The Fubini product and its applications

The Fubini product of operator spaces provide a powerful tool for analysing properties of tensor products. In this paper we review the the theory of Fubini products and apply it to the problem of computing invariant parts of dynamical systems. In particular, we study the invariant translation approximation property of discrete groups.


Introduction
The Fubini product of C * -algebras was first defined and studied by J. Tomiyama [19,20,21,22] and Wassermann [23]. It is now a standard tool in the study of operator algebras and operator spaces. See for instance [10,1,13,11,15,14,3]. Unfortunately, to our knowledge, a comprehensive treatment of the subject is missing from the literature.
In this paper, we try to the remedy the situation. In Section 2, we give a survey of the theory of Fubini products and its applications. Most of the results in Section 2 appear scattered in numerous articles, most notably [20,23,9,10,15,6,12]. After briefly recalling some definitions concerning operator spaces in Subsection 2.1, we define the Fubini product and prove its fundamental properties regarding functoriality, intersections, kernels, relative commutants, invariant elements, combinations in Subsections 2.2-2.8. In Subsection 2.9, we review the relation between the operator approximation property and the slice map property.
In Section 3, we apply the results in Section 2 to the study of groups with the invariant translation approximation property (ITAP) of J. Roe [17,Section 11.5.3]. In Subsection 3.1, we recall the definition of the uniform Roe algebra. In Subsection 3.2, we analyse the uniform Roe algebra of a product space. Finally, in Subsection 3.3, we study the ITAP of product groups. We show that for countable discrete groups G and H, if G has the approximation property (AP) of Haagerup-Kraus [6], the product G × H has the ITAP if and only if H has the ITAP.
Finally, in Section 4, we study the crossed product version of the Fubini product.

The Fubini Product
2.1. Operator Spaces. For the sake of completeness, we start with some definitions concerning operator spaces. See [4,16] for a complete treatment.
Lemma 2.12. Let K, L be subspaces of A * , B * respectively such that the closed unit ball of K and L are weak- * -dense in the unit balls of A * and B * respectively. Then for any S ⊆ A, T ⊆ B operator spaces the Fubini product F (S, T, A ⊗ B) equals the set The assumptions are for instance satisfied if K and L are the set of normal linear functionals in faithful representations of A and B.
Proof. Suppose that (φ ⊗ id B )(x) ∈ T for all φ ∈ L. We need to show that (φ ⊗ id B )(x) ∈ T for all φ ∈ A * . Let φ ∈ A * and let (φ n ) ⊆ L be a bounded sequence (or net) converging pointwise (i.e. in the weak- * -sense) to φ. Then (φ n ⊗ id B )(z) → (φ ⊗ id B )(z) for all z ∈ A ⊙ B and using norm boundedness of φ n ⊗ id B , an ε/3-argument shows that the same holds for z ∈ A ⊗ B. Since Definition 2.13. We say that (S, T, A ⊗ B) has the slice map property if Proof. For x ∈ F (S, T, A ⊗ B) and φ ∈ A * and ψ ∈ B * , we have Corollary 2.16 ((cf. [10, Lemma 2])). For i = 1, 2, let S ⊆ A i and T ⊆ B i be operator spaces. Suppose that completely bounded maps σ 1 : A 1 → A 2 , σ 2 : A 2 → A 1 and τ 1 : B 1 → B 2 , τ 2 : B 2 → B 1 satisfy, for i = 1, 2, σ i (s) = s for s ∈ S and τ i (t) = t for t ∈ T , then σ 1 ⊗ τ 1 restricts to an isomorphism More generally, for families of operator spaces {S α ⊆ A} and {T β ⊆ B}, we have Proof. Clear from the definitions.
has the slice map property, then Proof. We have .
Proof. Follows from Lemma 2.18 and Proposition 2.20.
2.6. Relative Commutants. As a corollary we obtain the following.
holds if and only if the triple (S ′ ∩ A, T ′ ∩ B, A ⊗ B) has the slice map property.
Proof. Follows from Proposition 2.23, since under the unitality conditions, we have Then σ g is completely bounded and Moreover, we have Now Theorem 2.22 completes the proof, since Proof. Clear since any element of T * extends to an element of B * .
has the slice map property, then by Lemma 2.28 and Lemma 2.18. Proof. Follows from Lemma 2.29 and the following commutative diagram of inclusions: The Operator Approximation Property. In this subsection, we review the connection between the operator approximation property and the slice map property. This is partly for completeness and partly because we use it in Section 3. However, we have nothing new to add here to the excellent work of Kraus [15] and Haagerup-Kraus [6]. Let A and B be operator spaces and let x ∈ A ⊗ B. Define Then Definition 2.36. We say that A has the operator approximation property It follows, for any   Proof. See the original article [6] or [3,Section 12.4].

The Invariant Translation Approximation Property
In this section, we apply the results in Section 2 to the problem of studying the invariant part of the uniform Roe algebra.

