On the isomorphism problem for non-minimal transformations with discrete spectrum

The article addresses the isomorphism problem for non-minimal topological dynamical systems with discrete spectrum, giving a solution under appropriate topological constraints. Moreover, it is shown that trivial systems, group rotations and their products, up to factors, make up all systems with discrete spectrum. These results are then translated into corresponding results for non-ergodic measure-preserving systems with discrete spectrum.

1. Introduction. The isomorphism problem is one of the most important problems in the theory of dynamical systems, first formulated by von Neumann in [17, pp. 592-593], his seminal work on the Koopman operator method and on dynamical systems with "pure point spectrum" (or "discrete spectrum"). Von Neumann, in particular, asked whether unitary equivalence of the associated Koopman operators ("spectral isomorphy") implied the existence of a point isomorphism between two systems ("point isomorphy"). In [17,Satz IV.5], he showed that two ergodic measure-preserving dynamical systems with discrete spectrum on standard probability spaces are point isomorphic if and only if the point spectra of their Koopman operators coincide, which in turn is equivalent to their spectral equivalence. These first results on the isomorphism problem considerably shaped the ensuing development of ergodic theory. The next step in this direction was the Halmos-von Neumann article [10] in which the authors gave a more complete solution to the isomorphism problem by addressing three different aspects: • Uniqueness: For which class of dynamical systems is a given isomorphism invariant Γ complete, meaning that two systems of the class (X, φ) and (Y, ψ) are isomorphic if and only if Γ(X, φ) = Γ(Y, ψ)? • Representation: What are canonical representatives of isomorphy classes of dynamical systems? • Realization: Given an isomorphism invariant Γ, what is the class of objects that occur as Γ(X, φ) for some dynamical system (X, φ)? In addition to the uniqueness theorem from [17] for the isomorphism invariant given by the point spectrum, the Halmos-von Neumann representation theorem showed that for each isomorphy class of ergodic dynamical systems with discrete spectrum L p (X), 1 ≤ p ≤ ∞, via T φ f := f • φ for f ∈ L p (X). With this definition, T φ is a Markov embedding, i.e., T φ |f | = |T φ f | for all f ∈ L p (X), T φ 1 = 1, and T φ 1 = 1 where T φ denotes the adjoint operator of T φ . Two measure-preserving dynamical systems (X, φ) and (Y, ψ) are point isomorphic if there exists an essentially invertible, measurable, measure-preserving map θ : X → Y such that θ • φ = ψ • θ. They are Markov isomorphic if there is an invertible Markov embedding S : L 1 (Y) → L 1 (X) such that T φ S = ST ψ . If S is merely a Markov embedding, we call (Y, ψ) a Markov factor of (X, φ). If X and Y are standard probability spaces, these notions of isomorphy and factors coincide by von Neumann's theorem [5,Theorem 7.20].
If G is a compact topological group and a ∈ G we define φ a : G → G, φ a (g) := ag and call the dynamical system (G, φ a ) the group rotation with a. We may also abbreviate (G, φ a ) by writing (G, a). Since the Haar measure m on G is invariant under rotation, the rotation can also be considered as a measure-preserving dynamical system (G, m; a).
Remark 2.1. For a measure-preserving dynamical system, we usually consider the associated Koopman operator on the L 1 -space instead of the L 2 -space, following the philosophy advocated in [5]: When only using the Banach lattice structure of the L p -spaces, it is natural to work on the biggest of them, the L 1 -space, unless Hilbert space methods are also explicitly needed. Standard interpolation arguments show that this choice is justified. And indeed, this article will not require any Hilbert space structure.
If T is a linear operator on a vector space E, we denote by T k its nth Cesàro mean and drop T from the notation if there is no room for ambiguity. Furthermore, we call fix(T ) := {x ∈ E | T x = x} the fixed space of T . If F ⊂ E is a T -invariant subspace, we set fix F (T ) := fix(T | F ). If (K, φ) is a topological dynamical system, the fixed space fix(T φ ) of its Koopman operator is a C * -subalgebra of C(K). Similarly, if (X, φ) is a measure-preserving dynamical system, fix L ∞ (X) (T φ ) is a C * -subalgebra of L ∞ (X). By the Gelfand representation theorem (cf. [16, Theorem I. 4.4]) there is a compact space L such that fix L ∞ (X) (T φ ) ∼ = C(L). The space L is necessarily extremally disconnected: Since fix L 1 (X) (T φ ) is a closed sublattice of L 1 (X), the representation theorem for AL-spaces (see [15,Theorem II.8.5]) shows that there is a compact space M and a Borel probability measure µ M on M such that But by [15,Theorem II.9.3], C(L) is isomorphic to a dual Banach lattice if and only if L is hyperstonean. In particular, L is extremally disconnected. This will be crucial for Theorem 4.10.
2.1. Operators with discrete spectrum. We start with a power-bounded operator T on a Banach space E, i.e., an operator satisfying sup n∈N T n < ∞, and briefly recall the definition of discrete spectrum and the Jacobs semigroup generated by T . This semigroup was first considered by Jacobs in [11, Definition III.1].
Definition 2.2. Let E be a Banach space and T ∈ L (E) a power-bounded operator on E.
