Measure-preserving semiflows and one-parameter Koopman semigroups
aa r X i v : . [ m a t h . D S ] J un MEASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETERKOOPMAN SEMIGROUPS
NIKOLAI EDEKO, MORITZ GERLACH AND VIKTORIA KÜHNER
Dedicated to Rainer Nagel on the occasion of his th birthday. Abstract.
For a finite measure space X, we characterize strongly continuousMarkov lattice semigroups on L p (X) by showing that their generator A actsas a derivation on the dense subspace D ( A ) ∩ L ∞ (X). We then use this tocharacterize Koopman semigroups on L p (X) if X is a standard probabilityspace. In addition, we show that every measurable and measure-preservingflow on a standard probability space is isomorphic to a continuous flow on acompact Borel probability space. In this article we address mainly the following two issues. First, we characterizestrongly continuous Markov lattice semigroups ( T ( t )) t ≥ on L p (X) by propertiesof their generators for a finite measure space X = ( X, Σ , µ ). We will show thata strongly continuous semigroup on L p (X) is a Markov lattice semigroup if andonly if its generator A acts as a derivation on D ( A ) ∩ L ∞ (X), ∈ D ( A ) and thesemigroup is locally bounded on L ∞ (X). Similar results have been established byR. Nagel and R. Derndinger in [2, Satz 2.5] for semigroups on C( K ), see also [8,Section B-II.3], and recently by T. ter Elst and M. Lemańczyk in [4] for unitarygroups on L (X).Second, we show that such semigroups are always similar to a semigroup of Koop-man operators. More precisely, we construct a compact space K and a Borel mea-sure ν such that L ( X, Σ , µ ) is isometrically Banach lattice isomorphic to L ( K, ν )and, via this isomorphism, the semigroup ( T ( t )) t ≥ is similar to a semigroup ofKoopman operators on L ( K, ν ) induced by a continuous semiflow ( ϕ t ) t ≥ on K .Furthermore, in case that the space L ( X, Σ , µ ) is separable, we show that K canbe chosen to be metrizable. Similar results have been already obtained for stronglycontinuous representations of locally compact groups on L p (X) as bi-Markov em-beddings, see [7, Theorem 5.14].The article is organized as follows. In the second part of the introduction, wespecify our notation and recall some basic facts we use throughout the article.In Section 1, we prove our main result, Theorem 1.1, that characterizes stronglyMarkov semigroups of lattice homomorphisms by the condition that their generatoracts as a derivation, followed by a version for semigroups that are not necessarilyMarkov. In Section 2 we then use these results to characterize Koopman semigroupsand in particular obtain [4, Theorem 1.1] in which ter Elst and Lemańczyk proveda corresponding result for unitary operator groups as Corollary 2.6. In Section 3,we turn to the construction of topological models. Finally, in Section 4, we considerergodic, measure-preserving flows and give a new proof for the fact that they containat most countably many non-ergodic mappings, provided that their induced groupof Koopman operators is strongly continuous on L (X). This has previously beenproven in [9, Theorem 1] for R k -actions. Let us recall some concepts and fix the notation used in this article. For a measurespace X = ( X, Σ , µ ) and 1 ≤ p ≤ ∞ we denote byL p (X) := L p (X; C ) = L p (X; R ) ⊗ iL p (X; R )the corresponding complex L p -space. This is a complex Banach lattice in the senseof [3, Definition 7.2]. The lattice operations in L p (X; R ) are denoted by ∨ and ∧ and we write f + and f − for the positive part f ∨ − f ) ∨ f ∈ L p (X; R ). We call X separable if L p (X) is separable for one (and henceall) p ∈ [1 , ∞ ). When considering more than one measure on X , we may distinguishbetween a measurable function f : X → C and its equivalence class with respectto a measure µ on X by writing [ f ] µ for the equivalence class of f . If f ∈ L ∞ (X)is an essentially bounded function, M f will denote its associated multiplicationoperator on L p (X), p ∈ [1 , ∞ ]. Let X = ( X, Σ , µ ) and Y = ( Y, Σ ′ , µ ′ ) be finitemeasure spaces and T : L p (X) → L p (Y) a linear operator. The operator T is called positive if it is real, i.e., if T L p (X; R ) ⊆ L p (Y; R ), and its restriction to the Banachlattice L p (X; R ) is positive. The operator T is said to be a lattice homomorphism if | T f | = T | f | for each f ∈ L p (X). In particular, every lattice homomorphism ispositive and fulfills T ( f + ) = ( T f ) + and T ( f − ) = ( T f ) − for each f ∈ L p (X; R ).The operator T is called a Markov operator if it is positive and T X = Y , and a bi-Markov operator if, additionally, T ′ Y = X . It is called a Koopman operator ifthere is a measurable map ϕ : Y → X such that ϕ − maps null-sets into null-setsand T f = f ◦ ϕ for all f ∈ L p (X). In this case, we denote the operator by T ϕ . Notethat every Koopman operator is a Markov lattice homomorphism.Let ( T ( t )) t ≥ be a C -semigroup on L p (X) for some 1 ≤ p < ∞ . We denote itsgenerator by A and the domain of A by D ( A ). The semigroup ( T ( t )) t ≥ is called a lattice semigroup if each operator T ( t ) is a lattice homomorphism. If, additionally,for each t ≥ T ( t ) is a (bi-)Markov operator, ( T ( t )) t ≥ is called a (bi-)Markov lattice semigroup . It is called a Koopman semigroup , if for each t ≥ T ( t ) is a Koopman operator.Consider the equivalence relation M ∼ N if M = N µ -almost everywhereon Σ. Then the set of equivalence classes Σ(X) := Σ / ∼ is called the measurealgebra of the measure space X and is a Boolean algebra with respect to the setoperations union, intersection and complementation. For the sake of simplicity,we do not distinguish notationally between elements of Σ and Σ(X). A mapping θ : Σ(X) → Σ(X) is called a