Uniform Roe Algebras.
Definition 3.1. We say that a (countable discrete) metric space is of bounded geometry if for any R > 0, there is N R < ∞ such that all balls of radius at most R have at most N R elements.  Proof. The action is given by It follows that Thus we identify A(X, S) ⊆ B(l 2 X ⊗ H). Proof. Only the last inclusion needs checking. Take a ∈ A R,M (X, S). For each positive integer n ≥ 1, we define a n ∈ A(X, S) as follows. For x 1 , x 2 ∈ X, if d(x 1 , x 2 ) > R, then let a n (x 1 , x 2 ) = 0, and if d(x 1 , x 2 ) ≤ R, then choose a n (x 1 , x 2 ) ∈ S to satisfy a n (x 1 , x 2 ) − a(x 1 , x 2 ) ≤ 1/n. Then a n ∈ A R,M+1 (X, S) and a n −a ≤ N R /n by Lemma 3.3. Hence the sequence a n converges to a in B(l 2 X ⊗ H) as n → ∞ and thus a ∈ A(X, S).
If X and Y are of bounded geometry, then so is X × Y .

Lemma 3.7. Let X and Y be metric spaces of bounded geometry and let S ⊆ B(H) be a subset.
Then for R, R ′ > 0 and M , M ′ > 0 we have a natural inclusion where R ′′ := max{R, R ′ } and M ′′ := M · M ′ . In particular, we have Proof. Take a ∈ A R,M (X) and b ∈ A R ′ ,M ′ (Y, S) and let R ′′ = max{R, R ′ } and M ′′ = M · M ′ . For x 1 , x 2 ∈ X and y 1 , y 2 ∈ Y , we define (1) For x 1 , x 2 ∈ X and y 1 , Hence a ⊗ b belongs to A R ′′ ,M ′′ (X × Y, S). The last statement is clear.

Lemma 3.8. Let X and Y be metric spaces of bounded geometry and let S ⊆ B(H) be a subset. Then for R > 0 and M > 0, we have a natural inclusion
Proof. Take a ∈ A R,M (X×Y, S). Fix x 1 , x 2 ∈ X and consider b x1,x2 (y 1 , y 2 ) := a((x 1 , y 1 ), (x 2 , y 2 )).
First we show that b x1,x2 is an element of A R,M (Y, S).
(1) For x 1 , x 2 ∈ X with d(x 1 , x 2 ) > R and for any y 1 , y 2 ∈ Y , we have d(( Theorem 3.9. Let X and Y be metric spaces of bounded geometry and let S ⊆ B(H) be an operator space. Then we have a natural inclusion Proof. Follows from Lemmas 3.7 and 3.8 3.3. The Invariant Translation Approximation Property. Let G be a countable discrete group. Let |G| denote the metric space of bounded geometry on G associated to a proper length function. Then G acts on |G| isometrically by right translations.
Example 3.10. Let G be a countable discrete group. Then A(G) is the * -algebra generated by C[G] and l ∞ (G), and therefore C * where L(G) denote the von Neumann algebra generated by C * λ (G). See [3, Section 5.1]. Definition 3.11. Let S be an operator space. We say G has the invariant translation approximation property (ITAP) for S, if we have We say G has the (strong) invariant translation approximation property if it has the ITAP for (all operator spaces S ⊆ B(H)) C.
The following theorem connects the strong ITAP to the AP.

Theorem 3.13 (([24])). Let G be a countable discrete group. Then G has the AP if and only if G is exact and has the strong ITAP.
Proof. See [24]. u |G| → C * u |H| and E λ : C * λ (G) → C * λ (H). The restrictions of E L and E u to C * λ (G) are equal to E λ . Now let x ∈ L(H) ∩ C * u |H| then x ∈ L(G) ∩ C * u |G| = C * λ (G) and E L (x) = x as well as E u (x) = x. It follows that E λ (x) = x i.e. x ∈ C * λ (H) and thus H also has the ITAP. Now we consider products.
and finally Moreover, it is clear that l is proper. Finally, we have Proposition 3.16. Let G and H be countable discrete groups. If G × H has the ITAP, then G and H also have the ITAP and the triple Proof. By Theorem 3.9, we have H). Thus G and H have the ITAP by Lemma 2.5 (this is also immediate from Theorem 3.14). The last statement follows from Corollary 2.27. Theorem 3.17. Let G be a countable discrete group with the AP and let B be an operator space equipped with an action of a group H by completely bounded maps. Then we have Proof. Clearly, we have Since G has AP, it has the strong ITAP by Theorem 3.13, thus Moreover, the C * -algebra C * λ (G) has the strong OAP by Theorem 2.42, thus by Proposition 2.26, we have It follows that This completes the proof.
Theorem 3.18. Let G and H be countable discrete groups. If G has the AP and H has the ITAP, then G × H has the ITAP.
Proof. By Theorem 3.9, we have . Thus, by Theorem 3.17, we have

