simple functions. Any eleme n t of S ( K ) admits a represen tatio n whic h cannot b e simplified in a sense, a s the following lemma shows. Lemma 3.11. L et g ∈ S ( K ) . Then ther e exist a finite set s ⊆ B + and a c ol le ction of r e al numb ers { z r : r ⊆ s } such that: (i) g = P r ⊆ s z r 1 \ W ( r,s \ r ) ; (ii) ther e is no s ′ ( s such that z r = z r ′ whenever r ∩ s ′ = r ′ ∩ s ′ , \ b ∈ r b 6 = 0 and \ b ∈ r ′ b 6 = 0 . 16 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Pr o of. Of course, w e can write g as in (i). T o get a represen tation satisfying (ii), w e pro ceed b y induction on | s | . The case s = ∅ b eing ob vious, w e assume tha t s 6 = ∅ and that the induction h yp othesis ho lds. Assume that (ii) fails and fix s ′ ( s suc h t ha t z r = z r ′ whenev er r ∩ s ′ = r ′ ∩ s ′ , \ b ∈ r b 6 = 0 and \ b ∈ r ′ b 6 = 0 . F or any t ⊆ s ′ with T b ∈ t b 6 = 0 , let A t b e the collection of all r ⊆ s suc h that r ∩ s ′ = t and T b ∈ r b 6 = 0 . Then z r = z t for ev ery r ∈ A t and W ( t, s ′ \ t ) = [ r ∈A t W ( r, s \ r ) , as can b e easily c hec ke d. Hence g = X r ⊆ s z r 1 \ W ( r,s \ r ) = X t ⊆ s ′ X r ⊆ s r ∩ s ′ = t z r 1 \ W ( r,s \ r ) = X t ⊆ s ′ T b ∈ t b 6 = 0 X r ∈A t z r 1 \ W ( r,s \ r ) = = X t ⊆ s ′ T b ∈ t b 6 = 0 X r ∈A t z t 1 \ W ( r,s \ r ) = X t ⊆ s ′ T b ∈ t b 6 = 0 z t 1 \ W ( t,s ′ \ t ) = X t ⊆ s ′ z t 1 \ W ( t,s ′ \ t ) . Since | s ′ | < | s | , the induction hypothesis no w ensures that g admits a represen tation satisfying b oth (i) and (ii).  Definition 3.12. L et D b e a finite p artition of B + . We denote by A D ( K ) the set of al l g ∈ S ( K ) which c an b e written as g = X r ⊆ s z r 1 \ W ( r,s \ r ) for some finite set s ⊆ B + and some c ol le ction of r e al numb ers { z r : r ⊆ s } such that: (i) | T ∩ s | = 1 for every T ∈ D ; (ii) ther e is no s ′ ( s such that z r = z r ′ whenever r ∩ s ′ = r ′ ∩ s ′ , \ b ∈ r b 6 = 0 and \ b ∈ r ′ b 6 = 0 . Our next step is to pro v e t ha t the sets A D ( K ) defined ab ov e b elong to the tr ace of Ba( C ( K ) , w ) on S ( K ) (see Corollary 3.16 b elow). F rom no w on we fix a c ountable algebra Z on B + whic h separates the p oints of B + (lik e in the pr o of of Theorem 3 .1 0). Lemma 3.13. L e t D b e a finite p artition o f B + with D ⊆ Z and let g ∈ A D ( K ) . Then f o r e ach T 0 ∈ D ther e is T ∈ Z such that µ T \ T 0 ( g ) 6 = µ T ∪ T 0 ( g ) . Pr o of. Our pro of is by con tradiction. Supp ose there is T 0 ∈ D suc h that (3.9) µ T \ T 0 ( g ) = µ T ∪ T 0 ( g ) for ev ery T ∈ Z . A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 17 W rite g = P r ⊆ s z r 1 \ W ( r,s \ r ) as in Definition 3.12. Let s ′ := s \ T 0 ( s . In o rder to reac h a con t r adiction, w e claim