(i) The operator T has discrete spectrum if its Kronecker space given by the closed linear span where the closure is taken with respect to the weak operator topology and the semigroup operation is the composition of operators.
The following characterization of an operator having discrete spectrum can be found in [5,Theorem 16.36].
Theorem 2.3. The following assertions are equivalent.
(i) T has discrete spectrum.
(ii) J(T ) is a weakly/strongly compact group of invertible operators.
Remark 2.4. If T has discrete spectrum, it is mean ergodic and J(T ) is a compact abelian group on which the weak and strong operator topology coincide. It is metrizable if E is.
2.2. Systems with discrete spectrum. Next, we consider Koopman operators corresponding to dynamical systems. See [5,Chapters 4,7] for general information.
Definition 2.5. We say that a measure-preserving dynamical system (X, φ) has discrete spectrum if its Koopman operator T φ has discrete spectrum on L 1 (X). Similarly, we say that a topological dynamical system (K, φ) has discrete spectrum if T φ has discrete spectrum as an operator on C(K).
Example 2.6. If B is a compact space, the trivial dynamical system (B, id B ) has discrete spectrum. Also, if G is a compact group and a ∈ G, the measure-preserving dynamical system (G, m; a) has discrete spectrum and so does the topological dynamical system (G, a). As we will see in Corollary 4.7 and Corollary 4.11, trivial systems and group rotations are, up to factors, the building blocks of all transformations with discrete spectrum.
If (K, φ) is a topological dynamical system and T φ ∈ L (C(K)) has discrete spectrum, the Jacobs semigroup J(T φ ) is related to the Ellis semigroup E(K, φ) ⊂ K K defined as E(K, φ) := {φ n | n ∈ N}, see [5,Section 19.3]. The following wellknown result establishes this connection and gives a topological characterization of the operator theoretic notion of discrete spectrum. Proposition 2.7. Let (K, φ) be a topological dynamical system. For the Koopman operator T φ , the following assertions are equivalent.
(i) T φ has discrete spectrum.
(iv) (K, φ) is equicontinuous and invertible. Moreover, if these conditions are fulfilled, the map is an isomorphism of compact topological groups.
The equivalence of (i) and (ii) follows from Theorem 2.3 and [5,Theorem 4.13]. The equivalence of (ii) and (iii) follows via the canonical isomorphism θ → T θ , and for the equivalence of (iii) and (iv) see [7,Proposition 2.5].
3. Bundles of dynamical systems. Bundles, e.g. in differential geometry or algebraic topology, allow to decompose an object into smaller objects such that the small parts fit together in a structured way. This perspective is important when dealing with dynamical systems which are not "irreducible", i.e., not minimal or ergodic, We therefore start by studying bundles of topological dynamical systems.
Remark 3.2. For a dynamical system (K, φ) and a compact space B, a tuple (K, B, p; φ) is a bundle of dynamical systems if and only if p is a factor map from (K, φ) to (B, id B ). (α(t)z, t) be the associated rotation on the cylinder K. Then (K, B, p B ; φ α ) is a compact bundle of topological dynamical systems. If α ≡ a for some a ∈ T, the system (K, φ) is just the product of the torus rotation (T, φ a ) and the trivial system (B, id B ). Definition 3.4. A bundle morphism of bundles (K 1 , B 1 , p 1 ) and (K 2 , B 2 , p 2 ) is a pair (Θ, θ) consisting of continuous functions Θ : K 1 → K 2 and θ : B 1 → B 2 such that the following diagram commutes: A morphism of compact bundles of topological dynamical systems (K 1 , B 1 , p 1 ; φ 1 ) and (K 2 , B 2 , p 2 ; φ 2 ) is a morphism (Θ, θ) of the corresponding bundles such that Θ is, in addition, a morphism of topological dynamical systems. If Θ and θ are homeomorphisms, we call (Θ, θ) an isomorphism.

3.1.
Sections. An important tool for capturing structure in bundles relative to the base space are sections. We recall the following definition.
Although the existence of sections is guaranteed by the axiom of choice, there may not exist continuous sections in general.    In case there is no continuous section, the so-called pullback construction allows to construct bigger bundles admitting sections. We repeat this standard construction from category theory in our setting since it will play an important role. and denote the restriction of the canonical projection p M : M × K → M to r * K by π M and the restriction of id M × φ to r * K by r * φ. Then (r * K, M , π M ; r * φ) is a bundle of topological dynamical systems called the pullback bundle of (K, B, p; φ) under r and (K, φ) is a factor of (r * K, r * φ) with respect to the projection π K onto the second component. We obtain the following commutative diagram of dynamical systems: Remark 3.11. Given a bundle (K, B, p; φ), the pullback bundle (p * K, K, π; p * φ) admits a continuous section: This pullback bundle is constructed by gluing to each point in K its fiber and so the map s : K → p * K, k → (k, k) is a canonical continuous section. In particular, every bundle of topological dynamical systems is a factor of a bundle admitting a continuous section. Moreover, properties such as minimality and unique ergodicity of each fiber as well as global properties such as equicontinuity, invertibility and mean ergodicity are preserved under forming pullback bundles.

3.2.
Maximal trivial factor and mean ergodicity. The following proposition shows that, up to isomorphism, there is a one-to-one correspondence between unital C * -subalgebras of fix(T φ ) and trivial factors (B, id B ) of the system (K, φ).
