A generalized Powers averaging property for commutative crossed products
aa r X i v : . [ m a t h . OA ] J a n A GENERALIZED POWERS AVERAGING PROPERTYFOR COMMUTATIVE CROSSED PRODUCTS
TATTWAMASI AMRUTAM AND DAN URSU
Abstract.
We prove a generalized version of Powers’ averagingproperty that characterizes simplicity of reduced crossed products C ( X ) ⋊ λ G , where G is a countable discrete group, and X is acompact Hausdorff space which G acts on minimally by home-omorphisms. As a consequence, we generalize results of Hart-man and Kalantar on unique stationarity to the state space of C ( X ) ⋊ λ G and to Kawabe’s generalized space of amenable sub-groups Sub a ( X, G ). This further lets us generalize a result ofthe first named author and Kalantar on simplicity of intermedi-ate C*-algebras. We prove that if C ( Y ) ⊆ C ( X ) is an inclu-sion of unital commutative G -C*-algebras with X minimal and C ( Y ) ⋊ λ G simple, then any intermediate C*-algebra A satisfying C ( Y ) ⋊ λ G ⊆ A ⊆ C ( X ) ⋊ λ G is simple. Contents
1. Introduction 12. The space of generalized probability measures 63. Proof of generalized Powers averaging 114. Unique stationarity and applications 15References 211.
Introduction
The notion of Powers’ averaging property for discrete groups hasplayed an important role in recent years in questions about simplicityrelated to reduced group C*-algebras and reduced crossed products.In this paper, we introduce a generalized version of Powers’ averagingproperty for reduced crossed products of the form C ( X ) ⋊ λ G , and Mathematics Subject Classification.
Key words and phrases. group action, minimal, compact space, C*-algebra,crossed product, simple, Powers averaging property.Second author supported by NSERC Grant Number 411300719. prove that it is equivalent to simplicity of the crossed product. Wethen derive various consequences.First, we recall the notion of Powers’ averaging property, along witha brief history of recent applications. Let G be a countable discretegroup, and let P ( G ) denote the set of probability measures on G . Forconvenience, we will denote the finitely supported probability measureson G by P f ( G ). Recall that we canonically have an action of P ( G ) onany G -C*-algebra A as follows: given µ ∈ P ( G ) and a ∈ A , we let µa = X g ∈ G µ ( g )( g · a ) . The group G is said to be C*-simple if its reduced group C*-algebra C ∗ λ ( G ) is simple. It was shown independently in [Ken15, Theorem 6.3]and [Haa16, Theorem 4.5] that C*-simplicity is equivalent to an aver-aging property originally considered by Powers, which can most con-veniently be stated as follows: G is said to have Powers’ averagingproperty if for any a ∈ C ∗ λ ( G ), we have τ λ ( a ) ∈ { µa | µ ∈ P f ( G ) } . Here, τ λ denotes the canonical trace on C ∗ λ ( G ), and we are canonicallyviewing C ⊆ C ∗ λ ( G ). The set P f ( G ) above can be replaced by P ( G )instead. It is also clear that it suffices to check only the a ∈ C ∗ λ ( G )satisfying τ λ ( a ) = 0, as it is always possible to “normalize” an arbitrary a ∈ C ∗ λ ( G ) by considering a − τ λ ( a ).It was later shown by Hartman and Kalantar in the proof of [HK17,Theorem 5.1] that averaging with respect to all of P f ( G ) is not nec-essary, and that if G is countable, then Powers’ averaging propertyfor C ∗ λ ( G ) is equivalent to the existence of a single measure µ ∈ P ( G )(not guaranteed to have finite support) satisfying µ n a → τ λ ( a ) for all a ∈ C ∗ λ ( G ).A generalization of Powers’ averaging property to reduced (twisted)crossed products of unital C*-algebras and C*-simple groups was givenby [BK16, Section 3]. They showed that the same averaging propertyholds for elements a ∈ A ⋊ λ G satisfying E ( a ) = 0, where E : A ⋊ λ G → A denotes the canonical conditional expectation, i.e.0 ∈ { µa | µ ∈ P f ( G ) } . The ideas mentioned above were used by the first named author andKalantar in [AK20] to show that if G is C*-simple and the action of G on a compact Hausdorff space X is minimal, then not only is the OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 3 reduced crossed product C ( X ) ⋊ λ G simple, but so is any intermediateC*-algebra lying between C ∗ λ ( G ) and C ( X ) ⋊ λ G .Of course, if G is not C*-simple, then A ⋊ λ G can never have Powers’averaging property, but it is still possible for the crossed product to besimple. An easy example is C ( T ) ⋊ λ Z , where Z acts on the circle T by an irrational rotation. For this reason, we introduce a generalizedversion of Powers’ averaging which does turn out to be equivalent tosimplicity in the end. Let X be a compact Hausdorff space equippedwith an action of G by homeomorphisms.We define in Section 2, and in particular in Definition 2 .
3, the spaces P ( G, C ( X )) and P f ( G, C ( X )) of what we call generalized ( G, C ( X )) -probability measures . Given an inclusion of unital G -C*-algebras C ( X ) ⊆ A , the space P ( G, C ( X )) canonically admits a left action on A , and aright action on the state space S ( A ). With this, we are able to con-veniently generalize Powers’ averaging property to crossed products asfollows: Definition 1.1.
Let G be a countable discrete group acting on a com-pact Hausdorff space X by homeomorphisms, and let E : C ( X ) ⋊ λ G → C ( X ) denote the canonical conditional expectation. We say that C ( X ) ⋊ λ G has the generalized Powers’ averaging property if for every a ∈ C ( X ) ⋊ λ G with E ( a ) = 0, we have0 ∈ { µa | µ ∈ P f ( G, C ( X )) } . One might define other generalized analogues of Powers’ averagingproperty, for example requiring that E ( a ) lie in the above set givenany a ∈ C ( X ) ⋊ λ G not necessarily satisfying E ( a ) = 0. It is notimmediately obvious, however, that this is equivalent with the versionin Definition 1 .
1, as considering a − E ( a ) for an arbitrary a ∈ C ( X ) ⋊ λ G just tells us that for any ε >
0, there is a µ ∈ P f ( G, C ( X )) with theproperty that k µa − µ E ( a ) k < ε . However, unlike in the case where C ( X ) = C , we do not have that µ E ( a ) = E ( a ) in general. Hence, ourfirst main result is perhaps a bit surprising: Theorem 1.2.