tha t z r = z r ′ whenev er r ∩ s ′ = r ′ ∩ s ′ , \ b ∈ r b 6 = 0 and \ b ∈ r ′ b 6 = 0 . Indeed, assume that r 6 = r ′ and pro ceed b y induction on | r ∩ s ′ | = | r ′ ∩ s ′ | . Supp ose first that r ∩ s ′ = r ′ ∩ s ′ = ∅ . Then we ha v e r = ∅ and r ′ = T 0 ∩ s (or vice v ersa). By (3.9) w e ha v e µ ∅ ( g ) = µ T 0 ( g ). Note t ha t µ ∅ ( g ) = z ∅ and, writing T 0 ∩ s = { b 0 } , w e ha v e µ T 0 ( g ) = z ∅ (1 − λ ( b 0 )) + z T 0 ∩ s λ ( b 0 ). It f ollo ws that z ∅ = z ∅ (1 − λ ( b 0 )) + z T 0 ∩ s λ ( b 0 ) , hence z ∅ = z T 0 ∩ s , a s required. Supp ose now that r ∩ s ′ = r ′ ∩ s ′ 6 = ∅ , tog ether with T b ∈ r b 6 = 0 6 = T b ∈ r ′ b , and the inductiv e h yp o thesis. Since r 6 = r ′ , w e ha v e either b 0 ∈ r and b 0 6∈ r ′ or vice vers a. W e assume for instance that b 0 ∈ r and b 0 6∈ r ′ . Then r = { b 0 } ∪ ( r ∩ s ′ ) and r ′ = r ∩ s ′ . By the inductiv e h yp othesis, (3.10) z r 0 = z r 0 ∪{ b 0 } for ev ery r 0 ( r ′ . Set T 1 := T 0 ∪ [ { T ∈ D : T ∩ s ⊆ r ′ } ∈ Z and observ e that T 1 ∩ s = r . W riting w ( t, t ′ ) := \ b ∈ t b \ [ b ′ ∈ t ′ b ′ ∈ B for an y pair of finite sets t, t ′ ⊆ B + , w e ha v e (3.11) µ T 1 ( g ) = X r 0 ⊆ r z r 0 λ  w ( r 0 , r \ r 0 )  = X r 0 ⊆ r ′ z r 0 λ  w ( r 0 , r \ r 0 )  + X r 0 ⊆ r b 0 ∈ r 0 z r 0 λ  w ( r 0 , r ′ \ r 0 )  = = z r ′ λ  w ( r ′ , { b 0 } )  + X r 0 ( r ′ z r 0 λ  w ( r 0 , r \ r 0 )  + + z r λ  w ( r, ∅ )  + X r 0 ( r ′ z r 0 ∪{ b 0 } λ  w ( r 0 ∪ { b 0 } , r ′ \ r 0 )  . F or each r 0 ( r ′ , the elemen ts w ( r 0 , r \ r 0 ) and w ( r 0 ∪ { b 0 } , r ′ \ r 0 ) are disjoin t and their union is w ( r 0 , r ′ \ r 0 ), hence (3.10) yields z r 0 λ  w ( r 0 , r ′ \ r 0 )  = z r 0 λ  w ( r 0 , r \ r 0 )  + z r 0 ∪{ b 0 } λ  w ( r 0 ∪ { b 0 } , r ′ \ r 0 )  . 18 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ F rom (3.11) it follo ws that (3.12) µ T 1 ( g ) = z r ′ λ  w ( r ′ , { b 0 } )  + z r λ  w ( r, ∅ )  + X r 0 ( r ′ z r 0 λ  w ( r 0 , r ′ \ r 0 )  . Bearing in mind that ( T 1 \ T 0 ) ∩ s = r ′ , w e also ha v e (3.13) µ T 1 \ T 0 ( g ) = X r 0 ⊆ r ′ z r 0 λ  w ( r 0 , r ′ \ r 0 )  = z r ′ λ  w ( r ′ , ∅ )  + X r 0 ( r ′ z r 0 λ  w ( r 0 , r ′ \ r 0 )  . Since µ T 1 \ T 0 ( g ) = µ T 1 ( g ) ( b y (3 .9)), equalities (3 .12) and (3.13) yield z r ′ λ  w ( r ′ , { b 0 } )  + z r λ  w ( r, ∅ )  = z r ′ λ  w ( r ′ , ∅ )  , therefore z r λ ( T b ∈ r b ) = z r ′ λ ( T b ∈ r b ) and so z r = z r ′ . This finishes the pro of.  R emark 3.14 . Let g ∈ S ( K ) b e written as g = P r ⊆ s z r 1 \ W ( r,s \ r ) for some finite set s ⊆ B + and z r ∈ R . If T , T ′ ⊆ B + satisfy T ∩ s = T ′ ∩ s , then µ T ( g ) = µ T ′ ( g ). Pr o of. F or eve ry r ⊆ s we hav e r ⊆ T if and only if r ⊆ T ′ . In this case, µ T ( W ( r, s )) = λ \ b ∈ r b \ \ b ∈ s ∩ T b ! = µ T ′ ( W ( r, s )) . Hence µ T ( g ) = P r ⊆ T z r µ T ( W ( r, s )) = P r ⊆ T ′ z r µ T ′ ( W ( r, s )) = µ T ′ ( g ).  