Proof. Let A be a unital C * -subalgebra of fix(T φ ). By the Gelfand-Naimark theorem, there is a compact space B such that A ∼ = C(B). The induced C * -embedding C(B) → C(K) is given by a Koopman operator T p for a continuous map p : K → B. Because T p is injective, p is surjective. Moreover, one obtains from the commutativity of the two diagrams is indeed a bundle of topological dynamical systems such that T p (C(B)) = A. Now take two such bundles (K, B, p; φ) and (K, B , p ; φ) of dynamical systems. Then C(B) ∼ = A ∼ = C(B ) and this isomorphism is again given by a Koopman operator T θ : C(B) → C(B ) corresponding to a homeomorphism θ : B → B. This yields that (id, θ) is an isomorphism between the two bundles.
Remark 3.13. Proposition 3.12 allows to order the bundles corresponding to a system (K, φ) by saying that (K, ). The term finer is used here because the above inclusion induces a surjective map r : B 1 → B 2 . In light of Proposition 3.12, there is (up to isomorphy) a maximal trivial factor of (K, φ) associated to the fixed space fix(T φ ). We denote this factor by (L φ , id L φ ) with the corresponding factor map q φ : K → L φ , but omit φ from the notation if the context leaves no room for ambiguity.
The maximal trivial factor allows to decompose a topological dynamical system into closed, invariant subsets in a canonical way and will be important throughout. As a first illustration of its use, we characterize mean ergodicity, showing that the global notion of mean ergodicity is in fact equivalent to a notion of fiberwise unique ergodicity. The elegant proof of the implication (b) =⇒ (a) was proposed by M. Haase in personal communication and is now presented here in favor of the original proof.
Theorem 3.14. Let (K, φ) be a topological dynamical system and q : K → L the projection onto its maximal trivial factor. Then the following assertions are equivalent.
(a) The Koopman operator T φ is mean ergodic on C(K).
If we denote the mean ergodic projection of T φ by P , then Pf ∈ fix(T φ ) = T q (C(L)) and hence Pf is constant on each fiber. Therefore, f = Pf | K l is constant and fix(T φ l ) is one-dimensional. Thus, T φ l is mean ergodic since the Cesáro averages converge uniformly on K and in particular on K l . Hence, each fiber (K l , φ l ) is uniquely ergodic. Now assume that each fiber (K l , φ l ) is uniquely ergodic and let µ l denote the corresponding unique invariant probability measure. Using this and Lemma 3.15 below, we obtain that the graph of the map l → µ l is closed. Since this map takes values in the compact set M 1 φ (K), the closed graph theorem for compact spaces (see [4,Theorem XI.2.7]) yields that the map l → µ l is weak * -continuous. Since each fiber is uniquely ergodic, we also have and this depends continuously on x, showing that T φ is mean ergodic. The equivalence of (b) and (c) is well-known for each fiber. Assertion (d) implies that fix(T φ ) separates fix(T φ ) and hence that T φ is mean ergodic. Conversely, if T φ is mean ergodic, a short calculation shows that the inverse of the map in (d) is given by ν → L µ l dν where µ l is the unique φ-invariant probability measure on K.
Lemma 3.15. Let (K, φ) be a topological dynamical system, q : K → L the projection onto its maximal trivial factor, and µ ∈ M(K) a probability measure. Then supp(µ) ⊂ K l if and only if T q µ = δ l .
Proof. Assume that supp(µ) ⊂ K l . If g ∈ C(L) satisfies g(l) = 0, then T q g is zero on K l and hence on supp(µ), meaning that So supp(T q µ) ⊂ {l} and since T q µ is a probability measure, we conclude that is positive and such that f ≤ 1 and supp(f ) ∩ K l = ∅, then l ∈ q(supp(f )) and by Urysohn's lemma there is a function g ∈ C(L) equal to 1 on q(supp(f )) satisfying g(l) = 0. But then f ≤ T q g and hence 0 ≤ f, µ ≤ T q g, µ = g, T q µ = g, δ l = 0.
So f, µ = 0 and we conclude that supp(µ) ⊂ K l Remark 3. 16. In the proof of the implication (b) =⇒ (a), the fact that we considered fibers with respect to the maximal trivial factor L was not used. Indeed, let B be any trivial factor such that the corresponding fibers are uniquely ergodic. The existence of a continuous surjection r : L → B from Remark 3.13 then shows that each fiber K b is contained in a fiber K l . But since each fiber (K l , φ l ) is also uniquely ergodic by Theorem 3.14, it cannot contain more than one of the sets K b and so r has to be a homeomorphism. Therefore, any bundle of topological dynamical systems (K, B, p; φ) with uniquely ergodic fibers is automatically isomorphic to the bundle (K, L, q; φ) and we may hence assume that B = L and p = q.
