Characterisation of conditional weak mixing via ergodicity of the tensor product in Riesz Spaces
aa r X i v : . [ m a t h . F A ] J a n Characterisation of conditional weak mixing viaergodicity of the tensor product in Riesz Spaces ∗ Jonathan Homann ♯ Wen-Chi Kuo ♯ Bruce A. Watson ♯ †‡ ♯ School of MathematicsUniversity of the WitwatersrandPrivate Bag 3, P O WITS 2050, South AfricaJanuary 5, 2021
Abstract
We link conditional weak mixing and ergodicity of the tensor product in Riesz spaces.In particular, we characterise conditional weak mixing of a conditional expectation pre-serving system by the ergodicity of its tensor product with itself or other ergodic systems.In order to achieve this we characterise all band projections in the tensor product of twoDedekind complete Riesz spaces with weak order units.
In Eisner et al [5, Theorem 9.23], Krengel, [17, pg. 98] and Petersen, [19, Section2.6, Theorem 6.1], characterisation of weakly mixing systems via the tensor product ofergodic systems is presented. The concept of ergodicity was discussed in the Riesz spacesetting in [11]. ∗ AMS Subject Classification: 46A40; 47A35; 37A25; 60F05. Keywords: Riesz spaces; tensor products;band projections; conditional expectation operators; weak mixing; ergodicity. † Supported in part by the Centre for Applicable Analysis and Number Theory and by National ResearchFoundation of South Africa grant IFR170214222646 with grant no. 109289. ‡ e-mail: [email protected] n the classical setting, given a probability space (Ω , A , µ ) and a measure preservingtransformation τ : Ω → Ω, the measure preserving system (Ω , A , µ, τ ) is ergodic if andonly if lim n →∞ n n − X k =0 µ (cid:16) τ − k ( A ) ∩ B (cid:17) = µ ( A ) µ ( B )for all A, B ∈ A . Here (Ω , A , µ, τ ) is said to be weakly mixing iflim n →∞ n n − X k =0 (cid:12)(cid:12)(cid:12) µ (cid:16) τ − k ( A ) ∩ B (cid:17) − µ ( A ) µ ( B ) (cid:12)(cid:12)(cid:12) = 0 , for all A, B ∈ A .Various aspects of mixing processes have been considered in the Riesz space setting in[11, 15, 16] and other for some other aspects of stochastic processes in Riesz spaces seefor example [8]. We recall from [12] that an operator, T , on a Dedekind complete Rieszspace, E , with weak order unit is said to be a conditional expectation operator if T is apositive order continuous projection on E which maps weak order units to weak orderunits and has range R ( T ) a Dedekind complete Riesz subspace of E (i.e. R ( T ) is orderclosed in E ). We say that a conditional expectation operator, T , is strictly positive if T x = 0 with x ∈ E + implies x = 0.In [10] we generalised the notion of a measure preserving system to the Riesz spacesetting as follows. Let E be a Dedekind complete Riesz space with weak order unit,say, e , let T be a conditional expectation operator on E with T e = e and let S be aRiesz homomorphism on E with Se = e . If T SP e = T P e , for all band projections P on E , then ( E, T, S, e ) is called a conditional expectation preserving system. Due toFreudenthal’s Spectral Theorem, [23, Theorem 33.2], the condition
T SP e = T P e for allband projections P on E in the above is equivalent to T Sf = T f for all f ∈ E .In [11] we obtained that the conditional expectation preserving system ( E, T, S, e ) isergodic if and only if 1 n n − X k =0 T (cid:16)(cid:16) S k P e (cid:17) · Qe (cid:17) → T P e · T Qe, (1.1)in order, as n → ∞ , for all band projection P and Q on E . Further, we defined( E, T, S, e ) to be conditionally weak mixing if1 n n − X k =0 (cid:12)(cid:12)(cid:12) T (cid:16)(cid:16) S k P e (cid:17) · Qe (cid:17) − T P e · T Qe (cid:12)(cid:12)(cid:12) → , (1.2)in order, as n → ∞ , for all band projections P and Q on E . Here · denote multi-plication in the f -algebra E e = { f ∈ E | | f | ≤ ke for some k ∈ R + } , the subspace of E of e bounded elements of E , see [2, 22, 24]. The f -algebra structure on E e gives e · Qe = P Qe for all band projections P and Q on E . Further, if T is a conditional ex-pectation operator on E with T e = e , then T is also a conditional expectation operatoron E e . We refer the reader to Aliprantis and Border [1], Fremlin [6], Meyer-Nieberg [18],Venter [22] and Zaanen [23] and [24] for background on Riesz spaces and f -algebras.The main result of the present paper characterises conditional weak mixing in a Rieszspace by ergodicity in the tensor product as follows. Theorem 1.1