Boolean algebra homomorphism if θ ( ∅ ) = ∅ , θ ( X ) = X as well as θ ( A ∪ B ) = θ ( A ) ∪ θ ( B ) and θ ( A ∩ B ) = θ ( A ) ∩ θ ( B ) for all A, B ∈ Σ(X). If, in addition, a Boolean algebra homomorphism θ : Σ(X) → Σ(X) satisfies µ ( θ ( A )) = µ ( A ) for all A ∈ Σ(X), then θ is called a measure algebra homomorphism .Every measurable map ϕ : X → X such that ϕ − maps null-sets into null-setsinduces a measure algebra homomorphism ϕ ∗ : Σ(X) → Σ(X) via A ϕ − ( A ). Forfurther information on Σ(X) we refer to [3, Section 6.1]. We call a measure space( X, Σ , µ ) a Borel probability space if X can be equipped with a Polish topologysuch that Σ is the corresponding Borel σ -algebra. A measure space ( X, Σ , µ ) iscalled a standard probability space if there is a measurable, measure-preserving andessentially invertible map to a Borel probability space.Finally, we call a linear operator δ on L p (X) with domain D ( δ ) a derivation on D ( δ ) ∩ L ∞ (X) if D ( δ ) ∩ L ∞ (X) is an algebra (with respect to the pointwise multi-plication) and δ ( f · g ) = δf · g + f · δg for all f, g ∈ D ( δ ) ∩ L ∞ (X). EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 3 Characterization of Markov lattice semigroups on L p -spaces In this section we characterize strongly continuous Markov lattice semigroups onL p (X)-spaces, where X = ( X, Σ , µ ) is a finite measure space, by means of theirgenerators. The following theorem is our main result. Theorem 1.1.
Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L p (X) , where X is a finite measure space and ≤ p < ∞ . Then the followingassertions are equivalent. (i) ( T ( t )) t ≥ is a Markov lattice semigroup. (ii) For every t ≥ there exists a Boolean algebra homomorphism θ t : Σ(X) → Σ(X) such that T ( t ) M = θ t ( M ) for all M ∈ Σ(X) . (iii) The space L ∞ (X) is invariant under ( T ( t )) t ≥ , the map t
7→ k T ( t ) k L (L ∞ (X)) is locally bounded, ∈ D ( A ) and A is a derivation on D ( A ) ∩ L ∞ (X) . Remark 1.2. (i) Given a finite measure space X and a bounded operator S on L p (X) such that L ∞ (X) is invariant under S , it follows from the closedgraph theorem that the restriction of S to L ∞ (X) is a bounded operator.Therefore, the map t
7→ k T ( t ) k L (L ∞ (X)) is well-defined in (iii). As willbe shown in Lemma 2.5, the local boundedness condition is automaticallyfulfilled if T is an operator group.(ii) A semigroup ( T ( t )) t ≥ satisfying (i)-(iii) in Theorem 1.1 uniquely extendsto a strongly continuous Markov lattice semigroup on L q (X) for each 1 ≤ q < ∞ with k T ( t ) k L (L q (X)) = k T ( t ) k q L (L (X)) , use [3, Theorem 7.23]. In particular, it extends to the biggest L q -space,L (X), and we will therefore only consider semigroups on L (X) in Sec-tions 3 and 4. At this point, however, the dependence on p in Theorem 1.1cannot be eliminated as easily since it is only clear a posteriori that theimplication (iii) = ⇒ (i) can be reduced to the case p = 1.As a preparation for the proof of Theorem 1.1, recall the following lemma relatingthe algebra and the lattice structure of L ∞ (X). Lemma 1.3.
Let X be a finite measure space and T : L ∞ (X) → L ∞ (X) be abounded linear operator satisfying T = . Then the follwing assertions are equiv-alent.(i) T is multiplicative.(ii) T is a C ∗ -homomorphism.(iii) T is a lattice homomorphism. Proof.
Obviously, (ii) implies (i). The equivalence of (ii) and (iii) can be found in[3, Theorem 7.23]. (There, the operator is assumed to be conjugation-preservingbut this assumption is trivially superfluous.) The implication (i) = ⇒ (ii) followsfrom [3, Theorem 4.13], the analogous statement for spaces of continuous functions,by applying the Gelfand-Naimark theorem [3, Theorem 4.23]. (cid:3) The following continuity property will be essential for the proof of Theorem 1.1.
Lemma 1.4.
Let X be a measure space and B ⊆ L ∞ (X) be bounded. Then themultiplication L p (X) × B → L p (X), ( f, g ) f g is k · k p -continuous. Proof.
Let M be a bound for B . For f, u ∈ L p (X), g, v ∈ B and c > f g − uv = ( f − u ) g + u ( g − v )= ( f − u ) g + u [ | u |≤ c ] ( g − v ) + u [ | u | >c ] ( g − v ) EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 4 and so lim sup ( f,g ) → ( u,v ) k f g − uv k p ≤ M (cid:13)(cid:13) u [ | u | >c ] (cid:13)(cid:13) p → c → ∞ ) . (cid:3) Next, we apply Lemma 1.4 to a C -semigroup as in Theorem 1.1. Corollary 1.5.
Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L p (X) , where X is a finite measure space and ≤ p < ∞ . In addition, suppose thatthe space L ∞ (X) is invariant under ( T ( t )) t ≥ and that the map t
7→ k T ( t ) k L (L ∞ (X)) is locally bounded. Then, for all f ∈ L ∞ (X) and all g ∈ L p (X) the function [0 , ∞ ) → t L p (X) , t T ( t ) f · T ( t ) g is continuous. Moreover, for f, g ∈ D ( A ) ∩ L ∞ (X) this function is differentiableand the product rule dd t ( T ( t ) f · T ( t ) g ) = T ( t ) Af · T ( t ) g + T ( t ) f · T ( t ) Ag holds.Proof. It suffices to prove the second part since the first is a consequence of Lemma 1.4.Let f, g ∈ D ( A ) ∩ L ∞ (X) and t ≥
0. Use Lemma 1.4 and differentiate to obtaindd t ( T ( t ) f · T ( t ) g ) = lim h → h (cid:0) [ T ( t + h ) f − T ( t ) f ] T ( t ) g + T ( t + h ) f [ T ( t + h ) g − T ( t ) g ] (cid:1) = lim h → h (cid:0) [ T ( t + h ) f − T ( t ) f ] (cid:1) T ( t ) g + lim h → T ( t + h ) f · lim h → h [ T ( t + h ) g − T ( t ) g ]= (cid:18) dd t T ( t ) f (cid:19) · T ( t ) g + T ( t ) f · (cid:18) dd t T ( t ) g (cid:19) which proves the assertion. (cid:3) We are now able to prove Theorem 1.1.
Proof of Theorem 1.1.
The equivalence (i) ⇔ (ii) is proved almost exactly as in thecase of bi-Markov operators, see [3, Theorem 12.10].To prove the implication (i) = ⇒ (iii), first note that every operator T ( t ) is positive.Since T ( t ) = for each t ≥
0, this already implies that the semigroup preservesthe subspace L ∞ (X) and the restriction of each T ( t ) to L ∞ (X) is a contraction. Inparticular, it follows from Lemma 1.3 that every operator T ( t ) is multiplicative onL ∞ (X). By Corollary 1.5, for every f, g ∈ D ( A ) ∩ L ∞ (X)dd t T ( t )( f · g ) = dd t (cid:0) T ( t ) f · T ( t ) g (cid:1) = T ( t ) Af · T ( t ) g + T ( t ) f · T ( t ) Ag = T ( t ) [ Af · g + f · Ag ] . In particular, this shows f · g ∈ D ( A ) and A ( f · g ) = Af · g + f · Ag for t = 0. Thisproves that A is a derivation and clearly ∈ D ( A ).We now prove that (iii) implies (i). Because of the local boundedness of t T ( t ) k L (L ∞ (X)) , there exists a constant C > (cid:13)(cid:13)(cid:13)(cid:13) t Z t T ( s ) f d s (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ C k f k ∞ EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 5 for 0 < t ≤ f ∈ L ∞ (X). This implies that D := D ( A ) ∩ L ∞ (X) is a densesubspace of L p (X). We use this fact to show that each T ( t ) is multiplicative onL ∞ (X). For fixed f, g ∈ D and t > s β ( s ) := T ( t − s )[ T ( s ) f · T ( s ) g ]on [0 , t ]. Since β (0) = T ( t )( f · g ) and β ( t ) = T ( t ) f · T ( t ) g , it suffices to show that β is constant. To this end, consider the operator valued mappings P, Q : [0 , t ] → L (L p (X)) given by P ( s ) = T ( t − s ) and Q ( s ) = M T ( s ) f ◦ T ( s ), where M T ( s ) f denotesthe multiplication with the bounded function T ( s ) f . It follows from Corollary 1.5that Q is strongly continuous and that for each h ∈ D , s Q ( s ) h is differentiablewith derivativedd s Q ( s ) h = T ( s ) Af · T ( s ) h + T ( s ) f · T ( s ) Ah = A (cid:0) T ( s ) f · T ( s ) h (cid:1) . Here, the second equality follows from the fact that, by assumption, A is a derivationand D is invariant under each T ( t ). In particular, D is invariant under Q . Since P is also strongly continuous and s P ( s ) h is differentiable for all h ∈ D , it followsfrom [5, Lemma B.16] that β ′ ( s ) = − AT ( t − s )[ T ( s ) f · T ( s ) g ] + T ( t − s ) A [ T ( s ) f · T ( s ) g ] = 0for all s ∈ [0 , t ]. This shows that β is constant and thus that every T ( t ) is multi-plicative on D .Since the multiplication with a fixed bounded function induces a bounded operatoron L p (X) and D is k · k p -dense in L ∞ (X), fixing a function g ∈ D and usinga standard approximation argument shows that T ( f · g ) = T ( t ) f · T ( t ) g for all f ∈ L ∞ (X) and g ∈ D . Fixing f ∈ L ∞ (X) and repeating the argument shows that T ( t )( f · g ) = T ( t ) f · T ( t ) g for all f, g ∈ L ∞ (X), so T ( t ) is multiplicative on all ofL ∞ (X). Furthermore, A = 0 since A is a derivation, i.e., T ( t ) = for all t ≥ T ( t ) is a lattice homomorphism on L ∞ (X) andhence, by density and continuity, also on L p (X). (cid:3) Remark 1.6.
The assumption ∈ D ( A ) in (iii) of Theorem 1.1 is automaticallysatisfied if T ( t ) is an isometry for each t ≥
0. To see this, note that in the proofof Theorem 1.1 the assumption ∈ D ( A ) was only important for the implication(iii) = ⇒ (i). There, we showed that all the operators of the semigroup aremultiplicative on L ∞ (X) and therefore map characteristic functions to characteristicfunctions. If T ( t ) is an isometry, it follows that T ( t ) = for each t ≥ ∈ D ( A ). This assumption is also fulfilled if T ( t ) ′ = for each t ≥
0, since then h T ( t ) , i = h , i = µ ( X ) and so, T ( t ) being a characteristic function, T ( t ) = for each t ≥ Corollary 1.7.
Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L p (X) , where X is a finite measure space and ≤ p < ∞ . Then the followingassertions are equivalent. (i) ( T ( t )) t ≥ is a bi-Markov lattice semigroup. (ii) For every t ≥ there exists a measure algebra homomorphism θ t : Σ(X) → Σ(X) such that T ( t ) M = θ t ( M ) for all M ∈ Σ(X) . (iii) The space L ∞ (X) is invariant under ( T ( t )) t ≥ , the map t
7→ k T ( t ) k L (L ∞ (X)) is locally bounded, A is a derivation on D ( A ) ∩ L ∞ (X) and A ′ = 0 .Proof. For the equivalence of (i) and (ii), the reader is again referred to [3, Theorem12.10]. The equivalence of (i) and (iii) follows from Theorem 1.1 and Remark 1.6since T ( t ) ′ = for all t ≥ A ′ = 0. (cid:3) EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 6
In the following we discuss C -semigroups of lattice homomorphisms on L p (X) thatare not necessarily Markov. We show that their generator is a derivation perturbedby a bounded multiplication operator. Theorem 1.8.
Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L p (X) , where X is a finite measure space and ≤ p < ∞ . Assume that ∈ D ( A ) and q := A ∈ L ∞ (X) . Then the following assertions are equivalent. (i) ( S ( t )) t ≥ is a lattice semigroup. (ii) The function q is real-valued and A = B + q where B is the generator of aMarkov lattice C -semigroup ( T ( t )) t ≥ on L p (X) .If (ii) holds, then S ( t ) f = exp (cid:18)Z t T ( s ) q d s (cid:19) · T ( t ) f (1) for all t ≥ and f ∈ L p (X) .Proof. To show the equivalence of (i) and (ii), we first recall from [5, TheoremIII.1.3] that B := A − q is a generator of a C -semigroup ( T ( t )) t ≥ on L p (X) because B is a bounded perturbation of A . Since B = 0, T ( t ) = for all t ≥
0. Nowit follows from [8, Corollary C-II.5.8] (Kato’s identity) that ( S ( t )) t ≥ is a latticesemigroup if and only if D ( A ) is a sublattice of L p (X) and A | f | = Re(sign( f ) Af )for all f ∈ D ( A ). Since (i) implies that q is real-valued, A satisfies this condition ifand only if B does, which proves the equivalence of the assertions (i) and (ii).Now assume that (ii) holds. Since each T ( t ) is multiplicative on L ∞ (X) by Lemma 1.3, T ( t ) exp( g ) = T ( t ) P ∞ n =0 g n n ! = exp( T ( t ) g ) for each g ∈ L ∞ (X). Using this, oneproves by induction that (cid:0) T ( t )e t M q (cid:1) n = M exp (cid:0) t P nj =1 T ( jt ) q (cid:1) T ( nt )for each n ≥
1. Replace t by tn and note that n X j =1 tn T (cid:18) jtn (cid:19) q k·k p −−−−→ n →∞ Z t T ( s ) q d s. By Lemma 1.4, one obtains the convergenceM exp (cid:0)P nj =1 tn T ( jtn ) q (cid:1) −−−−→ n →∞ M exp (cid:0)R t T ( s ) q d s (cid:1) of multiplication operators in the strong operator topology on L (L p (X)). By theTrotter product formula S ( t ) f = lim n →∞ (cid:20) T (cid:18) tn (cid:19) exp (cid:18) tn M q (cid:19)(cid:21) n f = exp (cid:18)Z t T ( s ) q d s (cid:19) · T ( t ) f for all f ∈ L p (X). (cid:3) Koopman semigroups on L p -spaces Every Koopman semigroup on an L p (X)-space is a Markov lattice semigroup but theconverse is, in general, not true. However, it does hold if X is a standard probabilityspace: In the case of bi-Markov lattice homomorphisms, this is a classical theoremby von Neumann, cf. [3, Theorem 7.20]. Below, we give an operator-theoretic proofextending this theorem to Markov lattice homomorphisms on L p -spaces. We thenrelate this to the results on semigroups from the previous section. EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 7
Theorem 2.1.
Let
X = ( X, Σ X , µ X ) and Y = ( Y, Σ Y , µ Y ) be standard probabilityspaces and T : L p (X) → L p (Y) , ≤ p ≤ ∞ , a Markov lattice homomorphism (notnecessarily bi-Markov). Then there is a measurable map ϕ : Y → X such that T = T ϕ . If ϑ : Y → X is another such map, then ϕ = ϑ µ Y -almost everyhwere.Proof. We will factorize T in order to apply von Neumann’s theorem for bi-Markovlattice homomorphisms. Since L ∞ (X) is dense in L p (X) for any p ∈ [1 , ∞ ], we onlyneed to prove the case p = ∞ . Moreover, assume without loss of generality thatX = ( K, B ( K ) , µ K ) and Y = ( L, B ( L ) , µ L ) are Borel probability spaces.Set ν := ( T ′ Y ) µ and note that ( K, B ( K ) , ν ) is again a Borel probability space.Since ν ≪ µ , the map P : L p ( K, B ( K ) , µ K ) → L p ( K, B ( K ) , ν ), [ f ] µ K [ f ] ν is awell-defined, surjective and bounded operator. Moreover, for f ∈ L ∞ ( K, B ( K ) , µ K ), Z K | f | d ν = Z L T | f | d µ L = Z L | T f | d µ L (2)and so ker T ⊆ ker P . Therefore, there is an operator ˆ T : L ∞ ( K, B ( K ) , ν ) → L ∞ ( L, B ( L ) , µ L ) such that T = ˆ T P . Since P is surjective, ˆ T also is a Markovlattice homomorphism and it follows from (2) that ˆ T is, in fact, bi-Markov.Now, von Neumann’s theorem shows that ˆ T [ f ] ν = [ f ◦ ϕ ] µ L for a measurable andmeasure-preserving map ϕ : ( L, B ( L ) , µ L ) → ( K, B ( K ) , ν ) and so T = T ϕ . Theproof that ϕ is unique almost everyhwere is the same as for measure-preservingmaps, so we refer the reader to, e.g., [3, Lemma 6.9]. (cid:3) Corollary 2.2.
Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L p (X) , where X = ( X, Σ , µ ) is a standard probability space and ≤ p < ∞ . Thenthe equivalent assertions (i) , (ii) and (iii) of Theorem 1.1 are also equivalent to (iv) There exists a family ( ϕ t ) t ≥ of measurable maps on X such that T ( t ) f = f ◦ ϕ t for all f ∈ L p (X) and t ≥ .Proof. If assertion (i) of Theorem 1.1 holds, we obtain assertion (iv) by Theorem 2.1below. Conversely, if (iv) holds, then every T ( t ) is a Markov lattice homomorphism,thus assertion (i) holds. (cid:3) Remark 2.3.
Given a Koopman semigroup ( T ϕ t ) t ≥ on L p (X) as in Corollary 2.2, itis immediate from the semigroup property and Theorem 2.1 that ϕ = id X µ -almosteverywhere and ϕ t ◦ ϕ s = ϕ t + s µ -almost everywhere for all s, t ≥
0. Therefore, thefamily ( ϕ t ) t ≥ forms a semiflow modulo null-sets, see Section 3. Note, however,that it can in general not be made into a semiflow by simply discarding a null-setsince the identity ϕ t ◦ ϕ s = ϕ t + s might hold outside of a null-set depending on s and t . Corollary 2.4.
Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L p (X) , where X = ( X, Σ , µ ) is a standard probability space and ≤ p < ∞ . Thenthe equivalent assertions (i) , (ii) and (iii) of Corollary 1.7 are also equivalent to (iv) There exists a family ( ϕ t ) t ≥ of measurable and measure-preserving mapson X such that T ( t ) f = f ◦ ϕ t for all f ∈ L p (X) and t ≥ .Proof. Assume (i) of Corollary 1.7. Then Corollary 2.2 shows that there are mea-surable maps ϕ t : X → X such that T ( t ) f = f ◦ ϕ t . Moreover, for M ∈ Σ µ ( ϕ − t ( M )) = h ϕ − t ( M ) , X i = h T ( t ) M , X i = h M , X i = µ ( M )and so each ϕ t is measure-preserving. On the other hand, (iv) implies (ii) with θ t = ϕ ∗ t . (cid:3) EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 8
The next lemma shows that in the case of groups, the boundedness assumptionin Theorem 1.1 (iii) is superfluous. This allows us to recover [4, Theorem 1.1] asCorollary 2.6. The idea of the following proof is based on the proof of [4, Theorem2.5].
Lemma 2.5.
Let X be a finite measure space and ( T ( t )) t> be a semigroup onL ∞ (X), strongly continuous with respect to k · k p where 1 ≤ p < ∞ . Then themapping t
7→ k T ( t ) k L (L ∞ (X)) is locally bounded. Proof.
For f ∈ L ∞ (X) and t >
0, setting q := 1 − p k T ( t ) f k ∞ = sup (cid:8) |h T ( t ) f, g i| : g ∈ L (X) and k g k = 1 (cid:9) = sup {|h T ( t ) f, g i| : g ∈ L q (X) and k g k = 1 } Since ( T ( t )) t> is strongly continuous with respect to k ·k p , h T ( · ) f, g i is continuousfor g ∈ L q (X). Hence, k T ( · ) f k ∞ is lower semicontinuous, being the supremum ofcontinuous functions. Therefore, the sets A n := { t > k T ( t ) f k ∞ ≤ n } are closed.Since R > = S n A n , Baire’s category theorem yields that for every k ∈ N , thereare 0 < a k < b k ≤ k and M k ∈ N such that [ a k , b k ] ⊆ A M k . By the semigroupproperty, sup s ∈ [ a k + t,b k + t ] k T ( s ) f k ∞ = sup s ∈ [ a k + t,b k + t ] k T ( t ) T ( s − t ) f k ∞ ≤ sup s ∈ [ a k ,b k ] k T ( t ) k ∞ k T ( s ) f k ∞ ≤ M k k T ( t ) k ∞ for each t >
0. This shows that k T ( · ) f k ∞ is locally bounded for each f ∈ L ∞ (X)and the claim now follows by the principle of uniform boundedness. (cid:3) Corollary 2.6 ([4, Theorem 1.1]) . Let A be the generator of a unitary C -group ( T ( t )) t ∈ R on L (X) where X = ( X, Σ , µ ) is a standard probability space. Then thefollowing assertions are equivalent. (i) For every t ∈ R there exists an essentially invertible measurable and measure-preserving map ϕ t : X → X such that T ( t ) f = f ◦ ϕ t for all f ∈ L (X) . (ii) The space L ∞ (X) is invariant under ( T ( t )) t ≥ and A is a derivation on D ( A ) ∩ L ∞ (X) .Proof. The implication (i) = ⇒ (ii) is a consequence of Corollary 1.7. In orderto prove the converse implication, we observe that it follows from Lemma 2.5 andRemark 1.6 that A and ( T ( t )) t ≥ as well as − A and ( T ( − t )) t ≥ fulfill condition(iii) in Theorem 1.1. Corollary 2.2 therefore shows that T ( t ) = T ϕ t for measurablemaps ϕ t : X → X and t ≥
0. The essential invertibility of the maps ϕ t followsfrom [3, Proposition 7.12] and [3, Corollary 7.21]. Also, since each T ( t ) is unitaryand a Markov operator, one shows as in Corollary 2.4 that each ϕ t is measure-preserving. (cid:3) Corollary 2.7.