Let G be a countable discrete group acting on a compactHausdorff space X by homeomorphisms, and assume that the action isminimal. Let E : C ( X ) ⋊ λ G → C ( X ) denote the canonical conditionalexpectation. The following are equivalent: (1) C ( X ) ⋊ λ G is simple. (2) C ( X ) ⋊ λ G has the generalized Powers’ averaging property. (3) E ( a ) ∈ { µa | µ ∈ P f ( G, C ( X )) } for all a ∈ C ( X ) ⋊ λ G . T. AMRUTAM AND D. URSU (4) ν ( E ( a )) ∈ { µa | µ ∈ P f ( G, C ( X )) } for all a ∈ C ( X ) ⋊ λ G and ν ∈ P ( X ) . Next we generalize Hartman and Kalantar’s results. It is worth not-ing that they operate under a slightly different action of P ( G ) on any G -C*-algebra, with a convolution product given by µ ∗ a = X g ∈ G µ ( g )( g − · a )for any µ ∈ P ( G ), and a left action of P ( G ) on S ( A ) given by µ ∗ φ = X g ∈ G µ ( g )( g · φ ) . However, this is only a minor technicality to keep in mind, and it iseasy to rephrase their results (which we do) in terms of the actions weuse in our paper.As previously mentioned, they show that Powers’ averaging propertyfor C ∗ λ ( G ) is equivalent to the existence of a measure µ ∈ P ( G ) with theproperty that µ n a → τ λ ( a ) for any a ∈ C ∗ λ ( G ) [HK17, Theorem 5.1].As a consequence, the only state φ ∈ S ( C ∗ λ ( G )) that is µ -stationary (that is, satisfying φµ = φ ) is the canonical trace τ λ , and this is in facta characterization of C*-simplicity of G [HK17, Theorem 5.2]. Similarresult holds in the crossed product setting: Theorem 1.3.
Let G be a countable discrete group acting on a compactHausdorff space X by homeomorphisms, and assume that the action isminimal. Let E : C ( X ) ⋊ λ G → C ( X ) denote the canonical conditionalexpectation. If C ( X ) ⋊ λ G is simple, then there is a generalized measure µ ∈ P ( G, C ( X )) with the property that µ n a → whenever E ( a ) = 0 .Optionally, we may also require that µ have full support. Corollary 1.4.
Let G be a countable discrete group acting on a compactHausdorff space X by homeomorphisms, and assume that the action isminimal. Let E : C ( X ) ⋊ λ G → C ( X ) denote the canonical conditionalexpectation. Then the crossed product C ( X ) ⋊ λ G is simple if and onlyif there is some µ ∈ P ( G, C ( X )) with full support and with the propertythat any µ -stationary state on C ( X ) ⋊ λ G is of the form ν ◦ E for some µ -stationary ν ∈ P ( X ) . It is worth noting in Corollary 1 . µ ∈ P ( G, C ( X )), there is no guarantee that there be a unique µ -stationary measure ν ∈ P ( X ). If one could strengthen the averagingin Theorem 1 . OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 5 were some ν ∈ P ( X ) and µ ∈ P ( G, C ( X )) such that µ n a → ν ( E ( a ))for any a ∈ C ( X ) ⋊ λ G , then it would be possible to obtain uniquenessof ν as well. However, we were unable to prove such a result.Our first application of Powers’ averaging property is a natural gen-eralization of the main result in [AK20]. Theorem 1.5.
Let G be a countable discrete group, and assume that C ( Y ) ⊆ C ( X ) is an equivariant inclusion of commutative unital G -C*-algebras. Assume moreover that the action of G on X is minimal. If C ( Y ) ⋊ λ G is simple, then every intermediate C*-algebra A satisfying C ( Y ) ⋊ λ G ⊆ A ⊆ C ( X ) ⋊ λ G is simple. One other result of Hartman and Kalantar that we generalize is thefollowing. Denote the space of amenable subgroups of G by Sub a ( G ).This is naturally a compact Hausdorff space if we equip it with theChabauty topology, which is the topology induced by viewing thiscanonically as a subset of 2 G (the power set of G ), and it also car-ries a G -action by homeomorphisms given by conjugation. It has beenknown for a few years that the dynamics on this space characterizesC*-simplicity, with [Ken15, Theorem 4.1] essentially stating that G is C*-simple if and only if is the unique minimal component inSub a ( G ), and [HK17, Corollary 5.7] stating that C*-simplicity is equiv-alent to unique stationarity of δ { e } with respect to some µ ∈ P ( G ).The dynamical analogue for crossed products C ( X ) ⋊ λ G (where X is minimal) is a result of Kawabe [Kaw17, Theorem 6.1]. Consider thespaceSub a ( X, G ) := { ( x, H ) | x ∈ X, H ≤ G x , and H amenable } , where G x denotes the stabilizer subgroup of x . This is again a compactHausdorff space with G -action given by s · ( x, H ) = ( sx, sHs − ), andKawabe’s result amounts to saying that C ( X ) ⋊ λ G is simple if andonly if the only minimal component in Sub a ( X, G ) is X × . Thishints that there should also be a “unique stationarity result” involvingmeasures supported on this minimal component. Corollary 1.6.
Let G be a countable discrete group acting on a com-pact Hausdorff space X by homeomorphisms, and assume that the ac-tion is minimal. Let Sub a ( X, G ) denote Kawabe’s generalized spaceof amenable subgroups, and view C ( X ) ⊆ C (Sub a ( X, G )) as dual tothe canonical projection Sub a ( X, G ) ։ X mapping ( x, H ) to x . Thecrossed product C ( X ) ⋊ λ G is simple if and only if there is some T. AMRUTAM AND D. URSU µ ∈ P ( G, C ( X )) with the property that any µ -stationary measure in P (Sub a ( X, G )) is supported on X × . Acknowledgements
The authors owe a huge debt of gratitude to Mehrdad Kalantar andMatthew Kennedy, who are their advisors respectively, for many use-ful discussions surrounding the problem. The first named author isalso grateful to Narutaka Ozawa for many helpful discussions aboutthe averaging property. The authors also thank Sven Raum, Euse-bio Gardella, Shirley Geffen and Yongle Jiang for taking the time tocarefully read a near complete draft of this paper and giving helpfulfeedback.
Recent Development.
Upon completion of this paper, NarutakaOzawa sent us his preprint where he has shown (via different tech-niques) that, in the non-minimal case, a slight variation on the gener-alized Powers’ averaging property is equivalent to the action of G on X having the residual intersection property. We remark that for minimal G -spaces X , residual intersection property is equivalent to the crossedproduct C ( X ) ⋊ λ G being simple.2. The space of generalized probability measures
To establish notation, G will denote a countable discrete group, and X will denote a compact Hausdorff space which G acts on by homeo-morphisms. All C*-algebras and morphisms are assumed to be unital.We define the notion of a generalized probability measure. As mo-tivation, consider the case of a G -C*-algebra A . Given a fixed a ∈ A ,any probability measure µ ∈ P ( G ) provides a convenient way of repre-senting a convex combination of the elements { g · a | g ∈ G } . Namely,we may define µa := P g ∈ G µ ( g ) g · a .With this in mind, we want a space of generalized probability mea-sures which represents C(X)-convex combinations . For convenience, wefirst review this notion here:
Definition 2.1.