In a conditional expectation preserving system ( E, T, S, e ) , with T strictlypositive, the following are equivalent.1. ( E, T, S, e ) is conditionally weak mixing.2. ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) is conditionally weak mixing for each conditionallyweak mixing ( E , T , S , e ) with T strictly positive.3. ( E ⊗ E, T ⊗ T, S ⊗ S, e ⊗ e ) is conditionally weak mixing.4. ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) is ergodic for each ergodic ( E , T , S , e ) with T strictly positive.5. ( E ⊗ E, T ⊗ T, S ⊗ S, e ⊗ e ) is ergodic. For this result, proved in Section 4, we need to characterise band projections on tensorproducts of Dedekind complete Riesz spaces with weak order units, this will be donein Section 3. In Section 3 we present some preliminary material on conditional weakmixing and ergodicity in Riesz spaces.
From [14, Lemma 2.1] it follows that if E is a Dedekind complete Riesz space and ( f n )is an order convergent sequence in E with order limit 0 then 1 n n − X k =0 | f k | →
0, in order,as n → ∞ . Further, the order convergence of the sequence of partial sums n − X k =0 | f k | ! implies the order convergence of the sequence of partial sums n − X k =0 f k ! . Moreover, if( f n ) has order limit f , then 1 n n − X k =0 f k → f , in order, as n → ∞ . et ( E, T, S, e ) be a conditional expectation preserving system. For f ∈ E and n ∈ N ,we define S n f := 1 n n − X k =0 S k f, (2.1a) L S f := lim n →∞ S n f, (2.1b)where the above limits are order limits, if they exist.We say that f ∈ E is S -invariant if Sf = f . The set of all S -invariant f ∈ E willbe denoted I S := { f ∈ E | Sf = f } . The set of f ∈ E for which L S f exists will bedenoted E S . As shown in [11], I S ⊂ E S and L S f = f , for all f ∈ I S . Using this,Birkhoff’s (Bounded) Ergodic Theorem, [13, Theorem 3.7], gives that, in a conditionalexpectation preserving system, ( E, T, S, e ), for each f ∈ E , the sequence ( S n f ) is orderbounded in E if and only if f ∈ E S and, further to this, for each f ∈ E S , L S f = SL S f and T L S f = T f . Thus, L S : E S → I S . Finally, if E = E S then L S is a conditionalexpectation operator on E with L S e = e . By restricting out attention to E e , we havethat ( S n f ) is order bounded in E e for each f ∈ E e and so, by Birkhoff’s (Bounded)Ergodic Theorem, E e ⊂ E S , moreover, L S f ∈ E e , giving that L S | E e is a conditionalexpectation operator on E e .As in [11] we say that the conditional expectation preserving system ( E, T, S, e ) is ergodicif L S f ∈ R ( T ) for all f ∈ I S . As shown in [11], ( E, T, S, e ) is ergodic if and only L S f = T f for all f ∈ I S . We now recall some results from [11]. Corollary 2.1
The conditional expectation preserving system ( E, T, S, e ) with E = E S is ergodic if and only if T = L S . Corollary 2.2
The conditional expectation preserving system ( E, T, S, e ) is ergodic ifand only if n n − X k =0 T (cid:16)(cid:16) S k P e (cid:17) · Qe (cid:17) → T P e · T Qe, (2.2) in order as n → ∞ , for all band projections P and Q on E . In [19], a subset N of N is said to be of density zero if1 n n − X k =0 χ N ( k ) → n → ∞ , where χ N ( k ) = 0 if k ∈ N \ N and χ N ( k ) = 1 if k ∈ N . he Koopman-von Neumann Lemma [19, Lemma 6.2] asserts that if a sequence ( f n ) ofreal numbers is non-negative and bounded, then 1 n n − X k =0 f k → n → ∞ if and only ifthere is N , a subset of N of density zero, such that f n → n → ∞ , for n ∈ N \ N .This was extended to the Riesz space context in [10] as follows. Definition 2.3 (Density Zero Sequence of Band Projections)
A sequence ( P n ) ofband projections in a Riesz space E with weak order unit e is said to be of density zeroif n n − X k =0 P k e → , in order, as n → ∞ . With the above definition of density zero sequences of band projections we can now givean analogue of the Koopman-von Neumann Lemma in Riesz spaces.