Lemma 3.15. L et D b e a finite p artition of B + with D ⊆ Z and l e t g ∈ S ( K ) . Then g ∈ A D ( K ) if and only if the fo l lowing two statements hold: ( ⋆ ) for e ach T 0 ∈ D ther e is T ∈ Z such that µ T \ T 0 ( g ) 6 = µ T ∪ T 0 ( g ) ; ( ⋆⋆ ) f o r e ac h T 0 ∈ D and e ach finite p artition D 0 of T 0 with D 0 ⊆ Z , ther e i s Z 0 ∈ D 0 such that µ T \ Z ( g ) = µ T ∪ Z ( g ) for every Z ∈ D 0 \ { Z 0 } and T ∈ Z . Pr o of. “Onl y if” p ar t. Supp ose g ∈ A D ( K ) and write g = P r ⊆ s z r 1 \ W ( r,s \ r ) as in Definition 3.1 2 . Statemen t ( ⋆ ) holds b y Lemma 3.1 3. T o c hec k ( ⋆ ⋆ ), tak e T 0 ∈ D and fix a finite partition D 0 of T 0 with D 0 ⊆ Z . Since T 0 ∩ s is a singleton, there is Z 0 ∈ D 0 suc h that Z ∩ s = ∅ for ev ery Z ∈ D 0 \ { Z 0 } , and so for an y T ∈ Z w e ha v e ( T \ Z ) ∩ s = ( T ∪ Z ) ∩ s , hence µ T \ Z ( g ) = µ T ∪ Z ( g ) ( Remark 3.14). “If” p ar t. W rite g = P r ⊆ s z r 1 \ W ( r,s \ r ) for some finite set s ⊆ B + and some collection o f real nu m b ers { z r : r ⊆ s } . By Lemma 3.11, this represen ta t io n can b e c hosen in suc h a w a y that there is no s ′ ( s suc h that z r = z r ′ whenev er r ∩ s ′ = r ′ ∩ s ′ , \ b ∈ r b 6 = 0 and \ b ∈ r ′ b 6 = 0 . In order to prov e that g ∈ A D ( K ) w e only hav e to ch ec k that | T 0 ∩ s | = 1 for eve ry T 0 ∈ D . Observ e first that, fo r each T 0 ∈ D , condition ( ⋆ ) tells us that there is T ∈ Z suc h that µ T \ T 0 ( g ) 6 = µ T ∪ T 0 ( g ), hence ( T \ T 0 ) ∩ s 6 = ( T ∪ T 0 ) ∩ s (R emark 3.1 4) and so T 0 ∩ s 6 = ∅ . Th us, w e can find a finite partition D ′ ⊆ Z of B + finer than D suc h that | T ′ ∩ s | = 1 for ev ery T ′ ∈ D ′ . Therefore, g ∈ A D ′ ( K ). A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 19 Fix T 0 ∈ D and set D 0 := { T ′ ∈ D ′ : T ′ ⊆ T 0 } . By Lemm a 3.13 applied to g and D ′ , for each T ′ ∈ D 0 there is T ′′ ∈ Z suc h t ha t µ T ′′ \ T ′ ( g ) 6 = µ T ′′ ∪ T ′ ( g ). This fact and condition ( ⋆⋆ ) yield | D 0 | = 1, that is, D 0 = { T 0 } and so | T 0 ∩ s | = 1. As T 0 ∈ D is arbitrary , g ∈ A D ( K ) and the pro of is o v er.  Corollary 3.16. L et D b e a finite p artition of B + . Then A D ( K ) b elo n gs to the tr ac e of Ba( C ( K ) , w ) on S ( K ) . Pr o of. W e can assume without lo ss of g eneralit y (b y enlarging Z if necessary) that D ⊆ Z . Since Z is countable, Lemma 3.15 give s the result.  