A Generalization of Whyburn's Theorem, and Aperiodicity for Abelian C*-Inclusions
aa r X i v : . [ m a t h . OA ] N ov A GENERALIZATION OF WHYBURN’S THEOREM, ANDAPERIODICITY FOR ABELIAN C ∗ -INCLUSIONS VREJ ZARIKIAN
Abstract.
Let j : Y → X be a continuous surjection of compact metricspaces. Whyburn proved that j is irreducible , meaning that j ( F ) ( X for anyproper closed subset F ( Y , if and only if j is almost one-to-one , in the sensethat { y ∈ Y : j − ( j ( y )) = y } = Y. In this note we prove the following generalization: There exists a unique min-imal closed set K ⊆ Y such that j ( K ) = X if and only if { x ∈ X : card( j − ( x )) = 1 } = X. Translated to the language of operator algebras, this says that if
A ⊆ B is aunital inclusion of separable abelian C ∗ -algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extensionproperty of Nagy-Reznikoff holds. More generally, we prove that a unital in-clusion of (not necessarily separable) abelian C ∗ -algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwa´sniewski-Meyer). Introduction
Let j : Y → X be a continuous surjection of compact Hausdorff spaces. Following[2], we say that j is almost one-to-one if { y ∈ Y : j − ( j ( y )) = y } = Y. We say that j is irreducible if j ( F ) ( X for all proper closed subsets F ( Y . It iseasy to see that j one-to-one = ⇒ j almost one-to-one = ⇒ j irreducible . In general, an irreducible map need not be almost one-to-one (see Example 5.1below). If, however, Y is metrizable, then Whyburn’s Theorem [11, Theorem 2]states that irreducible maps are necessarily almost one-to-one, so that the two no-tions coincide.Irreducible maps and almost one-to-one maps arise naturally in topological dy-namics. Indeed, a minimal self-map of a compact Hausdorff space must be irre-ducible [5, Lemma 2.1]. Thus a minimal self-map of a compact metric space mustbe almost one-to-one, by Whyburn’s Theorem (see also [5, Theorem 2.7]). Date : Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.2010
Mathematics Subject Classification.
Primary 54E40; Secondary 46J10.
Key words and phrases. irreducible map, almost one-to-one map, quasicontinuous selection,abelian C ∗ -algebra, unique pseudo-expectation, almost extension property, aperiodicity. In Theorem 4.2 below, we generalize Whyburn’s Theorem, proving that for acontinuous surjection j : Y → X of compact metric spaces, there exists a uniqueminimal closed set K ⊆ Y such that j ( K ) = X if and only if { x ∈ X : card( j − ( x )) = 1 } = X. To see that this is a generalization, suppose that j is irreducible. Then there existsa unique minimal closed set K ⊆ Y such that j ( K ) = X , namely K = Y . ByTheorem 4.2, X = X , where X = { x ∈ X : card( j − ( x )) = 1 } . If Y = { y ∈ Y : j − ( j ( y )) = y } , then j ( Y ) = X and j ( Y ) = X = X . Thus Y = Y , by irreducibility. So j isalmost one-to-one.As the following example shows, Theorem 4.2 is a strict generalization of Why-burn’s Theorem. Example 1.1.