Theorem 2.4 (Koopman-von Neumann)
Let E be a Dedekind complete Riesz spacewith weak order unit and let ( f n ) be an order bounded sequence in the positive cone E + ,of E , then n n − X k =0 f k → , in order, as n → ∞ , if and only if there exists a density zerosequence of band projections ( P n ) such that ( I − P n ) f n → , in order, as n → ∞ . We recall the following characterisation theorem from [10].
Theorem 2.5
Given the conditional expectation preserving system ( E, T, S, e ) , then thefollowing statements are equivalent.1. ( E, T, S, e ) is conditionally weak mixing.2. n n − X k =0 (cid:12)(cid:12)(cid:12) T (cid:16)(cid:16) S k f (cid:17) · g (cid:17) − T f · T g (cid:12)(cid:12)(cid:12) → , in order, as n → ∞ , for all f, g ∈ E e .3. Given band projections P and Q on E , there is a sequence of density zero bandprojections, ( R n ) , in E , such that ( I − R n ) | T (( S n P e ) · Qe ) − T P e · T Qe | → , in order, as n → ∞ . s shown in [11]: Theorem 2.6
If a conditional expectation preserving system is conditionally weak mix-ing then it is ergodic.
For the general construction of the tensor product of Riesz spaces and f -algebras werefer the reader to [3, 4, 7, 9, 20, 21]. We proceed here using the approaches of [3, 7, 9].As shown in [9], if E and F are Archimedean Riesz spaces then a partial ordering can beinduced on E ⊗ F by the cone generated by E + ⊗ F + . Further, from the multilinearityof the tensor product, we have that if 0 ≥ f ∈ E and 0 ≥ g ∈ F then f ⊗ g ≥ E ⊗ F . Proposition 3.1
Let E and F be Archimedean Riesz spaces and x ∈ E and y ∈ F ,then ( x ⊗ y ) + = x + ⊗ y + + x − ⊗ y − , ( x ⊗ y ) − = x + ⊗ y − + x − ⊗ y + , | x ⊗ y | = | x | ⊗ | y | . Proof:
As ( x ⊗ y ) + = ( x ⊗ y ) ∨ (0 ⊗ x ⊗ y ) + = ( x ⊗ y ) ∨ (0 ⊗ (cid:0) x + ⊗ y + (cid:1) ∨ (cid:0) x − ⊗ y − (cid:1) ∨ (cid:0) − x + ⊗ y − (cid:1) ∨ (cid:0) − x − ⊗ y + (cid:1) ∨ x + ⊗ y + + x − ⊗ y − , where 0 ⊗ E ⊗ F . Proceeding in a similar way we obtain the result for ( x ⊗ y ) − .Finally | x ⊗ y | = ( x ⊗ y ) + + ( x ⊗ y ) − = x + ⊗ y + + x − ⊗ y − + x + ⊗ y − + x − ⊗ y + = x + ⊗ | y | + x − ⊗ | y | = | x | ⊗ | y | . Let E and F be Archimedean Riesz spaces. Suppose that ( x α ) α ∈ Λ is a net in E with x α → x in E , in order, and suppose that ( y β ) β ∈ Γ is a net in F with y β → y in F , inorder, then there exists a net ( u α ) α ∈ Λ in E such that u α ↓ | x α − x | ≤ u α ↓ v β ) β ∈ Γ in F such that v β ↓ | y β − y | ≤ v β ↓
0. Furthermore, wenote for later reference, since y β → y , in order, there exists Y ∈ F + such that | y β | ≤ Y for all β ∈ Γ. Thus we have that x α ⊗ y β → x ⊗ y in E ⊗ F , in order.As discussed in [9], for two Archimdedean Riesz spaces E and F , when constructing thetensor product space E ⊗ F , the Dedekind completion is taken. Consequently, E ⊗ F isa Dedekind complete Riesz space. roposition 3.2 Let E , F and E ⊗ F be Dedekind complete Riesz spaces. If P is aband projection on E and Q is a band projection on F , then P ⊗ Q is a band projectionon E ⊗ F .Proof: In a Riesz space, say G , if R is a positive, order continuous projection with R ≤ I ,where I is the identity map on G , then R is a band projection on G . Let