Our next task is to prov e that, under the assumption that c is a Kunen cardinal, the restriction of the suprem um norm to a ny set of the fo r m A D ( K ) is relativ ely Ba( C ( K ) , w )-measurable (Lemma 3.21). Lemma 3.17. L et Φ : B + → 2 Z b e the ma pping define d by Φ( b ) := (1 T ( b )) T ∈Z . Set Ω := Φ( B + ) and let Σ b e the tr ac e of Bo r el(2 Z ) on Ω . T hen for e ach n ∈ N the mapping Φ n : (( B + ) n , ⊗ n σ ( Z )) → (Ω n , ⊗ n Σ) , Φ n ( b 1 , . . . , b n ) := (Φ( b 1 ) , . . . , Φ( b n )) , is an isomo rphism of me a s ur able s p ac es . Pr o of. It suffices to pro v e the case n = 1. Clearly , Φ is one-to-one (b ecause Z separates the p oints of B + ) and σ ( Z )- Σ- measurable. On the other hand, for eac h T 0 ∈ Z w e ha v e Φ( T 0 ) = { (1 T ( b )) T ∈Z : b ∈ T 0 } = Ω ∩ { ( x T ) T ∈Z ∈ 2 Z : x T 0 = 1 } ∈ Σ , hence Φ − 1 is Σ- σ ( Z )-measurable.  Lemma 3.18. L et D b e a fi n ite p artition of B + with D ⊆ Z , let T 0 ∈ D and T ∈ Z . L et g ∈ A D ( K ) . Write g = P r ⊆ s z r 1 \ W ( r,s \ r ) as in Defi n ition 3.12 and T 0 ∩ s = { b 0 } . (1) The fol lowin g statements ar e e quivalen t: (i) b 0 ∈ T ; (ii) µ ¯ T ( g ) = µ ¯ T ∪ T 0 ( g ) for every ¯ T ∈ Z such that ¯ T ∩ T 0 = T ∩ T 0 . (2) The fol lowin g statements ar e e quivalen t: (i’) b 0 6∈ T ; (ii’) µ ¯ T ( g ) = µ ¯ T \ T 0 ( g ) for every ¯ T ∈ Z such that ¯ T ∩ T 0 = T ∩ T 0 . Pr o of. (i) ⇒ (ii) Let ¯ T ∈ Z b e suc h that ¯ T ∩ T 0 = T ∩ T 0 . Since b 0 ∈ T b y a ssumption, w e ha v e b 0 ∈ T ∩ T 0 ⊆ ¯ T a nd so ( ¯ T ∪ T 0 ) ∩ s = ¯ T ∩ s . Bearing in mind Remark 3.14, w e get µ ¯ T ( g ) = µ ¯ T ∪ T 0 ( g ). A similar argument yields (i’) ⇒ (ii’). No w, in order to pro v e (ii) ⇒ (i) and (ii’) ⇒ (i’), it is enough to c hec k that state- men t s (ii) and ( ii’) cannot hold sim ultaneously . T o this end, pic k T ∗ ∈ Z suc h that µ T ∗ \ T 0 ( g ) 6 = µ T ∗ ∪ T 0 ( g ) ( w e apply Lemma 3.13) and set ¯ T := ( T ∗ \ T 0 ) ∪ ( T ∩ T 0 ) ∈ Z . 20 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Clearly , ¯ T ∪ T 0 = T ∗ ∪ T 0 and ¯ T \ T 0 = T ∗ \ T 0 , hence w e hav e either µ ¯ T ∪ T 0 ( g ) 6 = µ ¯ T ( g ) or µ ¯ T \ T 0 ( g ) 6 = µ ¯ T ( g ). Since ¯ T ∩ T 0 = T ∩ T 0 , this sho ws that either (ii) or (ii’) fails.  R emark 3.19 . Let D = { T 1 , . . . , T n } b e a finite partition of B + with D ⊆ Z . (1) Let i ∈ { 1 , . . . , n } . G iv en T ∈ Z , since statemen ts (ii) a nd (ii’) in Lemma 3 .18 (applied to T i ) are indep endent o f the represen tation of g ∈ A D ( K ), there is a mapping ψ D ,T i ,T : A D ( K ) → { 0 , 1 } suc h that, f o r an y g = P r ⊆ s z r 1 \ W ( r,s \ r ) as in D efinition 3.12 and writing T i ∩ s = { b i } , we hav e ψ D ,T i ,T ( g ) := ( 1 if b i ∈ T , 0 if b i 6∈ T . ψ D ,T i ,T is measurable with resp ect to the trace of Ba( C ( K ) , w ) on A D ( K ), thanks t o Lemma 3.18 (applied to T i ). Define ψ D ,T i : A D ( K ) → 2 Z b y ψ D ,T i ( g ) := ( ψ D ,T i ,T ( g )) T ∈Z . Observ e that ψ D ,T i ( A D ( K )) ⊆ Ω , b ecause for an y g ∈ A D ( K ) as ab ov e w e ha v e ψ D ,T i ( g ) = Φ( b i ). (2) Th us, we can consider the mapping ψ D : A D ( K ) → Ω n , ψ D ( g ) := ( ψ D ,T 1 ( g ) , . . . , ψ D ,T n ( g )) . Clearly , ψ D is measurable with resp ect to ⊗ n Σ and the trace of Ba( C ( K ) , w ) on A D ( K ). (3) Let P ⊆ { 1 , . . . , n } . Define ζ n,P : ( B + ) n → R b y ζ n,P ( b ′ 1 , . . . , b ′ n ) := λ \ i ∈ P b ′ i ! . Then the mapping L D ,P : A D ( K ) → R g iv en by L D ,P := ζ n,P ◦ (Φ n ) − 1 ◦ ψ D satisfies L D ,P ( g ) = λ ( T i ∈ P b i ) f o r every g ∈ A D ( K ) as ab ov e. F rom no w on w e deal with the additional assumption that c is a Kunen cardinal. Lemma 3.20. Supp ose c is a Kunen c ar dinal. Then ther e is a c ountable algeb r a Z 0 on B + sep ar a ting the p o i n ts of B + such that, for e ach n ∈ N and P ⊆ { 1 , . . . , n } , the mapping ζ n,P is ⊗ n σ ( Z 0 ) -me asur able. Pr o of. Since | B + | = c is a Kunen cardinal, eac h ζ n,P is ⊗ n P ( B + )-measurable. Th us, w e can find a coun table family C of s ubsets of B + suc h t ha t ζ n,P is ⊗ n σ ( C )-measurable for ev ery n ∈ N and ev ery P ⊆ { 1 , . . . , n } . No w, it is enough to c ho ose any countable algebra Z 0 on B + whic h separates the p oints of B + and contains C .  Lemma 3.21. Supp ose c is a Kunen c ar dinal. L et D b e a finite p a rtition of B + . Then the r estriction of the supr emum norm to A D ( K ) is me asur a b le with r esp e c t to the tr ac e of Ba ( C ( K ) , w ) o n A D ( K ) . A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 21 Pr o of. W rite D = { T 1 , . . . , T n } . W e can supp ose without loss of generalit y (b y enlarg- ing Z if necessary) that D ⊆ Z and that all functions ζ n,P are ⊗ n σ ( Z )-measurable (see Lemma 3.20). F or each P ⊆ { 1 , . . . , n } , define N D ,P : A D ( K ) → { 0 , 1 } , N D ,P ( g ) := ( 1 if L D ,P ( g ) 6 = 0, 0 if L D ,P ( g ) = 0. Since L D ,P is measurable with resp ect to the trace of Ba( C ( K ) , w ) on A D ( K ) (com bine Lemma 3.17 and Remark 3.19), the same holds for N D ,P . Fix g ∈ A D ( K ) and write g = X r ⊆ s y r 1 d J ( r ) for some s ⊆ B + finite with | T i ∩ s | = 1 for ev ery i ∈ { 1 , . . . , n } and some collection of real n um b ers { y r : r ⊆ s } (see the pro of of Lemma 3.2). Lemma 3.6(iv) ensures that y r = θ C r ( g ) for ev ery r ⊆ s with J ( r ) 6 = 0 . Hence g = P r ⊆ s θ C r ( g )1 d J ( r ) and therefore (3.14) g = X r ′ ⊆ s X r ⊆ r ′ θ C r ( g ) ! 