Assume C ( X ) ⊆ A is an inclusion of unital C*-algebras, and let K ⊆ A . We say that K is C ( X ) -convex if, givenfinitely many f , . . . , f n ∈ C ( X ) with P ni =1 f i = 1, and any a , . . . , a n ∈ K , we have P ni =1 f i a i f i ∈ K . Such a sum is called a C ( X ) -convex com-bination of a , . . . , a n . OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 7
Remark 2.2.
The usual notion of C ( X )-convex combinations is slightlymore general, and deals with sums of the form P ni =1 f ∗ i a i f i , where P ni =1 f ∗ i f i = 1 and f i is no longer assumed to be positive. For ourpurposes, we will stick with the definition given in Definition 2 .
1, asworking with positive f i is in particular necessary for Lemma 4 . C ( X ) ⊆ A is an inclusion of unital G -C*-algebras, and a ∈ A ,we want our notion of generalized probability measures to represent C ( X )-convex combinations of { g · a | g ∈ G } . Definition 2.3.
Consider a formal sum µ = P i ∈ I f i s i f i with theproperties f i ≥ f i = 0, and P i ∈ I f i = 1. We say that µ isa generalized ( G, C ( X )) -probability measure , and denote the set ofall such generalized measures by P ( G, C ( X )). The set of all finite-sum generalized measures is denoted by P f ( G, C ( X )). Given a uni-tal G -C*-algebra A containing an equivariant copy of C ( X ), and any µ = P i ∈ I f i s i f i ∈ P ( G, C ( X )), we have a unital and completely posi-tive map on A given by µa = X i ∈ I f i ( s i · a ) f i . Moreover, this induces a right action on the state space S ( A ), given by( φµ )( a ) = φ ( µa ). Remark 2.4.
We note the reasoning behind the choice of terminologyand notation, namely “finite-sum” and P f ( G, C ( X )), as opposed to“compactly supported” and P c ( G, C ( X )). This is because of the factthat, given a generalized probability measure µ = P i ∈ I f i s i f i , it ispossible to have infinitely many elements i ∈ I with s i all being equal.In other words, it is possible to have an infinite sum (which doesn’tnecessarily simplify to a finite sum) that is still “compactly supported”on G .The rest of this section is dedicated to proving various technicalitiesand basic properties of the space P ( G, C ( X )). First, the following isan easy exercise in functional analysis: Lemma 2.5.
Assume X is a Banach space, and P i ∈ I x i is an infiniteunordered sum. Then this sum converges if and only if for all ε > ,there exists a finite set F ⊆ I such that for all finite sets J ⊆ I \ F ,we have (cid:13)(cid:13)(cid:13)P j ∈ J x j (cid:13)(cid:13)(cid:13) < ε . From this, we obtain two important results:
T. AMRUTAM AND D. URSU
Corollary 2.6.
Any µ = P i ∈ I f i s i f i in P ( G, C ( X )) has the propertythat I is countable. In particular, P ( G, C ( X )) is indeed a set.Proof. Consider the sum P i ∈ I f i = 1 and ε = n ( n ∈ N ) in Lemma 2 . f i can have norm at least n .Hence, at most countably many f i can be nonzero. (cid:3) Corollary 2.7.
Assume C ( X ) ⊆ A is an inclusion of G -C*-algebras, µ = P i ∈ I f i s i f i ∈ P ( G, C ( X )) , and a ∈ A . Then the sum given by P i ∈ I f i ( s i · a ) f i is convergent, or in other words, the value µa is well-defined. Moreover, the map a µa is unital and completely positive.Proof. We first prove this for positive a . Let ε >
0, and let F ⊆ I besuch that for all finite J ⊆ I \ F , we have (cid:13)(cid:13)(cid:13)P j ∈ J f j (cid:13)(cid:13)(cid:13) < ε . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ J f j ( s j · a ) f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k a k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ J f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < ε k a k . To see that general values of a work, let µ F = P i ∈ F f i s i f i for finite F ⊆ I . Writing a as a finite linear combination of four positive elements P k =1 c k a k , we have that each of the nets ( µ F a k ) F converges for each k .In particular, the net µ F a = X k =1 c k µ F a k must therefore also be convergent. The fact that a µa is completelypositive follows from the fact that a µ F a is completely positive foreach finite F ⊆ I . (cid:3) Similar to how P ( G ) is a convex semigroup, we have that the spaces P f ( G, C ( X )) and P ( G, C ( X )) also form semigroups, and moreover sat-isfy an appropriate notion of C ( X )-convexity. Proposition 2.8.
The space P f ( G, C ( X )) is C ( X ) -convex, in the sensethat given finitely many { g j } j ∈ J ⊆ C ( X ) with P j ∈ J g j = 1 and any { µ j } j ∈ J ⊆ P f ( G, C ( X )) with µ j = P i ∈ I j f i s i f i , we have that X j ∈ J g j µ j g j := X j ∈ J X i ∈ I j g j f i s i f i g j also lies in P f ( G, C ( X )) . The same is true for P ( G, C ( X )) , exceptthat J can be infinite.Proof. We prove the case of P ( G, C ( X )), as the case of P f ( G, C ( X )) isalmost the same except without needing to worry about convergence. OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 9
Observe that, given any finite subsets F ⊆ J and F j ⊆ I j for j ∈ F ,we have X j ∈ F X i ∈ F j g j f i = X j ∈ F g j X i ∈ F j f i ≤ X j ∈ F g j · ≤ . Moreover, given ε >
0, if we choose F ⊆ J finite with P j ∈ J g j ≥ − ε and finite F j ⊆ I j for j ∈ F with P i ∈ F j f i ≥ − ε , then X j ∈ F X i ∈ F j g j f i = X j ∈ F g j X i ∈ F j f i ≥ X j ∈ F g j · (1 − ε ) ≥ (1 − ε ) . This proves that P j ∈ J P i ∈ I j g j f i = 1. (cid:3) Remark 2.9.
Given an inclusion of G -C*-algebras C ( X ) ⊆ A , a ∈ A , finitely many { g j } j ∈ J ⊆ C ( X ) with P j ∈ J g j = 1, and { µ j } j ∈ J ⊆ P f ( G, C ( X )), we have that X j ∈ J g j µ j g j ( a ) = X j ∈ J g j µ j ( a ) g j . Consequently, { µa | µ ∈ P f ( G, C ( X )) } is C ( X )-convex as well. In fact, it is the smallest G -invariant, C ( X )-convex subset of A containing a .Now we define a semigroup structure on P f ( G, C ( X )) and P ( G, C ( X )). Proposition 2.10.