I E and I F be theidentity maps on E and F , respectively, and let I E ⊗ F be the identity map on E ⊗ F . It isclear that I E ⊗ F = I E ⊗ I F . Furthermore, let P and Q be band projections on E and F ,respectively. By the properties of the tensor product, we have that P ⊗ Q is positive andorder continuous. Furthermore, ( P ⊗ Q ) = ( P ⊗ Q ) ( P ⊗ Q ) = (cid:0) P (cid:1) ⊗ (cid:0) Q (cid:1) = P ⊗ Q ,so P ⊗ Q is a positive, order continuous projection. Finally, since P ≤ I E and Q ≤ I F ,we have that P ⊗ Q ≤ I E ⊗ I F = I E ⊗ F .We now characterise the band projections in the tensor product of two Dedekind com-plete f -algebras. As will be seen in the proof below, this characterisation relies on theapproximation property of Grobler and Labuschagne, [9, Theorem 2.12(c)]. Proposition 3.3
Suppose that E and F are Dedekind complete Riesz spaces with weakorder units e and f , respectively. Let G := E ⊗ F . If R is a band projection in G , thenthere exist ( P n ) and ( Q n ) , sequences of band projections in E and F , respectively, suchthat R = W n ∈ N P n ⊗ Q n .Proof: As G is a Dedekind complete Riesz space with weak order unit e ⊗ f , every bandin G is a principle band. Thus if B is a band in G then there exists h ∈ G + with B = B h and without loss of generality h ≤ e ⊗ f . Here h is a weak order unit for B and, thus,for 0 ≤ g ∈ G , Rg = W n ∈ N ( g ∧ ( √ nh )).Since h ∈ G + , by [9, Theorem 2.12(c)], for each n ∈ N there are e n,j ∈ E + and f n,j ∈ F + , j = 1 , . . . , N n for some N n ∈ N , such that0 ≤ h − N n X j =1 e n,j ⊗ f n,j ≤ n e ⊗ f. Further, as h ≤ e ⊗ f , we can choose e n,j ≤ e and f n,j ≤ f . Now, e n,j and f n,j can beapproximated from below by an e -step and f -step functions, (see [23] for details on an e -step function). In particular there band projections P n,j,i on E that form a partitionof I E for i = 1 , , ..., M n,j and Q n,j,i on F that partition I F for i = 1 , , ..., M n,j so that e n,j − n · N n e ≤ M n,j X i =1 a n,j,i P n,j,i e ≤ e n,j nd f n,j − n · N n f ≤ M n,j X i =1 b n,j,i Q n,j,i f ≤ f n,j . Here 0 ≤ a n,j,i ≤ ≤ b n,j,i ≤
1. Hence, by the multilinearity of the tensor product, e n,j ⊗ f n,j − n · N n e ⊗ f ≤ M n,j X i,k =1 a n,j,i b n,j,k P n,j,i e ⊗ Q n,j,k f ≤ e n,j ⊗ f n,j . Here P n,j,i ⊗ Q n,j,k , i, k ∈ { , . . . , M n,j } is a partition of I E ⊗ I F .Summing over j we have that N n X j =1 e n,j ⊗ f n,j − n e ⊗ f ≤ N n X j =1 M n,j X i,k =1 a n,j,i b n,j,k P n,j,i e ⊗ Q n,j,k f ≤ N n X j =1 e n,j ⊗ f n,j , and thus (cid:18) h − n e ⊗ f (cid:19) + ≤ N n X j =1 M n,j X i,k =1 c n,j,i P n,j,i e ⊗ Q n,j,k f ≤ h, where c n,j,i := a n,j,i b n,j,k and 0 ≤ c n,j,i ≤ h = _ n ∈ N N n X j =1 M n,j X i,k =1 c n,j,i P n,j,i e ⊗ Q n,j,k f, and the band generated by h is then the same as that generated by _ c n,j,i =0 P n,j,i e ⊗ Q n,j,k f giving R = _ c n,j,i =0 P n,j,i ⊗ Q n,j,k . Re-indexing gives the stated result.We note that as the representation of R = _ n ∈ N P n ⊗ Q n in Proposition 3.3 is as a supremum of a countable family of band projections, we canalso represent R as R = _ n ∈ N ˜ P n ⊗ ˜ Q n here, as previously, ˜ P n is a band projection on E and ˜ Q n is a band projection on F