1 \ W ( r ′ ,s \ r ′ ) (see again the pro of of Lemma 3.2). W rite T i ∩ s = { b i } for ev ery i ∈ { 1 , . . . , n } and observ e that for eac h P ⊆ { 1 , . . . , n } we hav e N D ,P ( g ) = ( 1 if T i ∈ P b i 6 = 0 , 0 if T i ∈ P b i = 0 . Define C ( Q ) := { T i : i ∈ Q } for ev ery Q ⊆ { 1 , . . . , n } . F rom (3.14) it follo ws that k g k = sup r ′ ⊆ s J ( r ′ ) 6 = 0      X r ⊆ r ′ θ C r ( g )      = sup P ⊆{ 1 ,.. .,n }      X Q ⊆ P θ C ( Q ) ( g )      · N D ,P ( g ) . As g ∈ A D ( K ) is arbitrary , the norm function k · k coincides with sup P ⊆{ 1 ,.. .,n }      X Q ⊆ P θ C ( Q ) ( · )      · N D ,P ( · ) on A D ( K ) and so k · k is measurable with resp ect to the tra ce of Ba( C ( K ) , w ) on A D ( K ). The pro of is o v er.  Finally , w e arriv e at the main result of this subsection: Theorem 3.22. Supp ose c is a Kunen c ar din al. The n the r estriction of the supr emum norm to S ( K ) is me asur a b le with r esp e ct to the tr ac e of Ba( C ( K ) , w ) on S ( K ) . 22 A. A VI L ´ ES, G. PLEBANEK, AND J. RODR ´ IGUEZ Pr o of. Let Π be the collection of all partitions of B + in to finitely many elemen ts of Z . By Lemma 3 .1 1, w e can write S ( K ) = S D ∈ Π A D ( K ). Since Π is coun table and eac h A D ( K ) b elongs to the trace of Ba( C ( K ) , w ) on S ( K ) (see Coro llary 3.16), the result follo ws from Lemma 3.21.  3.3. Some op en problems. (A) Let L b e a compact space. Is the Ba( C ( L ) , w )-measurability of the suprem um norm on C ( L ) equiv alent to the w ∗ -separabilit y of B C ( L ) ∗ or C ( L ) ∗ ? What ab out the compact space K considered in this pap er? (B) Let ( X , k · k ) b e a Banac h space and supp o se A ⊆ X is a norm dense set ( o r linear subspace) such that k · k is relativ ely Ba( X , w )- measurable on A . Do es this imply tha t k · k is Ba ( X , w )- measurable on X ? Note that the analogous question for the property “ B X ∗ is w ∗ -separable” has p ositiv e answ er (just apply the Hahn-Ba na c h theorem), while for the pro p ert y “ X ∗ is w ∗ -separable” it has negativ e answ er, see [5, Example 1.1]. (C) L et L b e a compact space and consider the ‘square’ mapping S : C ( L ) → C ( L ) , S ( g ) := g 2 . Whic h conditions on L ensure that S is Ba( C ( L ) , w )-measurable? Note that if L carries a strictly p ositiv e measure, sa y µ , then the suprem um norm o n C ( L ) can b e computed as k g k = lim n →∞  Z L f 2 n dµ  1 2 n , hence k · k is Ba( C ( L ) , w )-measurable whenev er S is Ba( C ( L ) , w )-measurable. What a b out the compact space K considered in this pap er? Reference s [1] A. Avil´ es, G. Plebanek, J. Ro dr ´ ıguez, Me asura bility in C (2 κ ) and Kunen c ar dinals , to app ear in Israel J. Math. [2] M. Dˇ zamonja, G. Plebanek, St rictly p ositive me asur es on Bo ole an algebr as , J. Sym b olic Logic 73 (2008), no. 4, 1416– 1432 . [3] G. A. Edga r, Me asur ability in a Banach sp ac e , Indiana Univ. Math. J. 26 (1977), no . 