Let Y = ([0 , × { } ) ∪ ( { } × [0 , ⊆ R ,X = [0 , ⊆ R , and j : Y → X be defined by j ( x, y ) = x. Then there exists a unique minimal closed set K ⊆ Y such that j ( K ) = X , namely K = [0 , × { } , and { x ∈ X : card( j − ( x )) = 1 } = [0 ,
1) = [0 ,
1] = X. Clearly j is not irreducible. It is not almost one-to-one because { ( x, y ) ∈ Y : j − ( j ( x, y )) = ( x, y ) } = [0 , × { } = [0 , × { } ( Y. When translated into the language of operator algebras, Theorem 4.2 says thatfor a unital inclusion
A ⊆ B of separable abelian C ∗ -algebras, there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension prop-erty of Nagy-Reznikoff holds. (All these terms will be defined in Section 6 below.)This answers a question posed to the author by Ruy Exel and Bartosz Kwa´sniewskifollowing his 2019 talk at the International Workshop on Operator Theory and itsApplications in Lisbon, Portugal.More generally, we prove (Theorem 6.2) that a unital inclusion A ⊆ B of (not nec-essarily separable) abelian C ∗ -algebras has a unique pseudo-expectation if and onlyif the inclusion is aperiodic (in the sense of Kwa´sniewski and Meyer). Aperiodicityis weaker than the almost extension property [7, Theorem 5.5], but stronger thanhaving a unique pseudo-expectation [7, Theorem 3.6]. For separable C ∗ -inclusions,aperiodicity and the almost extension property coincide [7, Theorem 5.5]. Thus GENERALIZATION OF WHYBURN’S THEOREM 3
Theorem 4.2 is actually a corollary of Theorem 6.2, modulo a non-trivial operator-algebraic result, as illustrated in the diagram below:almost extension property [7, Thm. 5.5] + aperiodicity [7, Thm. 3.6] + separable [7, Thm 5.5] ` h unique pseudo-expectation abelian (Thm. 6.2) ` h separable abelian (Thm. 4.2) q y Nonetheless, we include a direct, purely-topological proof of Theorem 4.2 that doesnot rely on Theorem 6.2 nor on [7, Theorem 5.5].
Acknowledgements:
We thank Alexis Alevras and Marcos Valdes for helpfuldiscussions about this material. 2.
Preliminaries
Let j : Y → X be a continuous surjection of compact Hausdorff spaces. Inthis section we show that there must exist a minimal closed set K ⊆ Y such that j ( K ) = X (Proposition 2.1) and we establish a criterion for determining when K is unique (Proposition 2.2). Proposition 2.1.
Let j : Y → X be a continuous surjection of compact Hausdorffspaces. Then there exists a minimal closed set K ⊆ Y such that j ( K ) = X .Proof. Let F = { F ⊆ Y : F is closed and j ( F ) = X } , a nonempty set partially ordered by reverse inclusion. Suppose L ⊆ F is anonempty totally ordered subset and define F = \ { F : F ∈ L} . If x ∈ X , then { F ∩ j − ( x ) : F ∈ L} is a collection of closed subsets of Y with the finite intersection property. Since Y is compact, F ∩ j − ( x ) = \ { F ∩ j − ( x ) : F ∈ L} 6 = ∅ . Since x was arbitrary, j ( F ) = X . So F ∈ F is an upper bound for L . By Zorn’sLemma, F has a maximal element. (cid:3) Proposition 2.2.
Let j : Y → X be a continuous surjection of compact Hausdorffspaces. Then the following are equivalent: i. There exists a unique minimal closed set K ⊆ Y such that j ( K ) = X . ii. If F = T { F ⊆ Y : F is closed and j ( F ) = X } , then j ( F ) = X .Proof. Let F = \ { F ⊆ Y : F is closed and j ( F ) = X } . (i = ⇒ ii) Suppose there exists a unique minimal closed set K ⊆ Y such that j ( K ) = X . If F ⊆ Y is closed and j ( F ) = X , then K ⊆ F . Thus K ⊆ F . Infact, F = K , since F ⊆ K by definition. Therefore j ( F ) = j ( K ) = X . (ii = ⇒ VREJ ZARIKIAN i) Conversely, suppose j ( F ) = X . Let K ⊆ Y be a minimal closed set such that j ( K ) = X . Then F ⊆ K by definition and so K = F by minimality. (cid:3) Quasicontinuous Selections
Let j : Y → X be a surjection. A selection of j is a function α : X → Y such that j ◦ α = id X . It is well-known that continuous surjections of compact metric spaceshave Borel selections (see [1, Theorem 3.4.1.], for example). On the other hand,a continuous selection is usually impossible. In this section, we draw attentionto a theorem of Crannell, Frantz, and LeMasurier, which states that continuoussurjections of compact metric spaces have quasicontinuous selections. This resultplays a critical role in our proof of Theorem 4.2 below. Definition 3.1 (quasiopen) . Let Y be a topological space. We say that A ⊆ Y is quasiopen if A ⊆ A ◦ . Definition 3.2 (quasicontinuous) . We say that a mapping j : Y → X of topologicalspaces is quasicontinuous if for every open set U ⊆ X , j − ( U ) ⊆ Y is quasiopen. Theorem 3.3 ([3], Corollary 5) . Let j : Y → X be a continuous surjection ofcompact metric spaces. Then j admits a quasicontinuous selection α : X → Y . A Generalization of Whyburn’s Theorem
In this section we state and prove our main result, Theorem 4.2. We start withan elementary lemma.
