Locally Trivial W*-Bundles

We prove that a tracially continuous W$^*$-bundle $\mathcal{M}$ over a compact Hausdorff space $X$ with all fibres isomorphic to the hyperfinite II$_1$-factor $\mathcal{R}$ that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group $\mathrm{Aut}({\mathcal{R}})$ shown by Popa and Takesaki. There is no restriction on the covering dimension of $X$.


Introduction
Tracially continuous W * -bundles were introduced by Ozawa in [20,Section 5]. They are similar in spirit to other notions of bundle in functional analysis, such as continuous C(X)-algebras [5,15] and Hilbert-C(X)-modules [5,26]. However, the fibres of a W * -bundle are tracial von Neumann algebras and the topology is a mixture of the topology on the base space and the 2-norm topology in the fibres. For example, the trivial W * -bundle over a compact Hausdorff space X with fibre the tracial von Neumann algebra M is given by C σ (X, M ), i.e. the norm bounded, 2-norm continuous maps X → M .
It was shown in [20,Corollary 16] that a strictly separable W * -bundle with all fibres isomorphic to the hyperfinite II 1 factor R over a base space X with finite covering dimension is isomorphic to C σ (X, R). This automatic triviality is reminiscent of similar statements in the the context of Hilbert C(X)-modules [5] and continuous C(X)-algebras with strongly self-absorbing fibres [12,4,3]. In the discussion that follows the proof, Ozawa raises the possibility of trivialisation results when X is infinite dimensional (see also [2,Question 3.14]). This leads one to ask what a non-trivial W * -bundle over an infinite dimensional space X with fibres isomorphic to R could look like.
We show, in our main result (Theorem 4.10), that such a bundle would already have to be non-trivial locally. More precisely, we consider W * -bundles M that are locally trivial in the sense that every point x ∈ X has a closed neighbourhood Y ⊆ X such that the restriction M Y is isomorphic to C σ (Y, R) as a W * -bundle. We prove that this implies M ∼ = C σ (X, R) as W * -bundles.
Local triviality no longer implies triviality for W * -bundles with non-hyperfinite fibres. Indeed, we show in Section 5 that there are non-trivial, but still locally trivial W * -bundles already over such simple spaces as S 1 . The fibres are given by II 1 factors with prescribed outer automorphism group, which were constructed in [14,8]. These II 1 factors do not absorb R tensorially.
The study of W * -bundles is motivated by work on the structure and classification of simple, nuclear C * -algebras. In the light of the recent developments [27,7,10], the classification, by means of K-theoretic invariants, of simple, separable, nuclear, unital, infinite-dimensional C * -algebras of finite nuclear dimension that satisfy the UCT is now complete. To identify finite nuclear dimension is, therefore, now a priority.
The first-named author is supported by an EPSRC Doctoral Training Grant (grant reference numbers EP/K503058/1 and EP/J500434). The second-named author was supported by the SFB 878 while he was employed by the University of Münster.
Given a simple, separable, nuclear, unital, infinite-dimensional C * -algebra A whose trace simplex T (A) is a Bauer simplex, Ozawa showed that a certain tracial completion A u of A is a W * -bundle over the space of extreme traces ∂ e T (A) with fibres all isomorphic to R. When A has finite nuclear dimension, this bundle is trivial by combining results of [30] and [20]. In the reverse direction, the results of [20,17,16] (see also [28,22]) and [2] (which builds on [18,23]) show that triviality of the bundle A u combines with strict comparison, a mild condition on positive elements analogous to the order on projections in a II 1 factor being determined by their trace, to give finite nuclear dimension. This equivalence of regularity properties for C * -algebras forms part of the Toms-Winter conjecture; see [27,Section 6] for a full discussion. The proof of our main result is based on the observation that each W * -bundle M gives rise to a bundle (B, p) in the sense of [9, Chapter 2, Section 13.1] where each fibre has the additional structure of a tracial von Neumann algebra (see Definition 3.2). The W * -bundle M can be recovered by considering bounded, continuous sections of (B, p). If M is locally trivial with all fibres isomorphic to R in the above sense, then (B, p) is locally trivial in the sense of algebraic topology and therefore associated to a principal bundle P B with structure group Aut(R) equipped with the u-topology. It follows from the contractibility of Aut(R) [21] that P B has to be trivialisable, which translates into the triviality of M.
