CONDITIONAL INDEPENDENCE AND TENSOR PRODUCTS OF ...

J. Korean Math. Soc. 38 (2001), No. 1, pp. 125–134

CONDITIONAL INDEPENDENCE AND TENSOR PRODUCTS OF CERTAIN HILBERT L∞ -MODULES Thomas Hoover and Alan Lambert Abstract. For independent σ-algebras A and B on X, L2 (X, A ∨ B), L2 (X × X, A × B), and the Hilbert space tensor product L2 (X, A) ⊗ L2 (X, B) are isomorphic. In this note, we show that various Hilbert C ∗ algebra tensor products provide the analogous roles when independence is weakened to conditional independence.

1. Introduction Suppose that (X, F, µ) is a probability space, and A and B are independent sub sigma algebras of F. One can then show by standard rectangle approximation that L2 (X × X, A × B, µ × µ) and L2 (X, A ∨ B, µ) are unitarily equivalent via a measure algebra isomorphism induced unitary operator. Moreover, there is a natural equivalence of these spaces with the (unique) ⊗ Hilbert space tensor product L2 (X, A, µ) Hilbert L2 (X, B, µ). In this note, we consider analogous situations for the case that A and B are assumed only to be conditionally independent given A ∩ B. The role of L2 spaces is taken by certain Hilbert C ∗ -modules. As conditional independence may be easily described in terms of conditional expectations, and these conditional expectations are the basic building blocks of the Hilbert modules being investigated, this seems like the proper setting for this analysis. Moreover, there is a strong connection between the von Neumann algebra generated by the composition operator determined by a measurable transformation of X, and a pair of conditionally independent sigmal algebras. This relation is examined briefly at the end of this note. Received February 16, 2000. 2000 Mathematics Subject Classification: 46L06, 47C15, 47C33. Key words and phrases: L∞ tensor-products, composition operators.

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2. Notation and conventions • Let (X, F, µ) be a probability space. Upper case script letters A, B, C, etc. will generally be used to name sub sigma algebras of F. Then A ∨ B represents the smallest sigma algebra in F which contains both A and B. • For S a collection of sets in F, σ(S) is the smallest sigma algebra containing all the sets in S. • For a sigma algebra A ⊂ F, E A is the conditional expectation operator. We will only be concerned with E A acting on L2 (F), so that for each
f ∈ L2 (F), E A f is the unique function in L2 (A) satisfying A E A f dµ =
A is the orthogonal A f dµ for every A ∈ A. It is worth noting that E 2 A ∞ ∞ projection onto L (X, A, µ) and that E L (F) = L (A). • f ∗ means f¯, the complex conjugate of f . 3. Tensor products of Hilbert C ∗ modules Chapter 4 of E. C. Lance’s text [4] provides a detailed treatment of the general theory of Hilbert C ∗ module tensor products. We will be concerned here with certain specific examples. In general, when applied to vector spaces, ⊗ refers only to the algebraic tensor product (i.e., tensor product over C). Recall that if E and F are Hilbert Y modules, where Y is a C ∗ algebra, and φ is a *homomorphism from Y to L(F) (the space of adjointable maps on F) then the quotient of E ⊗ F by the linear span of {e ⊗ φ(y)(f ) − ye ⊗ f : e ∈ E, f ∈ F, y ∈ Y} is a Y valued inner product space, where the inner product is given on elementary tensors by e1 ⊗ f1 , e2 ⊗ f2  = f1 , φ(e1 , e2 E )(f2 )F . The completion of this space is the interior tensor product of E and F with respect to φ. This space is a Hilbert Y module. We will only be concerned with the case that φ is the left representation of Y as multiplication operators on F. With E, F, and Y as above, we may also form the exterior tensor product of E and F. This inner product is given on elementary tensors by e1 ⊗ f1 , e2 ⊗ f2  = e1 , e2 E ⊗ f1 , f2 F .