for each n ∈ N , but, in addition, ( ˜ P m ⊗ ˜ Q m ) ∧ ( ˜ P n ⊗ ˜ Q n ) = 0 for all n = m . With thisadditional property we have that R = _ n ∈ N ˜ P n ⊗ ˜ Q n = X n ∈ N ˜ P n ⊗ ˜ Q n , and hence the following corollary. Corollary 3.4 If E and F are Dedekind complete Riesz spaces with weak order unitsand R is a band projection on the Dedekind complete Riesz space E ⊗ F , then thereare sequences of band projections (cid:16) ˜ P n (cid:17) on E and (cid:16) ˜ Q n (cid:17) on F such that (cid:16) ˜ P m ⊗ ˜ Q m (cid:17) ∧ (cid:16) ˜ P n ⊗ ˜ Q n (cid:17) = 0 for all m = n and R = ∞ X n =1 (cid:16) ˜ P n ⊗ ˜ Q n (cid:17) . We now show that the tensor product of two conditional expectation operators is againa conditional expectation operator.
Proposition 3.5
Let E i , i = 1 , , be Dedekind complete Riesz spaces with weak orderunits e i , i = 1 , , respectively. If T i is a conditional expectation operator on E i with e i = T i e i , i = 1 , , then T ⊗ T is a conditional expectation operator on E := E ⊗ E with ( T ⊗ T )( e ⊗ e ) = ( e ⊗ e ) .Proof: Let T := T ⊗ T . Since e and e are weak order units e := e ⊗ e is a weak orderunit of E ⊗ E , see [2] and [9]. Observe that T e = ( T ⊗ T ) ( e ⊗ e ) = T e ⊗ T e = e ⊗ e = e .Since T and T are positive, order continuous operators the map H ( f , f ) := ( T f , T f ),is positive and order continuous, hence the induced map T on E ⊗ E is positive andorder continuous, see [7] and [9]. As H = H the induced map T has T = T .It remains to show that R ( T ) is an order closed (under order limits) Riesz subspace ofthe Dedekind complete Riesz space E . To this end, suppose that ( h α ) is a net in R ( T )with order limit h in E . Since T is order continuous, the net ( T h α ) converges in orderto T h in E . However, as T is a projection and h α ∈ R ( T ) we have that T h α = h α → h in E . Thus, by uniqueness of order limits, h = T h ∈ R ( T ). Proposition 3.6
Let ( P n ) be a sequence of density zero band projections in a Dedekindcomplete Riesz space E with weak order unit e and ( Q n ) be a sequence of density zeroband projections in a Dedekind complete Riesz space F with weak order unit f , then ( P n ⊗ Q n ) is a sequence of density zero band projections in E ⊗ F . roof: By hypothesis, n P n − k =0 P k e →
0, in order, as n → ∞ and n P n − k =0 Q k f →
0, inorder, as n → ∞ . Note that h := e ⊗ f is a weak order unit in E ⊗ F . Let R n := P n ⊗ Q n for each n ∈ N , then R n is a band projection for each n and 0 ≤ R n = P n ⊗ Q n ≤ P n ⊗ I F ,so 1 n n − X k =0 R k h ≤ n n − X k =0 ( P k e ⊗ f ) = n n − X k =0 P k e ! ⊗ f → ⊗ f = 0 , in order, as n → ∞ . To access the required properties of the tensor product we work via bilinear maps.The following proposition follows directly from the properties of multiplication in the f -algebra E e . Proposition 4.1
Let E be a Dedekind complete Riesz space with weak order unit, e ,where e is also taken to be the algebraic unit of the f -algebra E e . Let J : E × E → E be defined by J ( f, g ) = f · g for all f, g ∈ E e , then J is bilinear, positive and ordercontinuous on E e . Remark 4.2