The paper is organized as follows: Section 2 contains the definitions of W *bundles and morphisms between them. We also recall the definition of fibres and generalise it to show that W * -bundles can be restricted to closed subsets of the base space giving restriction morphisms M → M Y . This allows us to define local triviality. In Section 3, we introduce the topological bundle (B, p) of tracial von Neumann algebras associated to a W * -bundle M over X. The topology on B is such that one can retrieve M as the C * -algebra of its bounded, continuous sections. This is analogous to the total space in the theory of C(X)-algebras (see for example [5]). In Section 4, we introduce the principal bundle P B and prove our main theorem. Finally, Section 5 concerns the construction of non-trivial, locally trivial bundles with non-hyperfinite fibres.
(C) The unit ball {a ∈ M : a ≤ 1} is complete with respect to the norm defined by a 2,u = E(a * a) 1/2 C(X) . Remark 2.2. This definition is a slight modification of Ozawa's original definition, which appears in [20,Section 5]. The difference is that we do not require the base space to be metrisable. Close examination of [20,Section 5] reveals that the metrisability of X is not necessary for much of the basic theory of W * -bundles. It is worth noting, however, that Owaza's condition of strictly separability for a W *bundle (see [20,Theorems 13 and 15]), which is equivalent to separability of M with respect to the · 2,u -norm, implies that C(X) is separable, so X is metrisable.
We shall abbreviate tracially continuous W * -bundle to W * -bundle or even bundle when there is no chance of confusion. We shall call X the base space of the bundle, M the section algebra of the bundle, and E the conditional expectation. We shall speak of the tracial axiom, the faithfulness axiom and the completeness axiom respectively. We shall often confuse the bundle itself with its section algebra, and speak of the bundle M.
It is instructive to consider the case where X is the one point space { * }. We identify C, C({ * }), and C1 M . Now the data for a W * -bundle over X reduces to a unital C * -algebra M and a state. The first two axioms then require this state to be a faithful trace. The effect of the third axiom is to ensure that the image of the unit ball of M under the GNS representation corresponding to this faithful trace is closed in the weak operator topology (see for example [24,Lemma A.3.3]). A W * -bundle over a one point space is, therefore, just a tracial von Neumann algebra: a (necessarily finite) von Neumann algebra together with a faithful, normal trace.
We now recall the definition of morphisms between W * -bundles and of trivial W * -bundles, which are implicit in [20,Section 5]. Definition 2.3. Let M i be a W * -bundle over X i with conditional expectation E i for i = 1, 2. A morphism is a unital * -homomorphism α : M 1 → M 2 such that α(C(X 1 )) ⊆ C(X 2 ) and the diagram commutes.
Definition 2.4. For a given compact Hausdorff space X and tracial von Neumann algebra (M, τ ), the trivial W * -bundle over X with fibre M is the C * -algebra of · -bounded, · 2,τ -continuous functions X → M , denoted C σ (X, M ), together with the following embedding and conditional expectation: • The embedding ι : . The axioms (T), (F) and (C) are satisfied.
In [20,Section 5], Ozawa defines the fibre of a W * -bundle M at x ∈ X to be the image of M under the GNS representation π x : M → H x corresponding to the trace a → E(a)(x). This trace induces a canonical faithful trace on the fibre of M at x. In the case of the trivial bundle C σ (X, M ), it is more natural to use the evaluation map eval x : C σ (X, M ) → M than the GNS representation to define the fibre at x. Since both * -homomorphisms have kernel {a ∈ C σ (X, M ) : E(a * a)(x) = 0}, the First Isomorphism Theorem gives us an isomorphism ϕ such that the diagram commutes. Hence, the two ways of defining fibres for a trivial bundle agree.
In this paper, we find it most convenient to view the fibre of a general W * -bundle at x ∈ X as the quotient M/I x , where I x = {a ∈ M : E(a * a)(x) = 0}. We denote the fibre of M at x by M x . We write τ x for the induced faithful trace on this quotient and we write a → a(x) for the canonical quotient map This justifies the notation.
We now fix a W * -bundle M and show how the norms on the bundle relate to the corresponding norms on the fibres.
is an isometric * -homomorphism. In particular, Proof. For each x ∈ X the map a → a(x) is a * -homomorphism. Hence, Φ is a * -homomorphism. Suppose a(x) = 0. Then a ∈ I x and so E(a * a)(x) = 0. Hence, a(x) = 0 for all x ∈ X implies that E(a * a) = 0 and, consequently, a = 0 by the faithfulness axiom. Therefore, Φ is an injective * -homomorphism and thus isometric.