Let A be the generator of a C -semigroup ( S ( t )) t ≥ on a space L p (X) , where X is a standard probability space and ≤ p < ∞ , such that ∈ D ( A ) and q := A ∈ L ∞ (X) . Then ( S ( t )) t ≥ is a lattice semigroup if and only if q ∈ L ∞ (X; R ) and there exists a family ( ϕ t ) t ≥ of measurable maps on X correspondingto a strongly continuous Koopman semigroup on L p (X) such that S ( t ) f = exp (cid:18)Z t q ◦ ϕ s d s (cid:19) · ( f ◦ ϕ t )(3) for all f ∈ L p (X) and t ≥ . EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS 9
Proof.
If ( S ( t )) t ≥ is a lattice semigroup, it follows from Theorem 1.8 that thereexists a Markov lattice semigroup ( T ( t )) t ≥ on L p (X) with generator ( A − q, D ( A ))such that (1) holds. The representation (3) hence follows from Corollary 2.2. Con-versely, every semigroup of the form (3) with a real-valued q is a lattice semi-group. (cid:3) Topological Model
We have seen in Corollary 2.2 that on a standard probability space X = ( X, Σ , µ )every strongly continuous Markov lattice semigroup ( T ( t )) t ≥ on L (X) is inducedby a family ( ϕ t ) t ≥ of measurable maps on X . Since ( T ( t )) t ≥ is a semigroup,one has ϕ = id X and ϕ s ◦ ϕ t = ϕ s + t almost everywhere, using the uniqueness inTheorem 2.1. Recall that a family ( ϕ t ) t ≥ of maps on a set X is called a semiflow if ϕ = id X and ϕ t ◦ ϕ s = ϕ s + t for s, t ≥
0. A semiflow on a topological space X is called continuous if the map Φ : X × R + → X , ( x, t ) ϕ t ( x ) is continuous.Similarly, a semiflow on a measure space is called measurable if Φ is measurable. Itis called measure-preserving if for each t ≥ ϕ t is measurable and measure-preserving. In that case, there is an induced semigroup of operators ( T ϕ t ) t ≥ onL (X) called the Koopman semigroup induced by the semiflow. This semigroupis weakly measurable if the semiflow is measurable but even then it need not bestrongly continuous: Consider, e.g., the semiflow on ( { , } , P ( { , } ) , ( δ + δ ))with ϕ = id { , } and ϕ t ≡ t >
0. For measurable and measure-preserving flows ( ϕ t ) t ∈ R (defined analogously to semiflows) on separable measurespaces, however, the induced Koopman group is in fact strongly continuous. Proposition 3.1.
Let X = ( X, Σ , µ ) be a separable, finite measure space and( ϕ t ) t ≥ a measurable and measure-preserving semiflow on X . Then the inducedKoopman semigroup ( T ( t )) t ≥ on L p (X) is strongly continuous on (0 , ∞ ) for 1 ≤ p < ∞ . If ( ϕ t ) t ∈ R is a measurable and measure-preserving flow on X , the inducedKoopman group on L p (X) is strongly continuous on all of R for 1 ≤ p < ∞ . Proof.
Take f ∈ L p (X). Then the function f ◦ Φ is measurable on X × R and foreach g ∈ L q (X) with q = 1 − p the map t
7→ h T ( t ) f, g i = Z X f (Φ( x, t )) g ( x ) d µ ( x )is measurable (cf. [10, Theorem 1.7.15]). Therefore, the semigroup ( T ( t )) t ≥ isweakly measurable and even strongly measurable by [6, Theorem 3.5.5] since L p (X)is separable. Since strongly measurable semigroups are strongly continuous on(0 , ∞ ) by [6, Theorem 10.2.3], ( T ( t )) t ≥ is indeed strongly continuous. The case offlows follows immediately. (cid:3) We now show that for such measurable flows, one can construct a continuous flowon a compact metric space with an invariant probability measure such that the twoflows are isomorphic. This will be done by first proving that every strongly con-tinuous Markov lattice semigroup is similar to a Koopman semigroup induced by acontinuous semiflow. An analogous result was recently proved for bi-Markov latticeembedding representations of locally compact groups by de Jeu and Rozendaal, see[7, Theorem 5.14].
Definition 3.2.