Following Fremlin’s paper on tensor products of Archimedean vector lat-tices, [7], let E be a Dedekind compete Riesz space with weak order unit e and let J beas in Proposition 4.1, then the induced map j : E e ⊗ E e → E e with additivity defined by j ( f ⊗ g + p ⊗ q ) = J ( f, g ) + J ( p, q ) = f g + pq is linear and order continuous. Let ( E i , T i , S i , e i ) , i = 1 , , be conditional expectation preserving systems. If their tensorproduct is ergodic, by (2.2) of Corollary 2.2 we have that1 n n − X k =0 T (( S k P e ) · Q e ) ⊗ T (( S k P e ) · Q e ) → ( T P e · T Q e ) ⊗ ( T P e · T Q e ) , (4.1)in order, as n → ∞ , for all P i and Q i band projections on E i , i = 1 , n n − X k =0 (cid:12)(cid:12)(cid:12) T (( S k P e ) · Q e ) ⊗ T (( S k P e ) · Q e )( T P e · T Q e ) ⊗ ( T P e · T Q e ) (cid:12)(cid:12)(cid:12) → , (4.2)in order, as n → ∞ , for all P i and Q i band projections on E i , i = 1 , emma 4.3 For conditional expectation preserving systems ( E, T, S, e ) and ( E , T , S , e ) ,with T and T strictly positive, if ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) is ergodic (resp. condi-tionally weak mixing) then ( E, T, S, e ) and ( E , T , S , e ) are ergodic (resp. conditionallyweak mixing).Proof: Let P and Q be band projections on E . We begin by taking P = P, Q = Q, P = I = Q in (4.1) to give, by Corollary 2.2,1 n n − X k =0 T (( S k P e ) · Qe ) ⊗ e → ( T P e · T Qe ) ⊗ e, and hence 1 n n − X k =0 T (( S k P e ) · Qe ) → T P e · T Qe, i.e. (
E, T, S, e ) is ergodic. To prove that ( E , T , S , e ) ergodic is similar.For the conditionally weak mixing case, taking the band projections as above and ap-plying this to (4.2), we have1 n n − X k =0 | T (( S k P e ) · Qe ) ⊗ e − ( T P e · T Qe ) ⊗ e | → , in order as n → ∞ . Thus1 n n − X k =0 | T (( S k P e ) · Qe ) − T P e · T Qe | → , in order as n → ∞ , and similarly for ( E , T , S , e ). Proof: (of Theorem 1.1) (5) ⇒ (1):Let P and Q be band projections on E . By Lemma 4.3 ( E, T, S, e ) is ergodic.Denote α k := T (( S k P e ) · Qe ) and α = T P e · T Qe , then (2.2) gives1 n n − X k =0 α k → α, (4.3)in order as n → ∞ . Now taking P = P = P and Q = Q = Q in (4.1) we have1 n n − X k =0 α k ⊗ α k → α ⊗ α, (4.4) n order as n → ∞ , by assuming (5). Furthermore, setting f k := α k − α ,1 n n − X k =0 f k ⊗ f k = 1 n n − X k =0 α k ⊗ α k − n n − X k =0 α k ! ⊗ α − α ⊗ n n − X k =0 α k ! + α ⊗ α → α ⊗ α − α ⊗ α − α ⊗ α + α ⊗ α = 0 , in order, as n → ∞ , where 0 ⊗ E ⊗ F .Define J as in Proposition 4.1 and j as in the remark immediately following Proposition4.1. We have 1 n n − X k =0 f k ⊗ f k →
0, in order, as n → ∞ , and so, by Remark 4.2,1 n n − X k =0 j ( f k ⊗ f k ) = j n n − X k =0 f k ⊗ f k ! → j (0 ⊗
0) = 0 , in order, as n → ∞ . However, j ( f k ⊗ f k ) = f k , so, by Lemma 2.4 (the Riesz spaceextension of the Koopman-von Neumann Lemma) there exists a density zero sequenceof band projections on E , say ( R n ), such that ( I − R n ) f k →
0, in order, as n → ∞ .Hence, ( I − R n ) | f k | →