3.3. Group bundles. We now introduce the main object of this article: bundles of topological dynamical systems for which each fiber is a group rotation.
is a bundle and each fiber G b carries a group structure such that (i) the multiplication of group bundles is a bundle morphism such that Θ is a group homomorphism restricted to each fiber. It is called a morphism of group rotation bundles if, in addition, Θ is a morphism of the corre- We call it subtrivializable and ι a (G-)subtrivialization if ι is merely an embedding. We say that two subtrivializations Example 3.18. As an example of a bundle of topological dynamical systems for which each fiber is a group rotation, yet no continuous section α : B → K exists, recall the bundle from Example 3.6 and equip it with the dynamic φ : K → K, z → −z. The fibers here may be interpreted as copies of (Z 2 , n → n + 1) and it was seen in Example 3.6 that this bundle does not admit any continuous sections.
Remark 3.19. Products and pullbacks of group rotation bundles canonically are again group rotation bundles. However, when passing to factors, the existence of continuous sections may be lost, as seen in Example 3.10. If, however, such a factor (G , B , p ; φ ) has a continuous section s : B → G , it is again a group rotation bundle.
Remark 3.20. The notion of group bundles is not new: It has been considered as a special case of locally compact groupoids in, e.g., [14,Chapter 1].
In order to decompose systems with discrete spectrum, we single out group rotation bundles for which each fiber is minimal. Recall the following characterization of minimal group rotations.
Remark 3.21. Let (G, B, p; φ) be a group rotation bundle such that each fiber is minimal. Then by [5,Theorem 10.13], every fiber is uniquely ergodic, the unique φ-invariant probability measure being the Haar measure m b on the group G b . Remark 3.16 yields that we may therefore assume that B = L and p = q where q : G → L is the projection onto the maximal trivial factor. Moreover, if m l denotes the Haar measure on G l , the map l → m l is weak * -continuous. If µ is a φ-invariant measure on G, we define the pushforward measure ν := q * µ on L and disintegrate µ as in the proof of Theorem 3.14 via µ = L m l dν.
This will be important for Theorem 4.10.
The remainder of this section is dedicated to dual group bundles and their properties, which will be somewhat technical but crucial for Section 5 where we generalize Pontryagin duality to bundles. where (G b ) * is the dual group of G b and denote by π B : G * → B the canonical projection onto B. Next, let h ∈ C c (G), F ∈ C(G) and > 0. Set The family of these sets forms a subbasis for a topology which we call the topology of compact convergence on G * .
With this topology, the projection π B is continuous as can be deduced from the continuity of the neutral element section e : B → G by invoking Urysohn's lemma and Tietze's extension theorem to construct appropriate functions h and F . Therefore, (G * , B, π B ) is a bundle which we call the dual bundle of (G, B, q) and also denote by (G, B, q) * . If (Θ, θ) : (G, L, q) → (H, L, p) is a morphism of group bundles such that θ is bijective, define its dual morphism (Θ * , θ −1 ) : (H * , B , q) → (G * , B, p) by setting Θ * : H * → G * , χ → (Θ π L (χ) ) * χ.
For later reference and the convenience of the reader, we list some basic properties of dual bundles. To this end, we recall the following notions.    (i) The evaluation map ev : G * ⊕ G → C, (χ, g) → χ(g) is continuous. In fact, a net (χ i ) i∈I converges to χ ∈ G * if and only if π B (χ i ) → π B (χ) and for every convergent net (g i ) i∈I with p(g i ) = π B (χ i ) and limit g ∈ G we have is an isomorphism of locally compact groups.
In particular, the notation G * b is unambiguous. (iii) If G is a locally compact group and L is a compact space, (G × L, L, π L ) * = (G * × L, L, π L ). (iv) If the bundle (G, B, p) is lower-semicontinuous, G * is a Hausdorff space.
(v) The dual morphism Θ * is continuous and is a morphism of group bundles. Proof. The first part of (i) follows from the definition of the topology on G * using local compactness to invoke Urysohn's lemma and Tietze's extension theorem which provide appropriate functions h and F . The second part of (i) is a simple proof by contradiction. For part (ii), it suffices to show that the two sets carry the same topology. This follows from (i) since it shows that the two topologies have the same convergent nets. By the same argument, (iii) follows directly from (i) and so does (v), since it suffices to show that Θ * is continuous. In (iv), we obtain the Hausdorff property from lower-semicontinuity and (i), showing that every convergent net in G * has a unique limit.
For part (vi) (which trivially implies (vii)), note that Θ * is injective because Θ is surjective. Let (χ i ) i∈I be a net in H * such that Φ * (χ i ) converges to η ∈ G * b . Then η(g) = η(g ) if Θ(g) = Θ(g ) and so η = χ • Θ for a function χ : H b → C. It is again multiplicative and continuous because H b carries the final topology with respect to

Representation.
The classical examples for systems with discrete spectrum are group rotations (G, a) and trivial systems (B, id B ) as seen in Example 2.6. In Corollary 4.7 we show that, in fact, every system with discrete spectrum is a canonical factor of a product (G, a) × (B, id B ) and therefore arises from these two basic systems. This is an easy consequence of our Halmos-von Neumann representation theorem for not necessarily minimal or ergodic systems with discrete spectrum, see Theorem 4.6 and Theorem 4.10.
We briefly recall the Halmos-von Neumann theorem for minimal topological systems (K, φ) and, because the proof of Theorem 4.6 below is based on it, sketch a proof using the Ellis (semi)group E(K, φ) := {φ k | k ∈ N} ⊂ K K introduced by Ellis as the enveloping semigroup, see [6].