Conditional independence and tensor products of certain Hilbert L∞ -modules 127

The completion of this space is then a Hilbert Y ⊗Y module (in this context Y ⊗ Y is the spatial, or minimal C ∗ algebra tensor product.). The specific tensor products encountered in this note are: 

⊗ : The interior tensor product with respect to left multiplication.

⊗

L∞ (C)

of two Hilbert L∞ (C) modules, ⊗



⊗ : The interior tensor product L∞ (C×C) of two Hilbert L∞ (C × C) modules, with respect to left multiplication. 

⊗ : The exterior tensor product modules.

⊗

L∞ (C)⊗L∞ (C)

of two Hilbert L∞ (C)

4. Conditional independence Chapter 2 of M. M. Rao’s text [5] provides a detailed treatment of the material herein outlined. Given three sub sigma algebras J , K, and L of F, J and K are conditionally independent given L if ∀J ∈ J , E K∨L χJ = E L χJ . This may be restated in operator form as E K∨L E J = E L E J . In the special case of C = A ∩ B: A and B are conditionally independent given C if and only if E A E B = E C ; equivalently E A E B = E B E A . We will employ the notation “ A and B are ci | C ” for conditional independence. We will also make repeated use of the following: Lemma 0. If a is A measurable and b is B measurable, and A and B are ci | C, then E C (ab) = (E B a) · (E A b) = (E C a) · (E C b).

Proof. E C (ab) = E A E B (ab) = E A [(E B a)b]       = EA EB EAa b = EA EAEB a b         = EAEB a · EAb = EB EAa · EAb     = EB a · EAb . On the other hand, E B a = E B E A a = E C a, and similarly E A b = E C b.

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5. The Hilbert L∞ -modules associated with nested sigma algebras For C ⊂ A ⊂ F, let L(C, A) = {f : f is A measurable and E C |f |2 ∈ L∞ }; 1/2

f L = f L(C,A) = E C | f |2 ∞ . It was shown in [3] that (L (C, A) ,  L ) is a Banach space satisfying (A)⊂L (C,A)⊂L2 (F). Moreover, L (C, A) is a Hilbert L∞ (A)-module which is a realization of the localization via E A of L∞ (F). (see [4] for a discussion of Hilbert C* modules in general and localization in particular.) Let A and B be sub sigma algebras of F, and let C = A ∩ B. Assume A and B are ci|C . We note that in the case that C is the trivial algebra consisting of all F-sets of measure 0 or 1, then conditional independence given C reduces to the probabilistic concept of independence. Also, 2 Since A and B are conditionally independent given C, for a∈ L (A) and b∈ L2 (B),       (Lemma 0) E C |ab|2 = E C |a|2 · E C |b|2 L∞

so L (C, A) · L (C, B) ⊂ L (C, F) .

  Moreover, L (C, A) is a complemented Hilbert L∞ (C) submodule of C, F because the projection E A is an adjointable operator on L (C, F). Consider the interior tensor product ⊗ of L (C, A) and L (C, B) (with respect to the representation of L∞ (C) as left multiplication on L (C, B)) where A and B are ci|C: def.

a1 ⊗ b1 , a2 ⊗ b2  = b1 , a1 , a2 L · b2 L   = E C (b∗1 · a1 , a2 L · b2 ) = E C b∗1 · E C (a∗1 · a2 ) · b2 = E C (a∗1 · a2 ) · E C (b∗1 · b2 ) . As noted in Lemma 0, since A and B are ci|C , we have (1)

a1 ⊗ b1 , a2 ⊗ b2  = E C (a∗1 · a2 · b∗1 · b2 ) = E C ((a1 b1 )∗ · (a2 b2 )) .

In order to form the interior tensor product, we must consider the subspace N of the algebraic tensor product generated by all elements of the form

Conditional independence and tensor products of certain Hilbert L∞ -modules 129

(ca) ⊗ b − a ⊗ (cb) and take the completion of (L (C, A) ⊗ L (C, B))/N with  respect to the norm determined by Eq. 1. This space is denoted L (C, A) ⊗ L (C, B) . 