Proposition 2.6. For fixed a ∈ M, the map x → a(x) 2,τx is continuous. Furthermore, we have Proof. The proposition follows from the observation that a(x) 2,τx = E(a * a)(x) 1/2 .
We can now prove the key result that the fibres of a W * -bundle are tracial von Neumann algebras. This result is due to Ozawa [20,Theorem 11], at least in the case where X is metrisable. The proof given here avoids the use of Pedersen's updown theorem in Ozawa's proof by showing completeness of the unit ball via the argument in [6, Proposition 10.1.12].
Theorem 2.7. For each x ∈ X, M x is a tracial von Neumann algebra.
Proof. Fix x ∈ X. We need to show that the closed unit ball {b ∈ M x : b ≤ 1} is complete with respect to the · 2,τx -norm. Let (b n ) ⊆ M x be a sequence that satisfies b n ≤ 1 for all n ∈ N and is a Cauchy sequence with respect to the · 2,τx -norm on M x . Since a Cauchy sequence will converge to the limit of any convergent sub-sequence, we may assume that b n+1 − b n 2,τx < 1 2 n without loss of generality.
We shall construct a sequence (a n ) ⊆ M inductively such that a n (x) = b n , (2.5) a n ≤ 1, (2.6) a n+1 − a n 2,u < 1 2 n (2.7) for all n ∈ N. Recall that with C * -algebras we may always lift elements from quotient algebras without increasing the norm. Let a 1 be any such lift of b 1 . Suppose now that a 1 , . . . , a n have been defined and have the desired properties. Let a ′ n+1 be any lift of b n+1 with a ′ n+1 ≤ 1. Since − a n (x) 2,τx < 1 2 n , we can, by continuity, find an open neighbourhood U of x such that (2.9) sup y∈U a ′ n+1 (y) − a n (y) 2,τy < 1 2 n .
The sequence (a n ) converges to some a ∈ M with a ≤ 1 because the unit ball of M is complete in the · 2,u -norm. Set b = a(x). Then (b n ) converges in · 2,τx -norm to b by (2.4).
We now turn to the definition of the restriction of a W * -bundle M over X to a closed subset Y . The result will be a W * -bundle M Y over Y together with a morphism of W * -bundles M → M Y . The procedure is closely modelled on the definition of the fibres as quotients. Indeed, when Y is a singleton {x}, the result is the fibre M x viewed as a W * -bundle over a one point space.  Proof. We first show that I Y is a norm-closed, two-sided ideal of M. Since I Y = x∈Y I x , it is enough to note that each I x is a norm-closed, two-sided ideal. This is standard: I x is the kernel ideal of the trace a → E(a)(x).

The induced embedding arises because
where the vertical maps are the central embedding and the horizontal maps are the quotient maps. Since Hence there is a unital completely positive map E Y such that the diagram where the horizontal maps are the quotient maps, commutes.
All that remains is to prove (C) for M Y . For this, we'll need to pass to fibres. Write a → a| Y for the canonical map M → M Y . Let y ∈ Y . Since I Y ⊆ I y , the fibre map a → a(y) factors through the restriction map a → a| Y , and we have a commuting diagram for the central embeddings and a commuting diagram for the conditional expectations Hence, we can identify the fibre of M at y with the fibre of M Y at y. We obtain the following analogues of (2.3) and (2.4): be a sequence that satisfies b n ≤ 1 for all n ∈ N and is a Cauchy sequence with respect to the · 2,u -norm on M Y . We need to find b ∈ M Y with b ≤ 1 such that (b n ) converges to b in the · 2,u -norm on M Y . Since a Cauchy sequence will converge to the limit of any convergent sub-sequence, we may assume that b n+1 − b n 2,u < 1 2 n without loss of generality.
In the same way as in the proof of Theorem 2.7, we now inductively construct a sequence (a n ) ⊆ M such that The role of the point x in the proof of Theorem 2.7 is taken over by the compact set Y . We use compactness of Y to obtain an open set U ⊇ Y such that (2.9) holds and Urysohn's Lemma to obtain the continuous function f : The sequence (a n ) converges to some a ∈ M with a ≤ 1 because the unit ball of M is complete in the · 2,u -norm. We set b = a| Y . The convergence of (b n ) to b follows by (2.16).