The space P ( G, C ( X )) is a semigroup under thefollowing multiplication: given µ = P i ∈ I f i s i f i and ν = P j ∈ J g j t j g j , let µν := X i ∈ I X j ∈ J ( f i ( s i g j ))( s i t j )(( s i g j ) f i ) . Moreover, P f ( G, C ( X )) is a subsemigroup of P ( G, C ( X )) .Proof. Observe that, given any finite subsets F I ⊆ I and F J ⊆ J , wehave X i ∈ F I X j ∈ F J ( f i ( s i g j )) = X i ∈ F I f i s i X j ∈ F J g j ≤ X i ∈ F I f i s i ≤ . Moreover, any finite subset of I × J is contained in a set of the form F I × F J . Finally, given ε >
0, if one chooses F I and F J to be such that P i ∈ F I f i ≥ − ε and P j ∈ F J g j ≥ − ε , then we have X i ∈ F I X j ∈ F J ( f i ( s i g j )) = X i ∈ F I f i s i X j ∈ F J g j ≥ X i ∈ F I f i s i (1 − ε ) ≥ (1 − ε ) . This proves that P i ∈ I P j ∈ J ( f i ( s i g j )) = 1, and so this multiplicationon P ( G, C ( X )) is well-defined. Associativity is tedious but not hardto check. The fact that P f ( G, C ( X )) is a subsemigroup is clear. (cid:3) Remark 2.11.
The multiplication on P ( G, C ( X )) is defined in sucha way so that if C ( X ) ⊆ A is an inclusion of unital G -C*-algebras, µ , µ ∈ P ( G, C ( X )), and a ∈ A , then( µ µ )( a ) = µ ( µ a ) . In other words, we canonically have a left semigroup action of P ( G, C ( X ))on A , and consequently a right semigroup action on S ( A ).Let C ( X ) ⊆ A be an inclusion of unital G -C*-algebras, and let µ ∈ P ( G, C ( X )) be a generalized measure. We say that a state φ ∈ S ( A ) is µ -stationary if φµ = φ , and denote the set of all µ -stationarystates on A by S µ ( A ). Observe that this definition makes sense evenfor C*-subalgebras that don’t necessarily contain C ( X ), but are atleast µ -invariant. It is not hard to see that µ -stationary states alwaysexist and can be extended to a larger C*-algebra. The proof is a meremodification of [HK17, Proposition 4.2]. We include it for the sake ofcompleteness. Proposition 2.12.
Suppose that C ( X ) ⊆ A is an inclusion of unital G -C*-algebras, µ ∈ P ( G, C ( X )) , and B ⊆ A is a µ -invariant uni-tal C*-subalgebra. Then every µ -stationary state τ ∈ S µ ( B ) can beextended to a µ -stationary state η ∈ S µ ( A ) . In particular, S µ ( A ) isalways nonempty.Proof. Let K = { ζ ∈ S ( A ) | ζ | B = τ } , a compact convex set. For any µ ∈ P ( G, C ( X )), the map Φ µ : K → K defined by Φ µ ( ζ ) = ζ µ is anaffine continuous map. It is well-known that Φ µ has a fixed point, say η . Then, η ∈ S µ ( A ) and η | B = τ . To see that S µ ( A ) is nonempty, let B = C . (cid:3) We wish to define an appropriate notion of full support for measuresin P ( G, C ( X )). For this, the following observation will come in useful. Lemma 2.13.
Assume X is a Banach space, and P i ∈ I x i is an infiniteunordered sum that converges in norm. Then given any J ⊆ I , the sum P j ∈ J x j also converges.Proof. We know by Lemma 2 . ε >
0, there is a finitesubset F ⊆ I such that for any finite subset E ⊆ I \ F , we have OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 11 k P i ∈ E x i k < ε . But then, letting F ′ = F ∩ J , it is clear that for anyfinite set E ′ ⊆ J \ F ′ , we also have (cid:13)(cid:13)(cid:13)P j ∈ E ′ x j (cid:13)(cid:13)(cid:13) < ε . (cid:3) Definition 2.14.
We say that a generalized measure µ ∈ P ( G, C ( X ))has full support if, writing µ = X s ∈ G X i ∈ I s f i sf i , we have that for each s ∈ G , there is some δ > P i ∈ I s f i ≥ δ .Equivalently (by compactness of X ), given any s ∈ G and x ∈ X , wecan find i ∈ I s such that f i ( x ) > Proof of generalized Powers averaging
In this section, we prove Theorem 1 .
2. To give a brief overview, wefirst recall how this is proven in the case of the usual reduced groupC*-algebra.Let X be a G -space, that is a compact Hausdorff space equipped withan action of G by homeomorphisms. Recall that a measure ν ∈ P ( X )is said to be contractible if Gν w* contains all Dirac masses. A G -boundary is a G - space X with the additional property that everymeasure ν ∈ P ( X ) is contractible (note that if every measure ν ∈ P ( X )is contractible, then it follows that X is minimal). From the perspectiveof convexity, this is saying that P ( X ) is irreducible as a compact convex G -space. It is shown in [Ken15] that C*-simplicity of G is equivalentto { τ λ } being the unique G -boundary in S ( C ∗ λ ( G )). A similar resultcan consequently be achieved in the entire dual space of C ∗ λ ( G ), anda Hahn-Banach separation argument yields that this is equivalent toPowers’ averaging property.There is a generalized notion of boundaries introduced in [KS19, Sec-tion 7], which is used for dealing with noncommutative crossed products A ⋊ λ G . However, this notion is more technical, as it involves work-ing with matrix convex sets and matrix state spaces. It is possible todevelop a similar notion, but using only usual convex sets instead ofmatrix convex ones, and use this in the case of commutative crossedproducts C ( X ) ⋊ λ G . However, [Kaw17], which deals with provingequivalences of simplicity of such crossed products, does not developsuch a theory of generalized boundaries. Instead, the generalized no-tion of boundary necessary here is developed in [Nag20, Section 3],albeit from the perspective of compact sets and their measures rather than from convex sets. Briefly, we summarize the main notion andresults that we will use here.Assume C ( X ) ⊆ C ( Y ) is an inclusion of commutative unital G -C*-algebras. We say that Y is a ( G, X ) -boundary if whenever ν ∈ P ( Y ) issuch that ν | C ( X ) is contractible, then ν is contractible. Moreover, thespectrum of the G -injective envelope I G ( C ( X )), denoted ∂ F ( G, X ), isuniversal among (
G, X )-boundaries.The following lemma shows that, although measures on minimalspaces need not be contractible in general, they have the weaker prop-erty that arbitrary measures can still be pushed to Dirac masses using P ( G, C ( X )). Lemma 3.1.