Theorem 1. L(C, A) ⊗ L(C, B) and L(C, A ∨ B) are isometrically isomorphic as Hilbert L∞ -modules. Proof. For finite sequences {ai } and {bi } in L (C, A) and L (C, B), respectively, we have (via Eq. 1)  ai · bi , aj · bj L(C,A∨B) Σai · bi , Σai · bi L(C,A∨B) = i,j

=



E C ((ai · bi )∗ · (aj · bj ))

i,j

=



ai ⊗ bi , aj ⊗ bj 

i,j

= Σai ⊗ bi , Σai ⊗ bi L(C,A)⊗ L(C,B) . This insures that the map S:Σai ⊗ bi → Σai · bi extends to an L∞ (C)module isometry from L (C, A) ⊗ L (C, B) onto L (C, A ∨ B). To see that S is in fact an adjointable map, note that for ai ∈ L (C, A) and bi ∈ L (C, B) , i = 1, 2 we have 

  = a1 · b1 , a2 · b2 L(C,A) a1 · b1 , S a2 ⊗ b2 L(C,A∨B)

= E C (a∗1 a2 · b∗1 b2 ) = E C (a∗1 a2 ) E C (b∗1 b2 ) = a1 ⊗ b1 , a2 ⊗ b2 )L(C,A∨B) ; so indeed, S is adjointable, and S* (= S-1 ) is given (on a fundamental set) by S*(a · b) = a⊗ b. It was shown in [2] that L(C, A) = L2 (X, A, µ) if and only if C is generated by a finite partition. Indeed, if C = σ (Ci : 1 ≤ i ≤n), where {C1 , · · · , Cn } is a measurable partition of X, then  1   |f |2 dµ χCi ; E C |f |2 = Σ µ(Ci ) Ci so  1 2 |f |2 dµ, f L = max 1≤i≤n µ(Ci ) Ci Thus the L2 (A) and L(C, A) norms are equivalent. If C is the trivial sigma algebra σ(X), then E C is the unconditional expectation and A and B are

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actually independent. In this case, of course the L2 and L norms are identical. The previous few lines may be formalized as follows. Corollary 2. Suppose that A and B are ci|C where C is finitely generated. Then L2 (X, A, µ)⊗ L2 (X, B, µ) = L2 (X, A ∨ B, µ). If C is ⊗ ⊗ trivial then L2 (X, A, µ) Hilbert L2 (X, B, µ) = L2 (X, A ∨ B, µ), where Hilbert denotes the (unique) Hilbert space tensor product. Remark. If in fact C = σ(C1 , C2 , · · · , Cn ), where the Ci ’s are pairwise disjoint then we may identify L∞ (C) with Cn via the isomorphism n (λ1 , · · · , λn ) → λi χCi . We can then view L (C, A) and L(C, B) as i=1

Hilbert Cn -modules. Suppose for the moment that A and B have intersection C but are not necessarily ci|C. It is easy to see that if a is A-measurable and b is B  -measurable and we define a (x, y) = a(x) and b (x, y) = b(y), and let  F = a · b : X × X → C, then for any sigma algebras P and Q contained in A and B, respectively, (E P×Q F )(x, y) = (E P a)(x) · (E Q b)(y). Lemma 3. A × C and C × B are ci|C × C with respect to the measure space (X × X, A × B, µ × µ). Proof. Let a, b, and F be as in the remark immediately preceding the statement of the lemma. Then E C×B F (x, y) = (E C a)(x) · (E B b)(y) = (E C a)(x) · b(y). We may then apply the same reasoning to deduce that        E A×C E C×B F (x, y) = E A E C a (x) · E C b (y)     (sinceE A E C = E C ) = E C a (x) · E C b (y)  C×C  F (x, y) . = E Since the set of such F’s generates all A × B measurable functions, we see that E A×C E C×B = E C×C . But (A × C)∩(C × B) =C × C, which guarantees that (A × C) and (C × B) are ci| C × C. Corollary 4. Let A ∩ B = C (no assumption of conditional independ ence). Then L(C × C, A × C) ⊗ L(C × C, C × B) and L (C × C, A ×B) are isometrically isomorphic as Hilbert L∞ (C × C)-modules.