The morphism claim follows from the commuting diagrams (2.11) and (2.12).
We end this section with the definition of local triviality for a W * -bundle. We note in particular that the isomorphism class of the fibres for a locally trivial bundle is locally constant.

The Topological Bundle
In this section we shall show how to combine the fibres of a W * -bundle M to produce a bundle (B, p) in the sense of [9, Chapter 2, Section 13.1]. The W *bundle, more precisely its section algebra, can be recovered as the collection of bounded, continuous sections of (B, p). This builds on known results in the context of continuous fields of Hilbert spaces [5, Section 1.2] and Banach bundles [9, Chapter 2, Section 13.4]. We begin by recalling the general definition of a bundle from [9, Chapter 2, Section 13.1].
Definition 3.1. A bundle over a Hausdorff topological space X is a pair (B, p) where B is a Hausdorff topological space and p : B → X is a continuous, open surjection. The fibre at x ∈ X is the set p −1 (x).
By abuse of notation, we shall often speak of the bundle B instead of the bundle (B, p). We shall employ B x as an alternative notation for the fibre at x. We denote the set We now describe the additional structure necessary for a bundle to be a topological bundle of tracial von Neumann algebras. For this definition it is best to view tracial von Neumann algebras abstractly as C * -algebras with a tracial state such that the unit ball is complete with respect to the 2-norm (see [24,Lemma A.3.3]); an isomorphism of tracial von Neumann algebras is a trace preserving isomorphism of the C * -algebras. (iv) The map X → B which sends x to the to the additive identity 0 x of B x is continuous and so is the analogous map X → B which sends x to the to the multiplicative identity 1 x of B x . (v) The map τ : B → C obtained by combining the traces on each fibre is continuous and so is the map · 2 : B → C arising from combining the 2-norms from each fibre.
For the last two axioms the map · : B → [0, ∞] obtained by combining the C *norms from each fibre plays an auxillary role. We shall write B ≤r for the subspace We say that two topological bundles of tracial von Neumann algebras (B i , p i ) for i = 1, 2 are isomorphic if there are homeomorphisms ψ and ϕ such that the diagram is an isomorphism of tracial von Neumann algebras.
Remark 3.3. The axioms are modelled on the definition of a Banach bundle given in [9, Chapter 2, Section 13.4]. Note, however, that the fibres are not complete in the · 2 -norm. We only have · 2 -norm completeness of the · -norm closed unit ball.
The basic example of a topological bundle of tracial von Neumann algebras is (X × M, π 1 ) , where X is a Hausdorff space, M is a tracial von Neumann algebra, the topology on X ×M is the product of the topology of X and the 2-norm topology on M , and π 1 : X × M → X is the projection onto the first coordinate. This is the trivial bundle over X with fibre M . We can now define local triviality for topological bundles of tracial von Neumann algebras.
Definition 3.4. Let (B, p) be a topological bundle of tracial von Neumann algebras over the Hausdorff space X. We say (B, p) is locally trivial if every x ∈ X has an open neighbourhood U such that (p −1 (U ), p| p −1 (U) ) is isomorphic to a trivial bundle over U . Let M be a W * -bundle over the compact Hausdorff space X. Set B = x∈X M x and define p : B → X by p(b) = x whenever b ∈ M x . Note that, for each x ∈ X, the fibre p −1 (x) can be identified with M x and, therefore, endowed with operations, a norm and a trace that make it a tracial von Neumann algebra. In the following proposition, we define a topology on B so that (B, p) is a topological bundle of tracial von Neumann algebras. We then check that isomorphic W * -bundles give rise to isomorphic topological bundles.
Proposition 3.6. Let M be a W * -bundle over the compact Hausdorff space X. Set B = x∈X M x and define p : Moreover, if b ∈ B and a ∈ M is chosen with a(p(b)) = b, then the collection of V (a, ǫ, U ) as ǫ ranges over positive reals and U ranges over a neighbourhood basis of p(b) is a neighbourhood basis of b. (b) When B is endowed with the topology generated by B, (B, p) is a topological bundle of tracial von Neumann algebras.
for all This proves that B does form the basis for a topology on B, and also gives the required neighbourhood basis for b ∈ B.
(b) The topology defined by B is easily seen to be Hausdorff.