Assume X is a minimal G -space, and fix any x ∈ X .There is a net ( µ λ ) ⊆ P f ( G, C ( X )) with the property that for any ν ∈ P ( X ) , we have νµ λ w* −→ δ x .Proof. Fix an open neighbourhood V of x . Observe that, given any y ∈ X , we have that Gy is dense in X by minimality. In particular,there exists some s ∈ G with the property that sy ∈ V , or equivalently, y ∈ s − V . It follows that the sets sV form an open cover of X , and sothere is some finite subcover s V, . . . , s n V . Now let F , . . . , F n ∈ C ( X )be a partition of unity subordinate to this open cover, let f i = F / i ,and let µ V = n X i =1 f i s i f i . It is not hard to see that, given any ν ∈ P ( X ), νµ V is a measure withsupport contained in V . It follows that the net ( νµ V ), indexed by openneighbourhoods of x ordered under reverse inclusion, converges weak*to δ x . (cid:3) This allows us to push arbitrary measures towards the trivial bound-ary in S ( C ( X ) ⋊ λ G ) in the case of simple crossed products. Proposition 3.2.
Let X be a minimal G -space, and assume that thecrossed product C ( X ) ⋊ λ G is simple. Then given any state φ ∈ S ( C ( X ) ⋊ λ G ) , we have that { ν ◦ E | ν ∈ P ( X ) } ⊆ { φµ | µ ∈ P f ( G, C ( X )) } w* . Proof.
For convenience, denote the latter set above by K . By G -invariance, convexity, and weak*-closure, it suffices to prove that K contains δ x ◦ E for any single point x ∈ X . OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 13
To this end, let I G ( C ( X )) = C ( ∂ F ( G, X )) be the G -injective enve-lope of C ( X ). Extend the state φ to a state e φ on C ( ∂ F ( G, X )) ⋊ λ G . ByLemma 3 .
1, we can find a net ( µ λ ) ⊆ P f ( G, C ( X )) with the propertythat e φ | C ( X ) µ λ → δ x for some x ∈ X . Dropping to a subnet if necessary,we have that e φµ λ → ψ ∈ S ( C ( ∂ F ( G, X )) ⋊ λ G ) with the property that ψ | C ( X ) = δ x . Observe that ψ | C ( X ) ⋊ λ G ∈ K .Minimality tells us that ψ | C ( X ) is contractible, and so ψ | C ( ∂ F ( G,X )) is contractible as well by [Nag20, Theorem 3.2]. This tells us thatthere is a net ( g i ) with ψ | C ( ∂ F ( G,X )) g i → δ y for some y ∈ ∂ F ( G, X ).Again dropping to a subnet yields a state η ∈ S ( C ( ∂ F ( G, X )) ⋊ λ G )with the property that η | C ( ∂ F ( G,X )) = δ y . Observe once more that η | C ( X ) ⋊ λ G ∈ K .We claim that η | C ( X ) ⋊ λ G is the state we are looking for. Simplicityof C ( X ) ⋊ λ G implies that the action of G on ∂ F ( G, X ) is free by[Kaw17, Theorem 3.4]. From here, the rest is a common argument. Weknow that C ( ∂ F ( G, X )) lies in the multiplicative domain of η . Thus,it suffices to show that η ( λ t ) = 0 for t = e , as this will imply that forany f ∈ C ( ∂ F ( G, X )), we have η ( f λ t ) = f ( y ) η ( λ t ) = 0 . Now let f ∈ C ( ∂ F ( G, X )) be such that f ( y ) = 1 and f ( ty ) = 0. Thisis possible because ty = y . We have f ( y ) η ( λ t ) = η ( f λ t ) = η ( λ t ( t − · f )) = η ( λ t ) f ( ty ) . This forces η ( λ t ) = 0, as desired. (cid:3) The fact that arbitrary functionals on a C*-algebra are a finite linearcombination of states gives us a similar result on the entire dual space( C ( X ) ⋊ λ G ) ∗ . Proposition 3.3.
Let X be a minimal G -space, and assume that thecrossed product is simple. Then given any ω ∈ ( C ( X ) ⋊ λ G ) ∗ , we have { ω (1) ν ◦ E | ν ∈ P ( X ) } ⊆ { ωµ | µ ∈ P f ( G, C ( X )) } w* . Proof.
Again, for convenience, denote this latter set by K . Write ω = P i =1 c i φ i , a linear combination of four states. By Proposition 3 .
2, wecan find a net ( µ λ ) ⊆ P f ( G, C ( X )) with the property that φ µ λ → ν ◦ E . Dropping to a subnet if necessary, we may also assume that ( φ i µ λ )are all convergent to some φ ′ i for i ≥
2. In particular, ( ωµ λ ) convergesto some ω ′ ∈ K with the property that ω ′ = c ν ◦ E + P i =2 c i φ ′ i . Notingthat the set { ν ◦ E | ν ∈ P ( X ) } is weak*-closed and closed under the right action of P f ( G, C ( X )), repeating this averaging trick three moretimes nets us (no pun intended) that there is some element in K of theform η ◦ E satisfying η (1) = ω (1).Writing η = P i =1 d i ψ i as a linear combination of four states on C ( X ), fixing x ∈ X , and letting ( µ j ) be as in Lemma 3 .
1, we have that( η ◦ E ) µ j = ( ηµ j ) ◦ E converges to ω (1) δ x ◦ E , which must lie in K .Minimality of X and G -invariance, weak*-closure, and convexity of K yield that every ω (1) ν ◦ E lies in K as well. (cid:3) From here, it is an application of the Hahn-Banach separation argu-ment that gives us the strong generalized Powers’ averaging property.Conversely, lack of nontrivial ideals can be directly deduced even fromjust being able to average elements a ∈ C ( X ) ⋊ λ G satisfying E ( a ) = 0. Proof of Theorem . . (1) = ⇒ (4) Given that the extreme points of P ( X ) are the Dirac masses δ x , it suffices to prove the following: if a ∈ C ( X ) ⋊ λ G and x ∈ X , then E ( a )( x ) ∈ { µa | µ ∈ P f ( G, C ( X )) } . Assume otherwise, so that there is some a ∈ C ( X ) ⋊ λ G and x ∈ X forwhich this doesn’t hold. Then there is some functional ω ∈ ( C ( X ) ⋊ λ G ) ∗ and α ∈ R with the property thatRe ω ( E ( a )( x )) < α ≤ Re ω ( µa ) ∀ µ ∈ P f ( G, C ( X )) . However, given that ω ( E ( a )( x )) = ω (1) E ( a )( x ), and by Proposition 3 . ω ( µa ) can be made arbitrarily close to ω (1)( δ x ◦ E )( a ), this cannothappen.(4) = ⇒ (3) Our aim is to show that we may approximate E ( a )by C ( X )-convex combinations of E ( a )( x ), where x ∈ X . Let a ∈ C ( X ) ⋊ λ G , and let ε >
0. Given any x ∈ X , by continuity of E ( a ) ∈ C ( X ), there is some open neighbourhood U x of x for which | E ( a )( x ) − E ( a )( y ) | < ε for all y ∈ U x . By compactness, there is somefinite subcover U x , . . . , U x n of X . Let F i be a partition of unity sub-ordinate to the open cover, and let f i = F / i . Observe that, given any OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 15 x ∈ X , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 f i ( x )( E ( a )( x i )) f i ( x ) − E ( a )( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 f i ( x )( E ( a )( x i ) − E ( a )( x )) f i ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n X i =1 f i ( x ) | E ( a )( x i ) − E ( a )( x ) | < n X i =1 f i ( x ) · ε = ε Thus, we have k P ni =1 f i E ( a )( x i ) f i − E ( a ) k < ε . By Remark 2 .