Theorem 4.1. Let (K, φ) be a minimal topological dynamical system with discrete spectrum. Then (K, φ) is isomorphic to a minimal group rotation (G, φ a ) on an abelian compact group G. More precisely, for each x 0 ∈ K there is a unique isomorphism δ x0 : Proof. Pick a point x 0 ∈ K and consider the map Since K is minimal, δ x0 is injective. Moreover, δ x0 (E(K, φ)) is a closed, invariant subset of K which is not empty and hence δ x0 (E(K, φ)) = K. It is not difficult to check that the system (E(K, φ), φ) is isomorphic to (K, φ) via δ x0 .
Note that the isomorphism in Theorem 4.1 depends on the (non-canonical) choice of x 0 ∈ K. In order to extend this result to non-minimal systems, we need the following definition.  We abbreviate the Ellis semigroup bundle by E(K, B, p; φ) if the context leaves no room for ambiguity. We also note that it is a group rotation bundle if it is compact and E(K, φ) is a group, in which case we call it the Ellis group bundle. We now give a criterion for the space E(K, B, p; φ) to be compact.
and endow C(X, Y ) with the topology of locally uniform convergence. Moreover, let A ⊂ C(X, Y ) be a compact subset. If F is lower-semicontinuous, then the quotient A × B/∼ F is a compact space.
Proof. Since the quotient of a compact space by a closed equivalence relation is again compact (cf. [2, Proposition 10.4.8]), it suffices to show that ∼ F is closed. So let ((f i , b i ), (g i , b i )) i∈I be a net in ∼ F with limit ((f, b), (g, b)) ∈ (C(X, Y ) × B) 2 . Pick x ∈ F (b). Since F is lower-semicontinuous and b i → b, there is a net (x i ) i∈I such that x i ∈ F (b i ) and x i → x. But since (f i ) i∈I and (g i ) i∈I converge locally uniformly, Since x ∈ F (b) was arbitrary, it follows that f | F (b) = g| F (b) and so ∼ F is closed. For (ii), assume that each fiber ( is open in K and so p(U ) is open.
Lemma 4.5. Let (K, φ) be a topological dynamical system with discrete spectrum and q : K → L the canonical projection onto the maximal trivial factor. Then each fiber (K l , φ l ) is minimal and has discrete spectrum.
Proof. Each fiber (K l , φ l ) has discrete spectrum since E(K l , φ l ) = {ψ| K l | ψ ∈ E(K, φ)}, use Proposition 2.7. Moreover, for x, y ∈ K l one has orb(x) = E(K l , φ l )x and orb(y) = E(K l , φ l )y. Since E(K l , φ l ) is a group, we conclude that either orb(x) = orb(y) or orb(x) ∩ orb(y) = ∅. However, by Remark 2.4 the system (K, φ) is mean ergodic and hence (K l , φ l ) is uniquely ergodic by Theorem 3.14. We now conclude from the Krylov-Bogoljubov Theorem that K l cannot contain two disjoint closed orbits. Consequently, orb(x) = orb(y) for all x, y ∈ K l and hence (K l , φ l ) is minimal.
Theorem 4.6. Let (K, φ) be a topological dynamical system with discrete spectrum and assume that the canonical projection q : K → L onto the maximal trivial factor admits a continuous section. Then (K, L, q; φ) is isomorphic to its Ellis group bundle.
Because (K, φ) has discrete spectrum, the map is continuous, hence Φ is continuous and an isomorphism of topological dynamical systems.
Example 3.18 shows that there are systems with discrete spectrum which are not isomorphic to a group rotation bundle. However, the following is still true.
Corollary 4.7. Let (K, φ) be a topological dynamical system with discrete spectrum. Then (K, φ) is a factor of a trivial group rotation bundle (G, a) × (B, id B ) where the group rotation (G, a) is minimal and can be chosen as (G, a) = (E(K, φ), φ).
Proof. Let (K, φ) be a topological dynamical system with discrete spectrum and q : K → L the projection onto its maximal trivial factor. As noted in Remark 3.11, the associated pullback system (q * K, K, π K , q * φ) also has discrete spectrum. Moreover, its fibers are uniquely ergodic and so Remark 3.16 shows that its maximal trivial factor is homeomorphic to K. This, combined with Remark 3.11 yields that the canonical projection onto its maximal trivial factor admits a continuous section s : K → q * K. By Theorem 4.6 we obtain that the bundle (q * K, K, π K ; q * φ) is isomorphic to its Ellis group bundle which is, by construction, a factor of the trivial group rotation bundle (E(q * K, q * φ), q * φ)×(K, id K ). We now consider the following maps: where p 2 : q * K → K denotes the projection onto the second component. Both Q and P are continuous and satisfy Q(φ k ) = (q * φ) k and P ((q * φ) k ) = φ k for all k ∈ N. Since φ and q * φ generate their respective Ellis groups, P and Q are mutually inverse. Hence, Remark 4.8. The group rotation (E(K, φ), φ) is the smallest group rotation that can be taken as (G, a) in Corollary 4.7 in the sense that any such group rotation (G, a) admits an epimorphism η : (G, a) → (E(K, φ), φ). This is true because a factor map θ : (G, a) × (B, id B ) → (K, φ) induces a continuous, surjective group homomorphism Remark 4.9. If (K, φ) has discrete spectrum and the canonical projection q : K → L admits a continuous section, the system is already isomorphic to its Ellis group bundle and hence, by definition of the latter, a factor of the system (E(K, φ), φ) × (L, id L ). In this case, one can take B = L in Corollary 4.7.