Hence p is continuous. It is clearly surjective. We now check the axioms of Definition 3.2 in turn, noting that a simple scaling argument shows that axioms (ii) and (viii) imply that the map  (v) We show the continuity of · 2 on B. Continuity of τ then follows by the polarisation identity together with the continuity of x → 1 x . Let b ∈ B and a ∈ M, x ∈ X be such that a(x) = b. Let ǫ > 0. By Proposition 2.6, the map y → a(y) 2 is continuous. Hence, there is an open set U ∋ x such that and b ′ 2 are · -norm bounded by K, and that  Proof. Assume α : M 1 → M 2 is an isomorphism of the W * -bundles. Then α restricts to an isomorphism C(X 1 ) → C(X 2 ), so induces a homeomorphism α t : X 2 → X 1 . Since E 2 (α(a))(x 2 ) = α(E 1 (a))(x 2 ) = E 1 (a)(α t (x 2 )), α induces an isomorphism between the fibres (M 1 ) α t (x2) and (M 2 ) x2 for each x 2 ∈ X 2 . Combining all these isomorphisms, we get a bijection ϕ : B 1 → B 2 such that (3.1) holds with ψ = (α t ) −1 . By considering the basic open neighbourhoods in B 1 and B 2 , we see that ϕ is a homeomorphism. Indeed, ϕ(V M1 (a, ǫ, U )) = V M2 (α(a), ǫ, ψ(U )) for all a ∈ M 1 , ǫ > 0, and U open in X 1 .
In the other direction, given a topological bundle of tracial von Neumann algebras over a compact Hausdorff space, we can define a W * -bundle. This comes from considering sections.  Let (B, p) be a topological bundle of tracial von Neumann algebras over the compact Hausdorff space X. The set of bounded sections of (B, p) endowed with fibrewise-defined operations and the uniform norm s = sup x∈X s(x) is a C *algebra isomorphic to the product x∈X p −1 (x). Since the fibres are tracial von Neumann algebras, the uniform 2-norm s 2,u = sup x∈X s(x) 2 is complete when restricted to the closed unit ball in uniform norm. Let M be the collection of bounded, continuous sections. Axioms (i)-(viii) ensure that M is a unital *subalgebra. The following proposition shows that continuity of sections is preserved under uniform-2-norm limits and, a fortiori, under uniform-norm limits. Therefore, M inherits the completeness properties of the algebra of bounded sections, in particular M is a C * -algebra. The additional data for a W * -bundle over X with section algebra M can now be easily defined and the axioms verified. We identify f ∈ C(X) with the scalar valued section x → f (x)1 x . Such scalar valued sections are clearly bounded and are continuous since scalar multiplication and the section x → 1 x are continuous. This gives an inclusion C(X) ⊆ Z(M). We define E : M → C(X) by s → τ • s. This is a conditional expectation from M onto the image of C(X) in M and induces the uniform 2-norm on M. Axiom (C) follows from Proposition 3.10. Axioms (T) and (F) follow fibrewise from the corresponding properties of a faithful trace.
As before, we check that our construction is compatible with our notions of isomorphism.
Proposition 3.11. Let (B i , p i ) be a topological bundle of tracial von Neumann algebras over the compact Hausdorff space X i for i = 1, 2. Let M i be the W *bundle over X i with conditional expectation E i that comes from (B i , p i ). If the topological bundles are isomorphic then the W * -bundles are isomorphic.
Proof. If the topological bundles are isomorphic and ϕ and ψ are as in (3.1) then s → ϕ • s • ψ −1 defines a bijection between the bounded, continuous section of p 1 : B 1 → X 1 and those of p 2 : B 2 → X 2 , that is a map α : M 1 → M 2 .
Since for each ) is an isomorphism of tracial von Neumann algebras, α is a * -homomorphism of C * -algebras. Furthermore, the following computations show that α is a morphism of W * -bundles.
Firstly, let f 1 ∈ C(X 1 ) ⊆ Z(M 1 ) and x 2 ∈ X 2 . Then . Secondly, let s ∈ M 1 and x 2 ∈ X. Then We now investigate the inverse nature of the two constructions considered in the section. One direction is essentially [20,Theorem 11]. The other direction reduces to the question of whether we can construct a bounded, continuous section through any point of the topological bundle.  Proof. (a) Let a ∈ M. By construction s a is a section of (B, p). We have a(x) Mx ≤ a M for all x ∈ X, so the section s a is bounded. Let W be open in B and x ∈ s −1 a (W ). Then s a (x) = a(x) ∈ W . By Proposition 3.6(a), there exists ǫ > 0 and an open neighbourhood U of x in X such that a(x) ∈ V (a, ǫ, U ) ⊆ W . It follows that x ∈ U ⊆ s −1 a (W ). Hence, s a is continuous. (b) Assume s : X → B is a continuous and bounded section. Let x 0 ∈ X and ǫ > 0. Choose a 0 ∈ M such that a 0 (x 0 ) = s(x 0 ). Since the function x → s(x) − a 0 (x) 2 is continuous, there is a neighbourhood U of x 0 such that By [20,Theorem 11], there exists a ∈ M such that a(x) = s(x) for all x ∈ X.