The Fubini Crossed Product
In this section, we study the crossed product version of the Fubini product.
4.1. The Fubini Crossed Product. Let G be a countable discrete group. A G-operator space is an operator space equipped with a completely isometric action of G.
Let A ⊆ B(H) be an operator space equipped with a completely isometric action α of G. Define a new action π of A on H ⊗ l 2 G by π(a)(v ⊗ δ g ) := α g −1 (a)v ⊗ δ g and let G act on H ⊗ l 2 G by The reduced crossed product A ⋊ r G is defined as the operator space spanned by {π(a)λ(g) ∈ B(H ⊗ l 2 (G)) | a ∈ A, g ∈ G}.
The slice map id A ⋊ r ψ is given by the restriction of the von Neumann slice map which maps A * * ⊗B(l 2 G) to A * * .
In fact, many of the formal properties of the Fubini product hold for Fubini crossed products, usually with the same proof.
Proof. Since the action of G on A is trivial, we have . By Goldstine's theorem for any functional ψ ∈ C * λ (G) * , there exists a bounded net ψ n ∈ B(l 2 (G)) * with ψ n ≤ ψ , converging to ψ in the weak * topology. Now it is easy to see that for any The proof follows from Proposition 4.7 and Lemma 4.6, since Example 4.9. Let l ∞ (G) act on l 2 G by multiplication. Then G acts on l ∞ (G) by left multiplication. As already pointed out C * u |G| ∼ = l ∞ (G) ⋊ r G; moreover, thinking of C as embedded into l ∞ (G) via the constant functions we have C * u (|G|) G = F (C, l ∞ (G) ⋊ r G). Thus G has the ITAP if and only if (C, l ∞ (G) ⋊ r G) has the slice map property. Proposition 4.10. Let G be a discrete group with the AP. Then for any S ⊆ A, we have F (S, A ⋊ r G) = S ⋊ r G.
Proof. Since G has the AP there exists a net (u α ) in M 0 A(G) ∩ c c (G) converging to 1 ∈ M 0 A(G) in the σ(M 0 A(G), Q(G)) topology. As explained in [24], this implies that the net of Schur multipliersM uα ∈ CB(C * u (|G|, A)) given byM uα ([a s,t ]) = [u α (st −1 )a s,t ] converges to the identity map in the point norm topology. Fixing a faithful representation A ֒→ B(K) we can think of A ⋊ r G ⊆ B(K ⊗ ℓ 2 (G)) as matrices indexed by G with entries in A. Since the δ s , · δ t is a normal functional on B(ℓ 2 (G)) whose slice map gives the s, t entry of the matrix in B(K ⊗ ℓ 2 (G)) we can characterise F (S, A ⋊ r G) as those matrices in A ⋊ r G with entries in S. Thus it is clear that F (S, A ⋊ r G) ⊆ C * u (|G|, A). Moreover, since each u α has finite support, it is easy to check thatM uα (F (S, A ⋊ r G)) is contained in span{π(s)λ(g) | s ∈ S, g ∈ G} ⊆ S ⋊ r G. It follows that S ⋊ r G ∋M α (x) → x for any x ∈ F (S, A ⋊ G), concluding the proof.  Proof. Let x ∈ F (S, A ⋊ r G). For any ψ ∈ B(l 2 G) * , we have (id A ⋊ r ψ)(x) ∈ S, thus (id A ⋊ r ψ)[(σ ⋊ r G)(x)] = σ[(id A ⋊ r ψ)(x)] belongs to T . Hence the lemma holds.
Lemma 4.13. Let S ⊆ A be G-operator spaces. Let σ : A → A be a completely bounded G-map. If σ restricts to the identity on S, then σ ⋊ r G restricts to the identity on F (S, A ⋊ r G).
Proof. For x ∈ F (S, A ⋊ r G) and ψ ∈ B(l 2 G) * , we have Thus (σ ⋊ r G)(x) = x. Corollary 4.14 ((cf. [10, Lemma 2])). Let S ⊆ A and S ⊆ B be G-operator spaces. If φ : A → B and ψ : B → A are completely bounded G-maps such that (ψ • φ) |S = id S and (φ • ψ) |S = id S , then there is a completely bounded G-isomorphism F (S, A ⋊ r G) → F (S, B ⋊ r G) which is the identity on S ⋊ r G.
Injective envelopes of C * -algebras, operator systems and operator spaces have been considered by various authors [7,8,2,5]. The G-injective envelope has only been defined for operator systems in the literature. However, the definitions and constructions are all analogous. First one has to find an injective extension of the given object in the appropriate category and then minimise it in such a way that uniqueness is automatic. Therefore in the following, we omit the proofs. For a G-operator space S, we denote the G-injective envelope by I G (S). We write I(S) if G is trivial.  We say that S has the universal slice map property for G if S ⋊ r G = F (S, G).  Proof. We only sketch the proof.
Since l ∞ (G) is G-injective, we see that F (C, G) ∼ = F (C, l ∞ (G) ⋊ r G) ∼ = (C * u |G|) G by Lemma 4.15 and Example 4.9. On the other hand, for any S we have S ⋊ r G ⊆ F (S, G) ⊆ I G (S) ⋊ r G ⊆ I(S ⋊ r G).