4.1. The measure-preserving case. Since the problem of finding continuous sections can be solved for topological models of measure spaces as shown below, we obtain a stronger result for measure-preserving systems. This is a generalization of the Halmos-von Neumann representation theorem to the non-ergodic case. It is proved by constructing a topological model and then applying Theorem 4.6. For background information on topological models, see [5,Chapter 12]. Theorem 4.10. Let (X, φ) be a measure-preserving system with discrete spectrum. Then (X, φ) is Markov-isomorphic to the rotation on a compact group rotation bundle. More precisely, there is a compact group rotation bundle (G, B, p; φ α ) with minimal fibers and a φ α -invariant measure µ G on G such that (X, φ) and (G, µ G ; φ α ) are Markov-isomorphic. Moreover, this group rotation bundle can be chosen such that the canonical map j : Kro C(G) (T φα ) → Kro L ∞ (G,µ G ) (T φα ) of corresponding Kronecker spaces is an isomorphism.
Proof. We define and note that this is a T φ -invariant, unital C * -subalgebra of L ∞ (X) being dense in L 1 (X) by [5,Lemma 17.3] since (X, φ) has discrete spectrum. The Gelfand representation theorem (cf. [16,Theorem I.4.4]) yields that there is a compact space K and a C * -isomorphism S : C(K) → A. The Riesz-Markov-Kakutani representation theorem shows that there is a unique Borel probability measure µ K on K such that Moreover, T := S −1 •T φ •S : C(K) → C(K) defines a C * -homomorphism and so (cf. [5,Theorem 4.13]) there is a continuous map ψ : K → K such that T = T ψ . The operator S is, by construction, an L 1 -isometry and S|f | = |Sf | for all f ∈ C(K) by [5,Theorem 7.23]. Since A is dense in L 1 (X), we conclude that S extends to a Markov embedding S : L 1 (K, µ K ) → L 1 (X).
The (topological) system (K, ψ) still has discrete spectrum by construction. Let L ψ denote the maximal trivial factor of (K, ψ). Then C(L ψ ) ∼ = fix(T ψ ) ∼ = fix L ∞ (X) (T φ ) and so L ψ is extremally disconnected as noted in Section 2. From Theorem 3.8 we therefore conclude that the canonical projection q : K → L ψ has a continuous section. Theorem 4.6 shows that there is an isomorphism θ : (K, ψ) → (G, α) where (G, α) is the rotation on some compact group rotation bundle with minimal fibers. Equipping (G, α) with the push-forward measure µ G := θ * µ K , we obtain that the system (X, φ) is isomorphic to the system (G, µ G ; α). Corollary 4.11. Let (X, φ) be a measure-preserving dynamical system with discrete spectrum and (L, ν; id L ) a topological model for fix L ∞ (X) (T φ ). Then (X, φ) is a Markov factor of the trivial group rotation bundle (J(T φ ), m; T φ ) × (L, ν; id L ).
Proof. This follows from Theorem 4.10 and Remark 4.9.
Remark 4.12. It is not difficult to see that if the measure space X is separable, the group rotation bundle in Theorem 4.10 can be chosen to be metrizable: Going back to the proof of Theorem 4.10, the algebra A needs to be replaced by a separable subalgebra B which is still dense in L 1 (X). Using that T φ is mean ergodic on A and that there hence is a projection P : A → fix A (T φ ), this can be done in such a way that fix B (T φ ) is generated by its characteristic functions. Therefore, its Gelfand representation space is totally disconnected and using Proposition 3.7 instead of Theorem 3.8, one can continue the proof of Theorem 4.10 analogously. Hence, if X is a standard probability space, one obtains versions of Theorem 4.10 and Corollary 4.11 with point isomorphy and point factors. However, the group rotation bundles involved are not canonical. Remark 4.13. We can also interpret the Halmos-von Neumann theorem in the following way: If (X, φ) is an ergodic, measure-preserving system with discrete spectrum, there is a compact, ergodic group rotation (G, a) and a Markov isomorphism S : L 1 (X) → L 1 (G, m) such that the diagram commutes, i.e., T φ acts like an ergodic rotation on scalar-valued functions. If (X, φ) is not ergodic, we can interpret Corollary 4.11 similarly: There is a compact, ergodic group rotation (G, a), a compact probability space (L, ν) and a Markov embedding S : L 1 (X) → L 1 (G×L, m×ν) such that T φa×id L S = ST φ . The rotation φ a induces a Koopman operator T φa on the vector-valued functions in L 1 (G, m; L 1 (L, ν)). Using the π-tensor product, we obtain also commutes, i.e., T φ acts like an ergodic rotation on vector-valued functions. We can interpret the topological Halmos-von Neumann theorem Theorem 4.1 and Corollary 4.7 analogously.

5.