(c) The map a → s a is a unital homomorphism of C * -algebras. It is injective by Proposition 2.5 and surjective by (b). For f ∈ C(X) ⊆ Z(M), s f is the scalar section x → f (x)1 x (see the discussion preceding Proposition 2.5) and, for arbitrary a ∈ M and x ∈ X, τ (s a (x)) = τ x (a(x)) = E(a)(x). Therefore a → s a is an isomorphism of W * -bundles.  s(p(b))−b 2 , p(b)) is continuous. We complete the proof by showing that the set of all such V (s, ǫ, U ) contains a neighbourhood basis for each point of B. Axiom (ii) for topological bundles gives that V (0, ǫ, U ) as ǫ ranges over the positive reals and U ranges over a neighbourhood basis for x ∈ X form a neighbourhood basis for 0 x . Let b 0 ∈ B and s 0 be a bounded, continuous section with s 0 (p(b 0 )) = b 0 . Since the map G : B → B given by b → s 0 (p(b)) − b is a homeomorphism of B, we see that V (s 0 , ǫ, U ) as ǫ ranges over the positive reals and U ranges over a neighbourhood basis for p(b 0 ) form a neighbourhood basis for b 0 .
Remark 3.14. In all the topological bundles (B, p) that we consider in this paper, the assumption that there is a bounded, continuous section through every point of the bundle space B will be satisfied. Indeed it holds by construction for the topological bundles coming from W * -bundles. It is also clear when (B, p) is a locally trivial fibre bundle over a compact Hausdorff space with fibre a fixed tracial von Neumann algebra M , since such bundles look locally like the projection map X × M → X.
We observe that the topological bundle corresponding to a trivial W * -algebra C σ (X, M ), where M is a fixed tracial von Neumann algebra and X is a compact Hausdorff space, is (X × M, π 1 ), where the topology on X × M is the product of the topology of X and the 2-norm topology on M and π 1 : X × M → X is the projection onto the first coordinate. Thus, the notion of triviality for a topological bundle of tracial von Neumann algebras matches up with that for a W * -bundle. We show below that the notions of restriction to a closed subset also match up and, therefore, so do the natural notions of local triviality.   M commutes. In particular, α preserves the uniform 2-norm. The argument to show that α is surjective has two parts. First, using a partition of unity argument as in [6, Lemma 10.1.11], one shows that, for any continuous section s : Y → B with s(y) ≤ 1 for all y ∈ Y and any ǫ > 0, there is a bounded, continuous section s : X → B with s(x) ≤ 1 for all x ∈ X and s(y) − s(y) 2 < ǫ. This implies that the · -norm closed unit ball of M Y has · 2,u -dense image in the · -norm closed unit ball of M . The completeness of the · -norm closed units balls in · 2,u -norm then implies that α is surjective.

Locally Trivial Bundles
In this section, we prove our main result: a locally trivial W * -bundle with all fibres isomorphic to the hyperfinite II 1 factor R is trivial.
In fact, the only property of the II 1 factor R that we shall need is that its automorphism group is contractible. We begin, therefore, with a brief discussion of possible topologies on the automorphism group Aut(M ) of a tracial von Neumann algebra M , and note that in the factor case they coincide.  Proof. It is clear that the construction yields a bijection between the underlying sets; the only issue to check is continuity. By Lemma 4.2 the u-topology agrees with the pointwise 2-norm topology. Suppose first thatφ is continuous, i.e.φ(x n ) converges toφ(x) pointwise in 2-norm for every net (x n ) in U that converges to x ∈ U . Let (a m ) be a net in M converging to a ∈ M in 2-norm. We have where we used that an automorphism preserves the trace and is therefore isometric for the 2-norm. This proves that ϕ is continuous. Now suppose that ϕ is continuous, then we have that φ(x n )(a) −φ(x)(a) 2 = ϕ(x n , a) − ϕ(x, a) 2 converges to zero for all a ∈ M .