Lemma 4.1.
Let j : Y → X be a continuous surjection of compact metric spaces.Then the mapping X → R : x diam( j − ( x )) is upper semi-continuous.Proof. Let ε >
0. Suppose that x n → x , where diam( j − ( x n )) ≥ ε for all n ∈ N .For each n ∈ N , there exist y n , y ′ n ∈ j − ( x n ) such that d ( y n , y ′ n ) > ε − /n . Passingto a subsequence, if necessary, we may assume that y n → y and y ′ n → y ′ . Then y, y ′ ∈ j − ( x ) and d ( y, y ′ ) ≥ ε . Thus diam( j − ( x )) ≥ ε . (cid:3) Theorem 4.2.
Let j : Y → X be a continuous surjection of compact metric spaces.Then the following are equivalent: i. There exists a unique minimal closed set K ⊆ Y such that j ( K ) = X . ii. { x ∈ X : card( j − ( x )) = 1 } is dense in X .Proof. (i = ⇒ ii) Suppose there exists a unique minimal closed set K ⊆ Y suchthat j ( K ) = X . Let ε >
0. Assume that { x ∈ X : diam( j − ( x )) < ε } is not densein X . Then there exists x ∈ X and r > B r ( x ) ⊆ { x ∈ X : diam( j − ( x )) ≥ ε } . Let α : X → Y be a quasicontinuous selection of j . Then α ( x ) ∈ j − ( B r ( x )) andso there exists δ > B δ ( α ( x )) ⊆ j − ( B r ( x )). Then α − ( B δ ( α ( x )) isa quasiopen set containing x . Thus U = α − ( B δ ( α ( x ))) ◦ is a nonempty open subset of X . Now α − ( B δ ( α ( x ))) = j ( B δ ( α ( x )) ∩ α ( X )) , GENERALIZATION OF WHYBURN’S THEOREM 5 and so there exists A ⊆ B δ ( α ( x )) ∩ α ( X ) such that j ( A ) = U . In particular,diam( A ) ≤ δ . On the other hand, U = j ( A ) ⊆ j ( B δ ( α ( x ))) ⊆ B r ( x ) ⊆ { x ∈ X : diam( j − ( x )) ≥ ε } . Thus for every y ∈ A , there exists y ′ ∈ Y such that j ( y ′ ) = j ( y ) and d ( y ′ , y ) > ε/ d ( y ′ , A ) ≥ ε/ − δ . Setting A ′ = { y ′ : y ∈ A } , we see that j ( A ′ ) = U and d ( A ′ , A ) ≥ ε/ − δ . Choosing δ < ε/
16, we conclude that d ( A ′ , A ) > ε/ A ∩ A ′ = ∅ . But then F = A ∪ j − ( U c ) and F ′ = A ′ ∪ j − ( U c )are closed sets such that j ( F ) = X , j ( F ′ ) = X , and j ( F ∩ F ′ ) = U c ( X , whichcontradicts Proposition 2.2. By Lemma 4.1, { x ∈ X : diam( j − ( x )) < ε } is anopen dense subset of X . Thus { x ∈ X : card( j − ( x )) = 1 } = ∞ \ n =1 { x ∈ X : diam( j − ( x )) < /n } is dense in X .(ii = ⇒ i) Suppose X = { x ∈ X : card( j − ( x )) = 1 } is dense in X . Define K = j − ( X ). If F ⊆ Y is closed and j ( F ) = X , then j − ( X ) ⊆ F , which implies K ⊆ F . Since X = X = j ( j − ( X )) ⊆ j ( K ) = j ( K ) , we see that K is the unique minimal closed subset of Y such that j ( K ) = X . (cid:3) The Split Interval
In Whyburn’s theorem, the assumption of metrizability cannot be omitted [4,Exercise 3.1.C]. Of course, the same is true for Theorem 4.2, as the following ex-ample illustrates.