Conditional independence and tensor products of certain Hilbert L∞ -modules 131

Proof. By Lemma 3 (A × C) and (C × B) are ci| C × C, so that Theorem 1 applies.

When the hypothesis for Corollary 4 is strengthened so that A and B are ci|C, then we have another representation of L (C × C, A × B), as presented later in Theorem 5. 

We now examine the exterior tensor product L (C, A) ⊗ L (C, B).  We continue to use the notational conventions f (x,y) = f (x), f (x, y) =   f (y), with the obvious interpretations of S and S for a set S of functions of one variable. As we are using the minimal C* algebra norm on L∞ (C) ⊗ L∞ (C), we identify this space as L∞ (C × C). Theorem 5. Suppose that A and B are sigma sub algebras of F with  intersection C. Then L(C, A)⊗ L(C, B) and L(C×C, A×B) are isometrically isomorphic as Hilbert L∞ (C)⊗ L∞ (C)-modules.



Proof. The inner product for L(C, A) ⊗ L(C, B) is given on elementary tensors by a1 ⊗ b1 , a2 ⊗ b2  = a1 , a2 L ⊗ b1 , b2 L = E C (a∗1 a2 ) ⊗ E C (b∗1 b2 ) , from which it follows that for finite sequences {ai } and {bi } , Σai ⊗ bi , Σai ⊗ bi  =



E C (a∗i aj ) ⊗ E C (b∗i bj ) .

i,j

Thus Σ ai ⊗ bi 2L(C,A)⊗ L(C,B)





C ∗ C ∗

= E (ai aj ) ⊗ E (bi bj )

;

i,j

this last norm being computed with respect to the minimal C* algebra tensor product completion of L∞ (C) ⊗ L∞ (C).

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Now, Σa i · b

i , Σa i · b

i L(C×C, C×B)   ∗  = E C×C a i · b

i a j · b

j i,j

=



 

∗

E C×C a ∗ i · aj · bi · bj

i,j

=



E C (a∗i · aj ) · E C (b∗i · bj )

i,j

(the last equality following from the remark preceding Lemma 3)  E C×C (a∗i · aj ) · E C×C (b∗i · bj )

. = i,j

This is precisely the image of i,j E C (a∗i aj ) ⊗ E C (b∗i bj ) in the representation of (the minimal C* algebra tensor product completion of) L∞ (C) ⊗ L∞ (C) as L∞ (C × C).

6. An application to composition operators Starting with the probability space (X, F, µ), suppose that T is a map−1 ∈ ping from X to X such that T −1 F ⊂ F and µ ◦ T −1 ! µ. If dµ◦T dµ ∞ L , then the composition operator C given by the formula Cf = f ◦ T is a bounded linear operator on L2 (X, F, µ). (In [6], R. K. Singh and J. S. Manhas give many of the basic properties of such operators). In [1], the von Neumann algebra, W, generated by C was studied. In particular, the following results were established: Proposition. Let W be the von Neumann algebra generated by C. Then i) ii) iii) iv) v)

there is a sigma algebra M in F such that [W1] = L2 (M) ;   for A = {F ∈ F : T −1 F = F }, [W 1] = L2 (A );   E M E A = E A E M ; and  E M∨A is a central projection of W. W is a factor if and only if M ∨ A = F and M ∩ A is the trivial sigma algebra.