4, 66 3–677 . [4] R. Engelking , Gener al top olo gy , Monografie Ma tematyczne, T om 6 0, PWN - Polish Scientific Publishers, W a rsaw, 1 977. [5] V. F onf, A. Pezzotta, C. Za nc o , Tiling infinite-dimensional norme d sp ac es , Bull. Lo nd. Math. So c. 29 (1997), no. 6, 713– 719. [6] R. F rank ie w ic z, G. Plebanek, O n nonac c essible filt ers in m e asur e algebr as and fu n ctionals on L ∞ ( λ ) ∗ , Studia Math. 108 (1994), 191 –200. [7] D.H. F remlin, Bor el sets in nonsep ar able Banach sp ac es , Ho kk aido Math. J. 9 (1980 ), no. 2, 179–1 83. [8] D.H. F re mlin, Me asur e algebr as , in: J.D. Monk (E d.), Handb o ok of Bo o lean Algebra s, vol. I I I, North-Holland, Amsterdam, 1989 , Chapter 22. A WEAK ∗ SEP ARABLE C ( K ) ∗ SP AC E WHOS E UN IT BALL IS N O T WEAK ∗ SEP ARABLE 23 [9] D.H. F remlin, Me asur e The ory. Volume 2: Br o ad Foundations , T or res F remlin, Colchester, 2001. [10] K. Kunen, Inac c essibility pr op erties of c ar dinals , Pro Quest LLC, Ann Arbo r, MI, 1968. Thesis (Ph.D.) Stanford Universit y . [11] Z. Lip ecki, Extensions of additive s et functions with values in a top olo gic al gr oup , Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 22 (1974), 19–27. [12] G. M¨ agerl and I. Namiok a, Interse ction numb ers and we ak ∗ sep ar ability of sp ac es of me asur es , Math. Ann. 249 (1980), 273–2 79. [13] D. Pla chky , Extr emal a nd mo no genic ad ditive set funct ions , Pro c. Amer. Math. So c. 54 (1976), 193–1 96. [14] J. Ro dr ´ ıguez, We ak Bair e me asur ability of the b al ls i n a Banach sp ac e , Studia Math. 185 (2008), no. 2, 169– 176. [15] H.P . Rosenthal, Some line ar top olo gic al pr op erties of L ∞ of a finit e me asure sp ac e , Bull. Amer. Math. So c. 75 (1969 ), 798–802. [16] Z. Semadeni, Banach sp ac es of c ont inuous funct ions , Monogr afie Matematyczne, T om 55, PWN - Polish Scien tific P ublishers, W arsaw, 1971. [17] M. T alag rand, E st-c e que l ∞ est un esp ac e me asur able? , Bull. Sci. Math. 1 03 (1 979), 255–258. [18] M. T alag rand, S´ ep ar abilit´ e vague dans l’esp ac e des mesur es sur un c omp act , Israe l J. Math. 37 (1980), 171 –180. [19] S. T o dor cevic, Emb e dding function sp ac es into ℓ ∞ /c 0 , J. Math. Anal. App. 384 (2011), no. 2, 246–2 51. Dep a r t amento de Ma tem ´ aticas, F a cul t ad d e Ma tem ´ aticas, Universidad de Mur cia, 30100 E spinardo (Murcia), Sp ain E-mail addr ess : avil eslo@u m.es Instytut Ma tema tyczny, Uniwersytet Wroc la wski, Wr oc la w, Pol and E-mail addr ess : grze s@math .uni.w roc.pl Dep a r t amento de Ma tem ´ atica Aplicada, F a cul t ad de Inform ´ atica, U niversidad de Murcia, 30 100 Espinardo (Murcia), S p ain E-mail addr ess : jose rr@um. es