Let X = ( X, Σ , µ ) and Y = ( Y, Σ ′ , µ ′ ) be finite measure spaces. Wesay that two measurable semiflows ( ϕ t ) t ≥ and ( ψ t ) t ≥ on X and Y are isomorphic if there is a measure-preserving and essentially invertible map ρ : X → Y suchthat ψ t ◦ ρ = ρ ◦ ϕ t almost everywhere for each t ≥
0. We say that two Markovlattice semigroups ( T ( t )) t ≥ and ( S ( t )) t ≥ on L (X) and L (Y) are Markov similar if there is an invertible bi-Markov lattice homomorphism Φ : L (X) → L (Y) such EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS10 that S ( t ) ◦ Φ = Φ ◦ T ( t ) for each t ≥
0. The same notions are defined for flows andoperator groups analogously.We call continuous flows (resp. Koopman semigroups) as described above topologicalmodels for measurable flows (resp. Markov lattice semigroups). See also [3, Section12.3] for this terminology and similar results in the time-discrete case. For thesake of simplicity, we restrict ourselves to the case p = 1. The idea of the proofof the following result was kindly provided to us by Markus Haase. The employedtechnique of topological models, however, dates back much further. Theorem 3.3 (Topological Model) . Let A be the generator of a C -semigroup ( T ( t )) t ≥ on a space L (X) , where X = ( X, Σ , µ ) is a finite measure space. Thenthere exist a compact space K , a continuous semiflow ( ψ t ) t ≥ on K and a strictlypositive Borel probability measure ν such that the semiflow ( ψ t ) t ≥ induces a Koopman-semigroup on L ( K, ν ) which is Markov similar to the semigroup ( T ( t )) t ≥ on L (X) . The measure ν is ( ψ t ) t ≥ -invariant if and only if ( T ( t )) t ≥ is a bi-Markovlattice semigroup.Proof. Consider A := { f ∈ L ∞ (X) : s T ( s ) f is k · k ∞ -continuous } . Since eachoperator T ( t ) is contractive on L ∞ (X) and multiplicative by Lemma 1.3, A is analgebra and clearly ∈ A . Furthermore, A is closed with respect to k · k ∞ andclosed under conjugation. Therefore, A is a commutative C ∗ -algebra invariantunder ( T ( t )) t ≥ .We show that A is dense in L (X). The strong continuity of ( T ( t )) t ≥ on L (X)implies that k · k -lim t ց t R t T ( r ) f d r = f for each f ∈ L ∞ (X). Therefore, itsuffices to show that R t T ( r ) f d r ∈ A for f ∈ L ∞ (X). For all 0 ≤ s ≤ t and f ∈ L ∞ (X) (cid:12)(cid:12)(cid:12)(cid:12) T ( s ) Z t T ( r ) f d r − Z t T ( r ) f d r (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z t T ( s + r ) f d r − Z t T ( r ) f d r (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z t + ss T ( r ) f d r − Z t T ( r ) f d r (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t + st T ( r ) f d r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s T ( r ) f d r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s k f k ∞ since each T ( t ) is k · k ∞ -contractive. This shows that s T ( s ) R t T ( r ) f d r iscontinuous at zero and hence on [0 , ∞ ) with respect to k · k ∞ . Therefore, A isdense in L (X).By a combination of the Gelfand-Naimark theorem and the Riesz representationtheorem as in [3, Section 12.3] or [7, Theorem 5.14] one obtains a compact space K , a ∗ -isomorphism Φ : A → C( K ) with Φ = , a unique probability measure ν on K such that Z X Φ − g d µ = Z K g d ν for all g ∈ C( K ) and a semiflow semiflow ( ψ t ) t ≥ on K such that T ( t ) | A = Φ − ◦ T ψ t ◦ Φ. By [3, Theorem 4.17], the semiflow ψ is continuous, cf. also [8, TheoremB-II.3.4].Moreover, Φ extends to a bi-Markov lattice homomorphism Φ : L (X) → L ( K, ν ).Let ( S ( t )) t ≥ denote the semigroup ( T ( t )) t ≥ induces on L ( K, ν ) via Φ. Then S ( t )[ f ] ν = [ f ◦ ψ t ] ν (4) EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS11 for all continuous functions f ∈ C( K ). By a standard approximation argument,this holds for all bounded, Baire-measurable functions, cf. [3, Theorem E.1]. Viamonotone approximation, (4) extends to all positive integrable functions and ishence valid for all [ f ] ν ∈ L ( K, ν ). Finally, ( T ( t )) t ≥ is bi-Markov if and only if( S ( t )) t ≥ is, if and only if Z K f d ν = Z K S ( t ) f d ν = Z K f ◦ ψ t d ν for all f ∈ L ( K, ν ). This is given if and only if ( ψ t ) t ≥ preserves ν . (cid:3) Proposition 3.4.
If, in the situation of Theorem 3.3, the measure space is sepa-rable, then the compact space K can be chosen to be metrizable. Proof.
Let A be the algebra A := { f ∈ L ∞ (X) : s T ( s ) f is k · k ∞ -continuous } from the proof of Theorem 3.3. Since L (X) is separable and A is dense in theformer, there is a countable dense subset D of A . We set D := { T ( t ) f : f ∈ D , t ∈ Q + } ⊆ A and denote by A the C ∗ -subalgebra of A generated by D . The algebra A isthen separable, dense in L (X) and since A ⊆ A , T ( t ) A ⊆ A for not only t ∈ Q + but t ∈ R + . To complete the proof, one can now proceed as in the proofof Theorem 3.3 with A replaced by A , obtaining a compact representation spacewhich is metrizable because C( K ) is separable. (cid:3) Remark 3.5.
With slight notational adjustments, the proofs of the previous tworesults also work for Markov lattice groups and continuous flows.
Corollary 3.6.
Let
X = ( X, Σ , µ ) be a standard probability space and ( ϕ t ) t ∈ R ameasurable and measure-preserving flow on X . Then there are a compact metricspace K , a continuous flow ( ψ t ) t ∈ R on K and a strictly positive ( ψ t ) t ∈ R -invariantBorel probability measure ν on K so that the flows ( ϕ t ) t ∈ R and ( ψ t ) t ∈ R are isomor-phic.Proof. By Proposition 3.1 and Theorem 1.1, the flow ( ϕ t ) t ∈ R induces a bi-Markovgroup on L (X) and so Remark 3.5 shows that there are a compact metric space K , a continuous flow ( ψ t ) t ∈ R and a strictly positive ( ψ t ) t ∈ R -invariant probabilitymeasure ν on K such that the groups ( T ( t )) t ∈ R and ( S ( t )) t ∈ R induced by the flows( ϕ t ) t ∈ R and ( ψ t ) t ∈ R are Markov similar via an invertible bi-Markov lattice homo-morphism Φ. Applying von Neumann’s theorem shows that there is a measurableand measure-preserving map ρ : Y → X such that Φ = T ρ and ρ is essentially invert-ible because Φ is invertible, see [3, Corollary 7.21]. The identity Φ ◦ T ( t ) = S ( t ) ◦ Φnow shows that ϕ t ◦ ρ = ρ ◦ ψ t ν -almost everywhere, see [3, Proposition 7.19]. (cid:3) Remark 3.7.