Realization and uniqueness. The topological Halmos-von Neumann theorem shows that every minimal dynamical system with discrete spectrum is isomorphic to a minimal group rotation (G, a). Therefore, minimal group rotations can be seen as the canonical representatives of minimal systems with discrete spectrum. Moreover, the Pontryagin duality theorem shows that (G, a) and (G * * , δ a ) are isomorphic which has two consequences: On the one hand, G * ∼ = G * (a) via χ → χ(a) and G * (a) = σ p (T φa ) where T φa denotes the Koopman operator of φ a , see [5,Propositions 14.22 and 14.24]. In particular, σ p (T φa ) is a subgroup of T and for the canonical inclusion ι : is endowed with the discrete topology. Therefore, the point spectrum σ p (T φa ) is a complete isomorphism invariant for the minimal group rotation (G, a).
Combined with the Halmos-von Neumann theorem, this shows that the point spectrum σ p (T φ ) is a complete isomorphism invariant for the entire class of minimal topological dynamical systems (K, φ) with discrete spectrum. On the other hand, the Pontryagin duality theorem also implies that every subgroup of T can be realized as σ p (T φa ) for some group rotation (G, a). This completes the picture, showing that minimal systems with discrete spectrum are, up to isomorphism, in one-to-one correspondence with subgroups of T.
In order to generalize these results to the non-minimal setting, we need to adapt the Pontryagin duality theorem to group rotation bundles using the preparations from Section 3.3. We start with the necessary terminology.
Construction 5.1 (Dual bundles). If (G, L, q; α) is a compact group rotation bundle with minimal fibers and discrete spectrum, the map yields a surjective morphism (ρ, id L ) of group bundles which induces, by Proposition 3.26, an embedding ρ * : G * → E(G, φ α ) * × L. Since E(G, φ α ) is compact, its dual group is discrete and so we also have the embedding where T carries the discrete topology. The composition ι : G * → T × L of these two maps is hence a subtrivialization of G * and we call (G, L, q; α) * := (G * , L, π L ; ι) the dual bundle of (G, L, q; α). (Note that G * is, in general, neither locally compact nor Hausdorff.) If, conversely, (G, L, q; ι) is a group bundle with a T-subtrivialization ι, we set α ι : L → G * , l → ι l and call (G, L, q; ι) * := (G * , L, π L ; α ι ) the dual bundle of (G, L, q; ι). We say that two group bundles with T-subtrivializations (G, L, q; ι) and (G , L , q ; ι ) are isomorphic if their respective subtrivializations are, i.e., if there is an isomorphism (Θ, θ) : (G, L, q) → (G , L , q ) such that the diagram commutes. If L = L = pt, this means that ι and ι have the same image.
Definition 5.2. Let (K, φ) be a topological dynamical system and q : K → L the projection onto its maximal trivial factor L. Then we define We denote the projection onto the second component by π L and equip Σ p (K, φ) with the subspace topology induced by C × L if C carries the discrete topology. The bundle (Σ p (K, φ), L, π L ) is then called the point spectrum bundle of (K, φ). We say that the point spectrum bundles of two systems are isomorphic if there is an isomorphism of their canonical subtrivializations, i.e., if there is a homeomorphism η : L φ → L ψ such that is a (well-defined) homeomorphism. We call (H, η) an isomorphism of the point spectrum bundles. and hence (G, L, q; ι) * ∼ = (ι(G) * , L, π L ; (id ι(G l ) ) l∈L ).
In particular, G and hence its dual are completely determined by ι(G). Now, if (G, L, π L ; ι) is the dual of a compact group rotation bundle (H, L, p; α) with minimal fibers and discrete spectrum, it follows from the introduction to this section that ι(G) * , L, π L , (id ι(G l ) ) l∈L = Σ p (H, φ α ) * , L, π L , (id Σ p,l (H,φα) ) l∈L .
So we see that the dual bundle of a group rotation bundle with discrete spectrum and minimal fibers is completely determined by the system's point spectrum bundle.
Lemma 5.4. Let (K, φ) be a topological dynamical system with discrete spectrum. Then its point spectrum bundle is lower-semicontinuous.
Proof. Suppose (λ, l) ∈ Σ p (T φ ) and let f ∈ C(K l ) be a corresponding eigenfunction. Since T φ has discrete spectrum, λT φ is mean ergodic. So as in the proof of Theorem 3.14, f can be extended to a global fixed functionf ∈ C(K) of λT φ . Since the map q : Proposition 5.5. Let (G, L, q; α) be a compact group rotation bundle with discrete spectrum and minimal fibers. Then it is isomorphic to its bi-dual bundle.
Proof. The following diagram commutes: Combining Remark 5.3 and Lemma 5.4, we see that (G, L, q) * is lower-semicontinuous. Proposition 3.26 then shows that G * embeds into E(G, φ α ) × L, is therefore locally compact and so G * * is Hausdorff. Since ρ is a surjective, continuous map between compact spaces, G carries the final topology with respect to ρ, which shows that the map g → δ g is continuous and bijective. Since G * * is Hausdorff, this shows that G ∼ = G * * and the claim follows.
We can now formulate the answer to the three aspects of the isomorphism problem already discussed in the introduction.  Conversely, if L is a compact space, every lower-semicontinuous sub-group bundle of (T × L, L, π L ) can be realized as the point spectrum bundle of a topological dynamical system with discrete spectrum.