Example 5.1.
Let ¨ I be the “split interval”, i.e., the set((0 , × { } ) ∪ ([0 , × { } ) ⊆ R equipped with the order topology arising from the lexicographic order( x, y ) < ( x ′ , y ′ ) ⇐⇒ ( x < x ′ ) ∨ (( x = x ′ ) ∧ ( y < y ′ )) . Let I = [0 ,
1] be the unit interval equipped with its usual topology. Then j : ¨ I → I defined by j ( x, y ) = x is an irreducible continuous surjection of compact Hausdorff spaces, but { x ∈ I : card( j − ( x )) = 1 } = { , } , which is not dense in I . Of course, ¨ I is not metrizable (it is not second countable).6. Aperiodicity for Abelian C ∗ -Inclusions Let
A ⊆ B be a unital inclusion of C ∗ -algebras (cf. [1]). Then A ⊆ B has the pure extension property (PEP) if every pure state on A extends uniquely to a purestate on B . A useful relaxation of the PEP is the almost extension property (AEP)of Nagy and Reznikoff [8]. Instead of insisting that every pure state on A extendsuniquely to a pure state on B , the requirement is that a weak-* dense collection ofpure states on A extends uniquely to pure states on B . A conditional expectation (CE) of B onto A is a completely positive map E : B → A such that E | A = id A .A pseudo-expectation (PsExp) for the C ∗ -inclusion A ⊆ B is a completely positive
VREJ ZARIKIAN map θ : B → I ( A ) such that θ | A = id A [9, Definition 1.3]. Here I ( A ) is Hamana’s injective envelope of A , the minimal injective operator system containing A . Everyconditional expectation is a pseudo-expectation, but there are C ∗ -inclusions withno conditional expectations, while pseudo-expectations always exist. We have therelations(6.1) PEP = ⇒ AEP = ⇒ ∃ ! PsExp = ⇒ ∃ at most one CE . Only the middle implication is not immediate from the definitions.It is somewhat surprising that the differences between the conditions in (6.1)above, in particular the difference between the almost extension property and hav-ing a unique pseudo-expectation, can be witnessed by abelian C ∗ -inclusions. In-deed, let A ⊆ B be a unital inclusion of abelian C ∗ -algebras. By the Gelfand-Naimark Theorem, A ∼ = C ( X ), the continuous complex-valued functions on a com-pact Hausdorff space X (cf. [1, Theorem 1.7.3]). Likewise B ∼ = C ( Y ) for somecompact Hausdorff space Y . Under these identifications, the inclusion map A → B corresponds to the map C ( X ) → C ( Y ) : f f ◦ j for some continuous surjection j : Y → X . It is easy to see that A ⊆ B has the almost extension property if andonly if { x ∈ X : card( j − ( x )) = 1 } = X. By [10, Corollary 3.21],
A ⊆ B has a unique pseudo-expectation if and only if thereexists a unique minimal closed set K ⊆ Y such that j ( K ) = X . Thus the abelian C ∗ -inclusion C ( I ) ⊆ C (¨ I ) corresponding to the continuous surjection j : ¨ I → I de-scribed in Example 5.1 above has a unique pseudo-expectation, but not the almostextension property. On the other hand, by Theorem 4.2, for an abelian C ∗ -inclusion C ( X ) ⊆ C ( Y ) with Y compact metric, the almost extension property and havinga unique pseudo-expectation are equivalent. The metrizability of Y correspondsprecisely to the separability of C ( Y ), and so Theorem 4.2 says that for separableabelian C ∗ -inclusions A ⊆ B , the almost extension property is equivalent to havinga unique pseudo-expectation.Recently, Kw´asniewski and Meyer substantially