0, in order, as n → ∞ .(1) ⇒ (4):We assume that ( E, T, S, e ) is conditionally weak mixing and ( E , T , S , e ) is ergodic.Let P and Q be band projections on E and P and Q be band projections on E . Now, K n := 1 n n − X j =0 ( T ⊗ T )(( S ⊗ S ) k ( P e ⊗ P e ) · ( Qe ⊗ Q e ))= 1 n n − X j =0 ( T ⊗ T )(( S k P e ⊗ S k P e ) · ( Qe ⊗ Q e ))= 1 n n − X j =0 ( T ⊗ T )(( S k P e · Qe ) ⊗ ( S k P e · Q e ))= 1 n n − X j =0 T ( S k P e · Qe ) ⊗ T ( S k P e · Q e )= 1 n n − X j =0 α k ⊗ β k , where α k := T (( S k P e ) · Qe ), α := T P e · T Qe , β k := T (( S k P e ) · Q e ) and β := P e · T Q e . Hence, | K n − α ⊗ β | ≤ α ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − X j =0 β k − β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + n n − X j =0 | α k − α | ⊗ β k → , in order, as n → ∞ , since n P n − j =0 β k − β → E , T , S , e ) is ergodic and n P n − j =0 | α k − α | → E, T, S, e ) is conditionally weak mixing. Thus K n → α ⊗ β in order as n → ∞ .Using (2.1a)-(2.1b), we have that L S ⊗ S ( P e ⊗ P e ) = lim n →∞ n P n − k =1 ( S ⊗ S ) k ( P e ⊗ P e ),where the order limit is taken,Recall the properties of a conditional expectation operator and averaging operators, see[12] for details on the averaging properties of conditional expectation operators, then( T ⊗ T )( L S ⊗ S ( P e ⊗ P e ) · ( Qe ⊗ Q e )) = ( T ⊗ T )( P e ⊗ P e ) · ( T ⊗ T )( Qe ⊗ Q e )and( T ⊗ T )( P e ⊗ P e ) · ( T ⊗ T )( Qe ⊗ Q e ) = ( T ⊗ T )(( T ⊗ T )( P e ⊗ P e ) · ( Qe ⊗ Q e )) . Combining the above gives( T ⊗ T )(( L S ⊗ S ( P e ⊗ P e ) − ( T ⊗ T )( P e ⊗ P e )) · ( Qe ⊗ Q e )) = 0 . If L S ⊗ S ( P e ⊗ P e ) − ( T ⊗ T )( P e ⊗ P e ) = 0 let Q ⊗ Q be a non-zero band projectioninto the bands generated by one of ( L S ⊗ S ( P e ⊗ P e ) − ( T ⊗ T )( P e ⊗ P e )) ± , whicheveris non-zero. The strict positivity of T ⊗ T yields the above impossible. Hence L S ⊗ S ( P e ⊗ P e ) = ( T ⊗ T )( P e ⊗ P e ) . However, L S ⊗ S and T ⊗ T are order continuous on E ⊗ E so we can now use Corollary3.4 to extend this equality to general band projections on E ⊗ E . Hence, ( E ⊗ E , S ⊗ S , T ⊗ T , e ⊗ e ) is ergodic by Corollary 2.2.(4) ⇒ (5):By Lemma 4.3 ( E, T, S, e ) is ergodic, so we can take ( E , T , S , e ) = ( E, T, S, e ).(3) ⇒ (5):Follows from Theorem 2.6.(1) ⇒ (2):If ( E, T, S, e ) and ( E , T , S , e ) are conditionally weak mixing then by (1) ⇒ (4) we have( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) ergodic for all ergodic ( E , T , S , e ) and if ( E , T , S , e )is conditionally weak mixing then, by (1) ⇒ (5), ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ S ) ergodic,so this implies that ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) ⊗ ( E , T , S , e ) = ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) ⊗ ( E , T , S , e ) is ergodic for all ergodic ( E , T , S , e ) and by (5) ⇒ (1), E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) is conditionally weak mixing. We refer the reader to [5]for the classical version of this case.(2) ⇒ (3):By Lemma 4.3, if ( E ⊗ E , T ⊗ T , S ⊗ S , e ⊗ e ) is conditionally weak mixing for eachconditionally weak mixing ( E , T , S , e ) with T strictly positive, then ( E, T, S, e ) isconditionally weak mixing, so, now taking ( E , T , S , e ) = ( E, T, S, e ), we have that( E ⊗ E, T ⊗ T, S ⊗ S, e ⊗ e ) is conditionally weak mixing. References [1]
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