Proof. The representation result is Theorem 4.6. Moreover, Remark 5.3 and Proposition 5.5 show that the point spectrum bundle is a complete isomorphism invariant for compact group rotation bundles with minimal fibers and discrete spectrum and the representation theorem allows to extend this to (K, φ) and (M, ψ). The last part follows, analogously to the minimal case, from Proposition 3.26(iv), Proposition 5.5 and Remark 5.3.
Remark 5.7. Note that the statement of Theorem 5.6 is false if the assumption of a continuous section is removed. Indeed, one obtains a counterexample from Example 3.18.
We obtain a similar result for measure-preserving systems with discrete spectrum using topological models. This requires the following definition.
Remark 5.9. Let (K, φ) be a topological dynamical system, µ a regular Borel measure on K, q : K → L the canonical projection onto the maximal trivial factor of (K, φ) and ν := q * µ. If the canonical map j : Kro C(K) (T φ ) → Kro L ∞ (K,µ) (T φ ) is an isomorphism, then Σ p (K, φ) = Σ p (K, µ; φ). This is in particular the case for the group rotation bundles constructed in Theorem 4.10.
Recall that a regular Borel measure µ on a (hyper)stonean space K is called normal if all rare sets are null-sets. If µ is a normal measure on K with full support, then the canonical embedding C(K) → L ∞ (K, µ) is an isomorphism, cf. [16,Corollary III.1.16]. After this reminder, we can state the analogue of Theorem 5.6 for measure-preserving systems.
(a) (Representation) The system (X, φ) is Markov-isomorphic to a compact group rotation bundle (G, µ G ; φ α ) with minimal fibers. (b) (Uniqueness) The systems (X, φ) and (Y, ψ) are Markov-isomorphic if and only if their point spectrum bundles are isomorphic. In that case, the systems are also point isomorphic, provided X and Y are standard probability spaces. (c) (Realization) The point spectrum bundle of (X, φ) is continuous. Conversely, if (L, ν) is a hyperstonean compact probability space, ν is normal, and supp ν = L and (Σ, L, p) is a continuous sub-group bundle of (T×L, L, p), then (Σ, L, p; ν) can be realized as the point spectrum bundle of a measure-preserving dynamical system with discrete spectrum.
Using the disintegration formula from Remark 3.21, one quickly checks that Ψ * µ H = µ G because θ −1 * η = ν. For part (c), let (L, ν) be a hyperstonean compact probability space such that ν is normal and supp ν = L and let (Σ, L, p) be a continuous sub-group bundle of (T × L, L, p). Let (G, L, π L , φ α ) be its dual group rotation bundle endowed with the measure µ G := L m l dν.
Since G * is isomorphic to the point spectrum bundle Σ p (G, φ α ) via an isomorphism Φ by Proposition 5.5, the map η : U λ → G * , l → Φ −1 (λ, l) is continuous. Extend η to all of L by setting η(l) to the trivial character in G * l for l ∈ L \ U λ and note that η is continuous since U is open and closed. Now, for l ∈ U λ , each fiber (G l , φ α,l ) of (G, φ α ) is a minimal group rotation and hence the eigenspace of the Koopman operator T φ α,l corresponding to λ is at most one-dimensional and therefore spanned by η(l) ∈ G * l . So for ν-almost every l ∈ U λ , there is a constant c l ∈ C such that f l = c l η(l) m l -almost everywhere. If we extend c to L by 0, [c] ∈ L ∞ (L, ν) since [f ] is in L ∞ (G, µ). But C(L) ∼ = L ∞ (L, ν) via the canonical embedding and so we may assume that c is continuous. If q : G → L is the projection onto L, using (i) of Proposition 3.26, we see that the functionf : G → C, x → c q(x) η q(x) (x) is in C(G), f =f µ-almost everywhere, and T φαf = λf by construction. Now let (X, φ) be a measure-preserving dynamical system with discrete spectrum. In order to show that its point spectrum bundle is upper-semicontinuous, we may switch to its model (G, µ G , φ α ) on a compact group rotation bundle (G, L, p; φ α ) constructed in Theorem 4.10. Take λ ∈ T. By Remark 5.9 and Lemma 5.4, the set U λ := {l ∈ L | (λ, l) ∈ Σ p (G, µ G , φ α )} = {l ∈ L | (λ, l) ∈ Σ p (G, φ α )} is open. Via the isomorphism Θ : Σ p (G, φ α ) ∼ = G * , we see that the function F : U λ → G * , l → Θ(λ, l) selecting the (unique) character on G l correponding to the eigenvalue λ is continuous. By (i) of Proposition 3.26, F defines a continuous function f : p −1 (U λ ) → C and we may extend f to a measurable function on all of G by 0. Then T φα f = λf and since the C(G)-and L ∞ (G, µ G )-Kronecker space for T φα are canonically isomorphic, we can find a continuous representative g ∈ C(G) for the class [f ]. Since |f | ≡ 1 a.e. on p −1 (U λ ), |g| ≡ 1 on p −1 (U λ ) = p −1 (U λ ), where the last equality holds because p is open by Lemma 4.4. Therefore, U λ ⊂ U λ and hence U λ = U λ . This shows that the point spectrum bundle of (X, φ) is uppersemicontinuous.