We will now construct the principal Aut(M )-bundle P B → X associated to a locally trivial topological bundle (B, p) over the Hausdorff space X. Since we do not assume that the reader is familiar with the notion of principal G-bundles for a topological group G, we highlight the main points below. A good reference for this material is [13, Chapter 4, Sections 2 and 3].
Definition 4.4. Let X be a topological space and let G be a topological group. A (right) G-space P together with a continuous G-map q : P → X (where G acts trivially on X) is called a principal G-bundle, if every point x ∈ X has a neighbourhood U ∋ x, such that there exists a G-equivariant homeomorphism φ U : Let (B, p) be a locally trivial topological bundle of tracial von Neumann algebras with fibre M . For any x ∈ X there is an open neighbourhood U ∋ x and homeomorphisms ϕ and ψ such that the diagram is the projection onto the first coordinate. By replacing ϕ with (ψ −1 × id M ) • ϕ, we get a commuting diagram of the form We call such a U a trivialising neighbourhood for (B, p).
Consider Aut(M ) as a topological group equipped with the u-topology. The principal Aut(M )-bundle P B is obtained by replacing the fibre M of B by the group Aut(M ) while preserving the transition maps. Write B x = p −1 (x) for the fibre at x and Iso(M 1 , M 2 ) for the set of isomorphisms between two von Neumann algebras. As a set we define Let V ⊆ X be another subset with U ∩ V = ∅ and such that there is a local trivialisation ϕ V : V × M → p −1 (V ). Note that are homeomorphisms, this definition is consistent. With this topology all maps ψ Ui : U i × Aut(M ) → q −1 (U i ) become homeomorphisms. It is straightforward to check that this topology does not depend on the choice of trivialising cover and that q : P B → X is a principal Aut(M )-bundle.
Each point x ∈ X has a neighbourhood U , such that the restriction P | U is isomorphic to the trivial bundle U × Aut(M ). It follows that there is a homeomorphism B| U → U × M compatible with the projection maps to U . Since the continuity conditions (i)-(viii) from Definition 3.2 can all be checked locally and are true for the trivial bundle, they hold for B as well. It follows that B is in fact a topological bundle of tracial von Neumann algebras. It is called the associated topological bundle.
We shall show that these two constructions are inverse to one another. We need the following well-known fact about principal bundles: Lemma 4.5. Let X be a topological space and let G be a topological group. Let q : P → X be a principal G-bundle. Suppose there exists a continuous section σ : X → P . Then P is isomorphic to the trivial principal G-bundle X × G.
Remark 4.6. In a similar fashion one can show that any continuous G-equivariant map ϕ : P → P ′ between principal bundles q : P → X and q ′ : P ′ → X such that q ′ • ϕ = q is in fact an isomorphism. Such a map is said to cover the identity on X.
Proposition 4.7. Let M be a II 1 factor and let X be a Hausdorff space. The associated bundle construction yields a bijection between isomorphism classes of locally trivial topological bundles of tracial von Neumann algebras with fibre M over X and isomorphism classes of principal Aut(M )-bundles over X.
Proof. Let (B, p) be a locally trivial topological bundle of tracial von Neumann algebras and denote by P B the corresponding principal Aut(M )-bundle. We need to check that the topological bundle associated to P B agrees with B. Consider the map (P B × M )/ ∼ → B given by [r, a] → r(a), where r ∈ Iso(M, B q(r) ) and a ∈ M . To see that this is a homeomorphism, it suffices to check that it is a bijective local homeomorphism. It is straightforward to see that it is bijective. Any choice of local trivialisation of B, over U ⊆ X say, induces a corresponding trivialisation of P B and we have ( where the inverse of the lower horizontal map is given by (x, a) → (x, [id M , a]).
Observe that both horizontal maps restrict to isomorphisms of tracial von Neumann algebras in each fibre. Hence they are isomorphisms of topological bundles in the sense of Definition 3.2. Let P be a principal Aut(M )-bundle. We have to check that the principal Aut(M )-bundle P B obtained from B = (P × M )/ ∼ agrees with P . By Remark 4.6 it suffices to construct a continuous Aut(M )-equivariant map P → P B covering the identity on X. This is defined by sending r ∈ P to the isomorphism Iso(M, B q(r) ) that maps a to [r, a] ∈ B. Continuity is again easy to check in local trivialisations.