Corollary 3.6 is similar to [1, Theorem 5] but for two importantdifferences: On the one hand, the authors of [1] work with a slightly strongernotion of isomorphism of flows. On the other hand, the models considered in [1]need not be compact. 4.
Ergodic flows
In this section, we give an operator-theoretic proof for the fact that a measure-preserving ergodic flow on a separable measure space comprises at most countablymany non-ergodic maps if it induces a strongly continuous group on L (X). Thisis a special case of [9, Theorem 1] where R k -actions where considered. First, recallthe following two properties. EASURE-PRESERVING SEMIFLOWS AND ONE-PARAMETER KOOPMAN SEMIGROUPS12
Definition 4.1.
Let X = ( X, Σ , µ ) be a finite measure space.(a) A measure-preserving semiflow ( ϕ t ) t ≥ on X is called ergodic , if for every A ∈ Σ one has A ⊆ ϕ − t ( A ) modulo null-sets for each t ≥ A = ∅ or A = X modulo null-sets.(b) A Markov lattice semigroup ( T ( t )) t ≥ on L (X) is called irreducible if thereare no nontrivial closed ( T ( t )) t ≥ -invariant ideals in L (X).Similar notions can be defined for measure-preserving flows and Markov latticegroups in a straightforward way and it is not difficult to see that a flow ( ϕ t ) t ∈ R isergodic if and only if one/both of the associated semiflows ( ϕ t ) ± t ≥ is/are ergodic.As a consequence of [3, Theorem 7.10], the two properties in Definition 4.1 areequivalent for a (semi)flow and its corresponding Koopman (semi)group. Obviously,if some ϕ s is ergodic (meaning that A ⊆ ϕ − s ( A ) implies µ ( A ) ∈ { , } ), so is thesemiflow ( ϕ t ) t ≥ , while the converse is not true in general. However, it followsfrom [9, Theorem 1] that for an ergodic flow ( ϕ t ) t ∈ R all but at most countablymany maps ϕ t are ergodic. In the following we give a short proof of this fact usingthe Perron-Frobenius spectral theory. We shall prove this for semiflows for whichthe induced Koopman semigroup is strongly continuous on L (X). In light of theprevious remarks, this implies [9, Theorem 1]. Theorem 4.2.
Let ( ϕ t ) t ≥ be a measure-preserving semiflow on a separable fi-nite measure space X = ( X, Σ , µ ) such that the corresponding Koopman semigroup ( T ( t )) t ≥ is strongly continuous on L (X) . If ( ϕ t ) t ≥ is ergodic, then at most count-ably many maps ϕ t are not ergodic.Proof. Let ( ϕ t ) t ≥ be measure-preserving and ergodic and its induced Koopmansemigroup ( T ( t )) t ≥ strongly continuous on L (X) with generator A . Since theeigenspaces of an isometry on L (X) are pairwise orthogonal, the point-spectrumof T ( t ) is countable for each t ≥ (X) is separable. The spectral in-clusion theorem [5, Theorem IV.3.6] implies that the boundary point spectrum G := P σ ( A ) ∩ i R is countable.For a fixed s > ϕ s is not ergodic if and only if dim fix( T ( s )) ≥
2, see[3, Proposition 7.15]. By the spectral mapping theorem for the point spectrum [5,Corollary IV.3.8]fix( T ( s )) = span { f ∈ L (X) : Af = i λf for some i λ ∈ G with e i λs = 1 } . Moreover, for an irreducible Markov lattice semigroup the common fixed space \ t ≥ fix( T ( t )) = ker A is one-dimensional since it is a Banach sublattice of L (X) and for real-valued f ∈ T t ≥ fix( T ( t )) all characteristic functions [ f>c ] , c ∈ R , are contained in thefixed space (use [3, Exercise 7.13]). Thus, dim fix( T ( s )) ≥ λ ∈ G such that λs = 2 πk for some k ∈ Z \ { } . Since G is countable, thiscan only be the case for at most countably many s > (cid:3) As a consequence of Theorem 4.2, Proposition 3.1 and the remarks made above, wenote the following corollary.
Corollary 4.3.
Let ( ϕ t ) t ∈ R be a measurable and measure-preserving flow on aseparable finite measure space X . If ( ϕ t ) t ∈ R is ergodic, then at most countablymany maps ϕ t are not ergodic. The assertion in Theorem 4.2 does not hold for non-separable spaces. Take forexample the Bohr compactification b R of the additive group of real numbers anddefine for g ∈ b R the translation ϕ g : b R → b R , h h + g . Then ( ϕ t ) t ∈ R is a EFERENCES 13 continuous flow on b R preserving the Haar measure m. Denote the correspondingKoopman group on L (b R , m) by ( T ( t )) t ∈ R . If f ∈ L (b R , m) is such that T ( t ) f = f ◦ ϕ t = f for all t ∈ R , then f ◦ ϕ g = f for all g ∈ b R since R is dense in b R . Itfollows that f is constant almost everywhere. Because T ( t ) f = f is equivalent to T ( − t ) f = f for each t ≥
0, this shows that the flow ( ϕ t ) t ≥ is ergodic. However, ϕ t is not ergodic since T ( t ) f = f for all periodic functions with period t . Acknowledgement.
We express our sincere gratitude towards the anonymousreferee for his insightful observations and detailed comments.
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Nikolai Edeko, Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle10, D-72076 Tübingen, Germany
E-mail address : [email protected] Moritz Gerlach, Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Straße24–25, 14476 Potsdam
E-mail address : [email protected] Viktoria Kühner, Mathematisches Institut, Universität Tübingen, Auf der Morgen-stelle 10, D-72076 Tübingen, Germany
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