9, wehave that E ( a ) ∈ { µa | µ ∈ P f ( G, C ( X )) } , as this set is closed under C ( X )-convex combinations.(3) = ⇒ (2) This direction is clear.(2) = ⇒ (1) Let I be any nontrivial ideal of C ( X ) ⋊ λ G , and let a beany nonzero element of I . Replacing a by a ∗ a , we may assume withoutloss of generality that a is a nonzero positive element. Faithfulness ofthe canonical expectation tells us that E ( a ) is also a nonzero positiveelement, and so there is some ε > U ⊆ X with theproperty that E ( a )( x ) > ε for all x ∈ U . Using the same trick as in theproof of Lemma 3 .
1, minimality of X gives us that X = s U ∪ · · · ∪ s n U for finitely many s i ∈ G . Hence, replacing a by s a + · · · + s n a , wemay assume without loss of generality that E ( a ) > ε . If we choose µ ∈ P ( G, C ( X )) so that k µ ( a − E ( a )) k < ε , then as this value is inparticular self-adjoint, we have that µ ( a − E ( a )) ≥ − ε . Consequently, µa = µ ( E ( a )) + µ ( a − E ( a )) ≥ ε − ε ε . In particular, µa ∈ I is invertible, which gives us that I is the entirecrossed product C ( X ) ⋊ λ G . (cid:3) Unique stationarity and applications
This section generalizes the various results in [HK17] on equivalencebetween C*-simplicity and unique stationarity of the canonical tracein C ∗ λ ( G ), along with its consequences.It is worth noting that one cannot expect simplicity of C ( X ) ⋊ λ G to be equivalent to unique stationarity of an element of S ( C ( X ) ⋊ λ G ),even with respect to a generalized measure µ ∈ P ( G, C ( X )). This is because of the fact that there may not exist a uniquely stationarystate on C ( X ), and any µ -stationary state on C ( X ) will extend to oneon the whole crossed product. The natural fix is to instead expectthat the µ -stationary states on C ( X ) ⋊ λ G all be of the form ν ◦ E ,where ν ranges over the µ -stationary measures ν ∈ P ( X ). It is alsoworth noting that one cannot expect to work with the usual notionof measure µ ∈ P ( G ), as this would again imply unique stationarityof τ λ ∈ S ( C ∗ λ ( G )). However, this is equivalent to C*-simplicity of G [HK17, Theorem 5.2], which is by no means necessary for the crossedproduct C ( X ) ⋊ λ G to be simple - take, for example, C ( T ) ⋊ λ Z , where Z acts on the circle T by an irrational rotation.We begin with the observation that averaging elements in the re-duced group C*-algebra C ∗ λ ( G ), even with respect to a generalizedmeasure µ ∈ P ( G, C ( X )), is enough to average elements in the crossedproduct C ( X ) ⋊ λ G . Lemma 4.1.
Let X be a minimal G -space, and let µ ∈ P ( G, C ( X )) .Then given any t ∈ G and f ∈ C ( X ) , we have k µ ( f λ t ) k ≤ k f k k µλ t k . Proof.
It is well-known that the crossed product ℓ ∞ ( G ) ⋊ λ G (the uni-form Roe algebra), can canonically be viewed as a C*-subalgebra of B ( ℓ ( G )). Fixing x ∈ X gives us a unital G -equivariant injective *-homomorphism ι : C ( X ) ֒ → ℓ ∞ ( G ), given by ι ( f )( t ) = f ( tx ). Thislets us view C ( X ) ⋊ λ G as a C*-subalgebra of B ( ℓ ( G )) as well.Write µ = P i ∈ I g i s i g i . Given ξ ∈ ℓ ( G ) and r ∈ G , we have(( µ ( f λ t )) ξ )( r ) = X i ∈ I g i ( s i f )( s i ts − i g i ) λ s i ts − i ξ ! ( r )= X i ∈ I g i ( rx ) f ( s − i rx ) g i ( s i t − s − i rx ) ξ ( s i t − s − i r ) . Now letting | ξ | ∈ ℓ ( G ) be given by | ξ | ( r ) = | ξ ( r ) | , we note that k| ξ |k = k ξ k . Moreover, we have k ( µ ( f λ t )) ξ k = X r ∈ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ I g i ( rx ) f ( s − i rx ) g i ( s i t − s − i rx ) ξ ( s i t − s − i r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k f k X r ∈ G X i ∈ I g i ( rx ) g i ( s i t − s − i rx ) (cid:12)(cid:12)(cid:12) ξ ( s i t − s − i r ) (cid:12)(cid:12)(cid:12)! = k f k k ( µλ t ) | ξ |k . OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 17
It follows that k µ ( f λ t ) k ≤ k f k k µλ t k . (cid:3) It is also an easy remark that Powers’ averaging property can bemade to work with finitely many elements at once.
Lemma 4.2.
Assume C ( X ) ⋊ λ G has Powers’ averaging property.Then given any a , . . . , a n ∈ C ( X ) ⋊ λ G satisfying E ( a i ) = 0 , and ε > , there is some µ ∈ P ( G, C ( X )) with the property that k µa i k < ε for all i = 1 , . . . , n .Proof. Let µ ∈ P ( G, C ( X )) be such that k µ a k < ε . Choosing µ k +1 ∈ P ( G, C ( X )) inductively by letting µ k +1 be such that k µ k +1 ( µ k . . . µ a k +1 ) k <ε , we see that µ = µ n . . . µ is the generalized measure we are lookingfor. (cid:3) Proof of Theorem . . First, we claim that there is such a measure thatworks for all elements a ∈ C ∗ λ ( G ) ⊆ C ( X ) ⋊ λ G satisfying τ λ ( a ) = 0.This is a near-verbatim repeat of the proof of [HK17, Theorem 5.1].We repeat the construction of µ here, along with the appropriate mod-ifications.Let ( n k ) be an increasing sequence of positive integers satisfying (cid:16)P ki =1 12 i (cid:17) n k < k , and let ( a i ) be any dense sequence in the unit ball ofker τ λ ⊆ C ∗ λ ( G ). Let µ ∈ P ( G, C ( X )) be anything. Using Lemma 4 . µ l for l ≥ k µ l µ k r . . . µ k a s k < l for all 1 ≤ s, k , . . . , k r < l and 0 ≤ r < n l . Here, by r = 0, we meanthat µ l µ k r . . . µ k a s becomes µ l a s . A tedious computation then showsthat µ = P ∞ l =1 12 l µ l will satisfy µ n a → a ∈ ker τ λ .To force µ to have full support, observe that if ( s n ) n ∈ N is an enumer-ation of G , then the measure ν = ∞ X n =1 n +1 s n n +1 ∈ P ( G, C ( X ))has full support. Then fixing any l and letting α > µ l by αν + (1 − α ) µ l and still satisfy the requiredapproximation properties above. Thus, without loss of generality, some µ l has full support, and hence so does µ .Now, to see that µ n a → a ∈ C ( X ) ⋊ λ G satisfies E ( a ) = 0,we first prove this for elements a = f λ t + · · · + f n λ t n , where t i = e . Note that by Lemma 4 .