Algebraic topology and sheaf theory provide tools for classifying principal Gbundles (see for example [13,Chapter 4,Section 12]). For our purpose, we need only the following theorem.
Theorem 4.8. Let X be a paracompact Hausdorff space and let G be a contractible topological group. Let q : P → X be a principal G-bundle. Then P is trivialisable.
Proof. The assumptions about P , X and G imply that q : P → X has a global section by [5,Lemma 4]. Now apply Lemma 4.5.  Theorem 4.10. A locally trivial W * -bundle with all fibres isomorphic to the hyperfinite II 1 factor R is trivial.

Non-trivial, locally trivial bundles
In this section we give examples of non-trivial, but still locally trivial W * -bundles over the circle S 1 . The construction is motivated by the following facts from the theory of principal bundles: For every topological group H there exists a topological space BH, such that isomorphism classes of principal H-bundles over a paracompact topological space X are in bijection with homotopy classes of maps X → BH (see [19,Theorem 3.1] and [13,Proposition 6]). In particular, Proposition 4.7 implies that isomorphism classes of locally trivial topological bundles of tracial von Neumann algebras with fibre a II 1 factor M over the circle S 1 are in bijection with [S 1 , BAut(M )]. The set [S 1 , BAut(M )] is in bijection with the set of conjugacy classes in the group a of path-components π 0 (Aut(M )). Since π 0 (Aut(M )) surjects onto π 0 (Out(M )), it therefore suffices to find factors, for which Out(M ) is not path-connected to obtain non-trivial examples. Factors M of type II 1 with Out(M ) isomorphic to a prescribed compact group have been constructed by Ioana, Peterson and Popa in [14] in the abelian case and by Vaes and Falguières in [8] for general compact groups.
We will use the construction from [8]: Fix a non-trivial finite group G. As sketched at the end of [8, Section 2], there exists a minimal action of G on R. By The group Out(M ) is defined as a quotient and could in principle be non-Hausdorff. However, since M is full, Out(M ) is Hausdorff. Therefore the continuous bijection G → Out(M ) induced by the action is a homeomorphism. Let θ : Aut(M ) → Out(M ) ∼ = G be induced by the quotient map and the above identification. Fix g ∈ G and let α ∈ Aut(M ) be an automorphism with θ(α) = g. This choice induces a group homomorphism Z → Aut(M ), which will also be denoted by α. Let that is, take the product R × M modulo the equivalence relation (t + n, m) ∼ (t, α(n)(m)) for all n ∈ Z. Together with the canonical quotient map B → S 1 , this is a topological bundle of tracial von Neumann algebras over S 1 in the sense of Definition 3.2 and is locally trivial in the sense of Definition 3.4. We can, therefore, via Theorem 3.13, define a locally trivial W * -bundle M which induces B.
Lemma 5.1. Let G be a finite group, let M be the II 1 factor with Out(M ) ∼ = G constructed above and let θ : Aut(M ) → Out(M ) ∼ = G be the quotient map. Let α ∈ Aut(M ) with g = θ(α) = e. Then the W * -bundle M associated to the topological bundle B given by (5.1) is non-trivial.
Proof. Let q : P → S 1 be the principal Aut(M )-bundle of B. Suppose for the sake of contradiction that M is trivialisable. By the results of Section 3, P is trivialisable. Consider Q = P × θ G defined as the quotient of the product P × G with respect to the equivalence relation (p · β, h) ∼ (p, θ(β) · h) for β ∈ Aut(M ). If P is trivialisable, so is Q, but Q → S 1 is a principal G-bundle over S 1 for the finite group G. By elementary covering space theory the isomorphism classes of these are in correspondence with the conjugacy classes of G.
More precisely, the conjugacy class associated to Q can be obtained as follows: Fix a basepoint q 0 ∈ Q and lift the quotient map [0, 1] → S 1 to a continuous path γ : [0, 1] → Q with γ(0) = q 0 . By the path lifting property such a lift exists and is unique. Since γ(0) and γ(1) lie in the same fibre, there is a unique h ∈ G, such that γ(0) = γ(1) · h. The conjugacy class of h ∈ G is independent of q 0 .
The bundle Q constructed above corresponds to the class of g ∈ G, whereas the trivial bundle corresponds to the conjugacy class of the neutral element e ∈ G, which only contains e, in contradiction with g = e. Therefore M can not be trivial.