advanced our understanding ofthe relationship between the almost extension property and having a unique pseudo-expectation. Namely they introduced the notion of aperiodicity for (arbitrary) C ∗ -inclusions [6, Definition 5.14], proved in [7, Theorems 5.5 and 3.6] that(6.2) AEP = ⇒ aperiodicity = ⇒ ∃ ! PsExp , and showed that for separable C ∗ -inclusions, the almost extension property is equiv-alent to aperiodicity [7, Theorem 5.5]. In Theorem 6.2 below, we prove thatfor abelian C ∗ -inclusions, aperiodicity is equivalent to having a unique pseudo-expectation. Combined with [7, Theorem 5.5], this gives another proof of Theorem4.2 above, our generalization of Whyburn’s Theorem.For a C ∗ -algebra A we denote by A +1 the positive norm-one elements of A , andby H ( A ) the non-zero hereditary subalgebras of A . Definition 6.1 (aperiodic bimodules and C ∗ -inclusions) . Let A be a unital C ∗ -algebra. Following [6]: GENERALIZATION OF WHYBURN’S THEOREM 7 • We say that a normed A -bimodule X is aperiodic if for all x ∈ X , ε > D ∈ H ( A ), there exists d ∈ D +1 such that k dxd k < ε. • We say that an inclusion
A ⊆ B of unital C ∗ -algebras is aperiodic if B / A is aperiodic as a normed A -module. Equivalently, if for all b ∈ B , ε > D ∈ H ( A ), there exists d ∈ D +1 and a ∈ A such that k dbd − a k < ε. Theorem 6.2.
For an abelian C ∗ -inclusion A ⊆ B , aperiodicity is equivalent tohaving a unique pseudo-expectation.Proof.
Let
A ⊆ B be an abelian C ∗ -inclusion with unique pseudo-expectation E : B → I ( A ). Then E is a ∗ -homomorphism and J = ker( E ) is the unique maximal A -disjoint ideal of B [10, Corollary 3.21]. To show that A ⊆ B is aperiodic, we mustshow that B / A is aperiodic as a normed A -module. Since the canonical maps( J + A ) / A → B / A → B / ( J + A )form an exact sequence of normed A -modules, it suffices to show that ( J + A ) / A ∼ = J and B / ( J + A ) ∼ = ( B / J ) / A are aperiodic as normed A -modules [6, Lemma5.12].The aperiodicity of the normed A -module ( B / J ) / A is equivalent to the aperi-odicity of the C ∗ -inclusion A ⊆ B / J . That, in turn, follows from the aperiodicityof the C ∗ -inclusion A ⊆ I ( A ) [7, Remark 3.17] and the fact that the induced map˙ E : B / J → I ( A ) is a ∗ -monomorphism.To show that J is aperiodic as a normed A -module, we identify the C ∗ -inclusion A ⊆ B with the C ∗ -inclusion C ( X ) ⊆ C ( Y ) arising from the continuous surjection j : Y → X . Then J = { g ∈ C ( Y ) : g | K = 0 } , where K ⊆ Y is the unique minimal closed set such that j ( K ) = X . Let g ∈ J , ε >
0, and
D ∈ H ( A ). Then D = { f ∈ C ( X ) : f | F = 0 } , where F ( X is a closed set. Define U = { y ∈ Y : | g ( y ) | < ε } , an open subsetof Y containing K . Assume that j ( U c ) ∪ F = X . Then K ′ = U c ∪ j − ( F ) is aclosed subset of Y such that j ( K ′ ) = X . It follows that K ⊆ K ′ , which implies K ⊆ j − ( F ), which in turn implies j ( K ) ⊆ F , a contradiction. So j ( U c ) ∪ F ( X .By Urysohn’s Lemma, there exists f ∈ C ( X ) +1 such that f | j ( U c ) ∪ F = 0. Then f ∈ D +1 and k ( f ◦ j ) g ( f ◦ j ) k < ε , which completes the proof. (cid:3) References
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Department of Mathematics, U. S. Naval Academy, Annapolis, MD 21402, USA.
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