1, we have k µ n a k ≤ n X i =1 k µ ( f i λ t i ) k ≤ n X i =1 k f i k k µλ t i k → . Now given an arbitrary a with E ( a ) = 0, and ε >
0, we may choose a as before with k a − a k < ε . Choosing N such that, given n ≥ N , wehave k µ n a k < ε , we also have k µ n a k ≤ k µ n a k + k µ n ( a − a ) k < ε + ε = 2 ε. (cid:3) Remark 4.3.
In the above proof, if we instead wanted to directlyconstruct a generalized measure µ ∈ P ( G, C ( X )) with the propertythat µ n a → a ∈ C ( X ) ⋊ λ G with E ( a ) = 0, as opposed to a ∈ C ∗ λ ( G ) with τ λ ( a ) = 0, we would have required separability of ker E ⊆ C ( X ) ⋊ λ G , which requires separability of C ( X ) (metrizability of X ).Proceeding with ker τ λ ⊆ C ∗ λ ( G ) first and then lifting the averagingto the entire crossed product avoids this extra assumption. There arenatural examples of spaces on which G acts that are not metrizable.For example, if G is not amenable, then the Furstenberg boundary ∂ F G is such a space [KK17, Corollary 3.17].For a minimal G -space X , it is well known that if µ ∈ P ( G ) has fullsupport, then any µ -stationary state on C ( X ) is faithful. A similarresult holds for generalized probability measures (with the definitionof full support given in Definition 2 . Lemma 4.4.
Let X be a minimal G -space. Let µ ∈ P ( G, C ( X )) be a generalized probability measure with full support. Then every µ -stationary state on C ( X ) is faithful.Proof. Let f ∈ C ( X ) be such that f ≥ f = 0. Let µ = P s ∈ G P i ∈ I s f i sf i be a generalized probability measure with full sup-port. Since f is nonzero, there exists x ∈ X such that f ( x ) > X being minimal that, for every x ∈ X , there exists s x ∈ G such that f ( s − x x ) >
0. Moreover, since µ has full support,there exists i x ∈ I s x such that f i ( x ) >
0. Therefore, µ ( f )( x ) = X s ∈ G X i ∈ I s f i ( x ) f ( s − x ) f i ( x ) > f i x ( x ) f ( s − x x ) f i x ( x ) > . By compactness of X , it follows that there exists a δ > µ ( f ) ≥ δ . Consequently, for any µ -stationary state τ on C ( X ), we see OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 19 that τ ( f ) = τ ( µ ( f )) ≥ δ. Hence, τ is faithful. (cid:3) Proof of Corollary . . Let X be a minimal G -space. Suppose that C ( X ) ⋊ λ G is simple. Let µ ∈ P ( G, C ( X )) be the generalized mea-sure obtained from Theorem 1 . τ be a µ -stationary state on C ( X ) ⋊ λ G . Then, for any a ∈ C ( X ) ⋊ λ G with E ( a ) = 0, we have that τ ( a ) = τ ( µ n a ) → τ (0) = 0 , and so for general a ∈ C ( X ) ⋊ λ G , we have τ ( a ) = τ ( E ( a )) + τ ( a − E ( a )) = τ ( E ( a )) . In other words, τ = τ | C ( X ) ◦ E .On the other hand, suppose that there exists a generalized probabil-ity measure µ ∈ P ( G, C ( X )) with full support along with the propertythat every µ -stationary state on C ( X ) ⋊ λ G is of the form ν ◦ E forsome µ -stationary ν ∈ P ( X ). By faithfulness of E and Lemma 4 . µ -stationary state on C ( X ) ⋊ λ G is faithful. This is enough toguarantee that the C ( X ) ⋊ λ G is simple - the proof is similar to [HK17,Proposition 4.9].Assume that there was a nontrivial ideal I ⊆ C ( X ) ⋊ λ G . Observethat the quotient map π : C ( X ) ⋊ λ G → ( C ( X ) ⋊ λ G ) /I is nonfaithful,as nontrivial ideals always contain nonzero positive elements. More-over, the quotient ( C ( X ) ⋊ λ G ) /I is canonically a G -C*-algebra, with t ∈ G acting by Ad π ( λ t ), and the quotient map π is G -equivariant.In particular, we still canonically have C ( X ) ⊆ ( C ( X ) ⋊ λ G ) /I (underthe quotient map π ) by minimality of X , and so there is at least one µ -stationary state φ ∈ S µ (( C ( X ) ⋊ λ G ) /I ) by Proposition 2 .
12. Thecomposition φ ◦ π is a µ -stationary state on C ( X ) ⋊ λ G that is notfaithful, contradicting our earlier conclusion. (cid:3) One should notice that G -simplicity doesn’t necessarily pass to sub-algebras and therefore, simplicity for invariant sub-algebras of simplecrossed products shouldn’t be expected to hold in general. Consider,for example, any simple C*-algebra A , any C ∗ -simple group G actingon A trivially, and any abelian C ∗ -subalgebra B ⊆ A . However, givenan inclusion of unital G -C*-algebras C ( Y ) ⊂ C ( X ) (via a factor map π : X → Y ), since any G -invariant C*-subalgebra A , C ( Y ) ⊂ A ⊂ C ( X ) is of the form C ( Z ) where Z is an equivariant factor of X , andminimality passes to factors, it follows from the characterization of Kawabe [Kaw17, Theorem 6.1] that C ( Z ) ⋊ λ Γ is simple. We followarguments similar to the proof of [AK20, Theorem 1.3] to deal withgeneral intermediate C*-subalgebras between C ( Y ) ⋊ λ G and C ( X ) ⋊ λ G , not necessarily of the above form. Proof of Theorem . . By Theorem 1 .