Conditional independence and tensor products of certain Hilbert L∞ -modules 133

(The square brackets above indicate closed linear span in L2 (X, F, µ) ). Let C = M ∩ A . Then (iii) above shows that M and A are ci|C. Thus the tensor product results apply (where = is used for Hilbert C∗-module equivalence): i) ii) iii)

L(C, M) ⊗ L(C, A ) = L(C, M ∨ A ) L(C × C, M × C) ⊗

L(C × C, C × A ) = L(C × C, M × A ) L(C, M) ⊗

L(C, A ) = L(C × C, M × A ).

Now the composition operator C actually leaves L(C, M) and L(C, A ) invariant; indeed, it is the identity operator on the latter. Consider the tensor product “equation” (i) above, and let R be the Hilbert L∞ (C)-module isomorphism implementing the “equality.” Define S on L(C, M) ⊗ L(C, A ) by (extending the elementary tensor map) S(m ⊗ a ) = m ◦ T ⊗ a . Then it is clear that RS = CR, and similar statements may be made for (ii) and (iii). We now show that in the case that W is a factor, these three module identifications coalesce and provide a special representation of T. In this case, we have C being the trivial sigma algebra, so that L∞ (C), L∞ (C × C), and L∞ (C) ⊗ L∞ (C) “are” C. Then (i) reduces to the identi⊗ fication of L2 (M) Hilbert L2 (A ) with L2 (M ∨ A ); while (iii) identifies this same tensor product with L2 (M × A ). The identification of L2 (M × A ) and L2 (M ∨ A ) may be realized at the set level by the mapping χM ×A → χM ∩A , and verifying (via independence) that this extends in the proper way to a unitary equivalence. This unitary equivalence then establishes a model for all composition operators generating factors. We note that when W is a factor, C|L2 (M) is irreducible, so that part b of the following result is truly a converse to part a. Theorem. a) Suppose that W is a factor (as above). Define S on X ×X by S(x, y) = (T x, y). Then the composition operator Cs on L2 (X × X, M ×A , µ × µ) is unitarily equivalent to CT on L2 (X, F, µ). b) Suppose that To is a transformation such that its corresponding com position operator CTo on L2 (Xo , Fo , µo ) is irreducible so that WTo is the  ring of all operators on L2 (Xo , Fo , µo ) . Let (X1 , F1 , µ1 ) be a probability space. Define T on Xo × X1 by T (x, y) = (To x, y). Then WT is a factor.

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Proof. Part a follows directly from the comments preceding the statement under consideration. As for part b, it is easy to see that the corresponding MT and A T sigma subalgebras of Fo × F1 are given by MT = To × F1

and A T = Fo × T1 ,

where the Ti ’s are the appropriate trivial sub sigma algebras. It then follows that MT ∨ A T = Fo × F1 and MT ∩ A T = To × T1 , this last algebra being the trivial algebra in Fo × F1 . Thus WT is a factor.

References [1] C. Burnap, T. Hoover, and A. Lambert, Von Neumann algebras associated with composition operators, Acta Scientiarum Mathematicarum (Szeged) 62 (1996), 565–582. [2] A. Lambert, Lp multipliers and nested sigma algebras Operator Theory, Advances and Applications 104 (1998), 147–153. , A Hilbert C*-module view of some spaces related to probabilistic conditional [3] expectation, to appear in Quaestiones Mathematicae. [4] E. C. Lance, Hilbert C ∗ -Modules, Cambridge University Press, Cambridge, 1995. [5] M. M. Rao, Conditional Measures and Applications, Marcel Dekker, New York, 1993. [6] R. K. Singh and J. S. Manhas, Composition Operators on Function Spaces, NorthHolland, Amsterdam, 1993.

Thomas Hoover Department of Mathematics University of Hawaii Honolulu, Hawaii, USA E-mail: [email protected] Alan Lambert Department of Mathematics University of North Carolina at Charlotte Charlotte, North Carolina, USA E-mail: [email protected]