3, there exists a generalized mea-sure µ ∈ P ( G, C ( Y )) with full support and the property that µ n a → a ∈ C ( Y ) ⋊ λ G is such that E ( a ) = 0. Observe that we canon-ically have P ( G, C ( Y )) ⊆ P ( G, C ( X )). Since X is minimal and µ hasfull support, it follows from Lemma 4 . µ -stationary state ν on C ( X ) is faithful, and since E is also faithful, it follows that every µ -stationary state on C ( X ) ⋊ λ G of the form ν ◦ E is faithful. We claimthat the proof is complete once we establish that every µ -stationarystate τ on C ( X ) ⋊ λ G is of the form ν ◦ E .Indeed, if this is the case, let A be any intermediate C*-algebra of theform C ( Y ) ⋊ λ G ⊆ A ⊆ C ( X ) ⋊ λ G . Suppose that I is a proper closedtwo-sided ideal of A . Then the action of G on A induces an action of G on A/I (as I is necessarily G -invariant). Moreover, by minimalityof Y , we also canonically have C ( Y ) ⊆ A/I . By Proposition 2 . µ -stationary state ϕ on A/I . Upon composing ϕ withthe canonical quotient map A → A/I , we obtain a µ -stationary state e ϕ on A which vanishes on I . Using Proposition 2 .
12 again, extend e ϕ to a µ -stationary state τ on C ( X ) ⋊ λ G . By our assumption, τ beingof the form ν ◦ E , is faithful. But τ | I = 0, which cannot occur if I isnontrivial.We return to the question of showing that every µ -stationary state on C ( X ) ⋊ λ G is indeed of the form ν ◦ E for some µ -stationary ν ∈ P ( X ).We claim that µ n a → a ∈ C ( X ) ⋊ λ G (not just C ( Y ) ⋊ λ G )satisfies E ( a ) = 0. To see this, first let f ∈ C ( X ) and t = e . Lemma 4 . k µ n ( f λ t ) k ≤ k f k k µ n λ t k → . It follows that for finite linear combinations a = f λ t + · · · + f n λ t n ,where f i ∈ C ( X ) and t i = e , we have µ n a → a ∈ C ( X ) ⋊ λ G with E ( a ) = 0, ε >
0, and a as before with theadditional property that k a − a k < ε . Then given N ∈ N such that k µ n a k < ε for any n ≥ N , we have k µ n a k ≤ k µ n a k + k µ n ( a − a ) k < ε + ε = 2 ε. It follows that µ n a →
0. The proof of Corollary 1 . µ -stationary state on C ( X ) ⋊ λ G is of the form we want. (cid:3) OWERS AVERAGING FOR COMMUTATIVE CROSSED PRODUCTS 21
With Theorem 1 . G on the space of amenable subgroups Sub a ( G ). Recall thatKawabe [Kaw17, Theorem 5.2] introduced the G -space Sub a ( X, G ) ofpairs ( x, H ), where x ∈ X and H is an amenable subgroup of the stabi-lizer group G x . Observe that the canonical projection Sub a ( X, G ) ։ X induces an inclusion C ( X ) ⊆ C (Sub a ( X, G )).
Proof of Corollary . . There is a G -equivariant, unital and completelypositive map θ : C ( X ) ⋊ λ G → C (Sub a ( X, G )) given by θ ( f λ t )( x, H ) = f ( x )1 H ( t ). A similar map can be found used in the proof of [Kaw17,Theorem 5.2], but a proof of the existence of such a map is not given.We briefly argue existence here. Given any ( x, H ) ∈ Sub a ( X, G ), itis not hard to show that there is a state φ ∈ S ( C ( X ) ⋊ λ G ) given by φ ( f λ t ) = f ( x )1 H ( t ). This gives us a continuous map from Sub a ( X, G )to S ( C ( X ) ⋊ λ G ), and θ : C ( X ) ⋊ λ G → C (Sub a ( X, G )) is dual to thismap.Choose a generalized measure µ ∈ P ( G, C ( X )) as in Corollary 1 . µ -stationary η in P (Sub a ( X, G )), we have that η ◦ θ : C ( X ) ⋊ λ G → C is necessarily of the form ν ◦ E . In particular, we notethat for t = e , η ( { ( x, H ) | t ∈ H } ) = η ( θ ( λ t )) = 0 . Countability of G gives us that S t = e { ( x, H ) | t ∈ H } is also a null set,or in other words, its complement X × has measure 1.Conversely, assume that the crossed product C ( X ) ⋊ λ G is not simple.Then by [Kaw17, Theorem 6.1], there must exist a closed G -invariantsubset of Z ⊆ Sub a ( X, G ) that does not intersect X × . Ob-serve that we still canonically have C ( X ) ⊆ C ( Z ) by minimality of X .Thus, for any µ ∈ P ( G, C ( X )), if we choose any µ -stationary state on C ( Z ) (such a state always exists by Proposition 2 . C (Sub a ( X, G )) ։ C ( Z ) gives us a µ -stationarystate on C (Sub a ( X, G )) with support disjoint from X × . (cid:3) References [AK20] Tattwamasi Amrutam and Mehrdad Kalantar,
On simplicity of interme-diate C*-algebras , Ergodic Theory and Dynamical Systems (2020),no. 12, 3181–3187.[BK16] Rasmus Sylvester Bryder and Matthew Kennedy, Reduced Twisted CrossedProducts over C*-Simple Groups , International Mathematics Research No-tices (2016), no. 6, 1638–1655. [Haa16] Uffe Haagerup,
A new look at C*-simplicity and the unique trace propertyof a group , Operator Algebras and Applications (Cham) (Toke M. Carlsen,Nadia S. Larsen, Sergey Neshveyev, and Christian Skau, eds.), SpringerInternational Publishing, 2016, pp. 167–176.[HK17] Yair Hartman and Mehrdad Kalantar,
Stationary C*-dynamical systems ,arXiv e-prints (2017), arXiv:1712.10133, to appear in Journal of the Eu-ropean Mathematical Society.[Kaw17] Takuya Kawabe,
Uniformly recurrent subgroups and the ideal structure ofreduced crossed products , arXiv e-prints (2017), arXiv:1701.03413.[Ken15] Matthew Kennedy,
An intrinsic characterization of C*-simplicity , arXive-prints (2015), arXiv:1509.01870, to appear in Annales Scientifiques del’École Normale Supérieure.[KK17] Mehrdad Kalantar and Matthew Kennedy,
Boundaries of reduced C*-algebras of discrete groups , Journal für die reine und angewandte Mathe-matik (Crelles Journal) (2017), no. 727, 247–267.[KS19] Matthew Kennedy and Christopher Schafhauser,
Noncommutative bound-aries and the ideal structure of reduced crossed products , Duke Mathemat-ical Journal (2019), no. 17, 3215–3260.[Nag20] Zahra Naghavi,
Furstenberg Boundary of Minimal Actions , Integral Equa-tions and Operator Theory (2020), no. 2. Department of Mathematics, University of Houston, USA
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