Self-Similar Graph C*-Algebras and Partial Crossed Products
aa r X i v : . [ m a t h . OA ] J un Self-Similar Graph C*-Algebras and Partial CrossedProducts
Ruy Exel ∗ Charles Starling † Abstract
In a recent paper, Pardo and the first named author introduced a class of C*-algebras which which are constructed from an action of a group on a graph. Thisclass was shown to include many C*-algebras of interest, including all Kirchbergalgebras in the UCT class. In this paper, we study the conditions under which thesealgebras can be realized as partial crossed products of commutative C*-algebras bygroups. In addition, for any n ≥ H in this class, the Cuntz algebra O n is isomorphic to a partial crossedproduct of a commutative C*-algebra by H . In this paper, we study C*-algebras arising from self-similar graph actions . A self-similargraph action consists of a group G , a directed graph E , a “cocycle” ϕ : G × E → G , anda length-preserving action of G on E ∗ , the set of finite paths through E . Furthermore, theaction of G on E ∗ should be “self-similar”, in the sense that g ( eα ) = ( ge )( ϕ ( g, e ) α ) , g ∈ G, e ∈ E , α ∈ E ∗ . The C*-algebra associated to (
G, E, ϕ ), denoted O G,E is then the universal C*-algebragenerated by a Cuntz-Krieger family for E and a unitary representation of G , subject torelations given by the action. The main question this work addresses is the following: whencan O G,E be written as a partial crossed product of a commutative C*-algebra by a group?Some of the most powerful tools to analyze the structure of a given C*-algebra becomeavailable when one is able to describe it as a crossed product. Questions about represen-tation theory, structure of ideals, simplicity, nuclearity, K-theory, KMS states, and many ∗ Partially supported by CNPq (Brazil). † Supported by CNPq (Brazil). O G,E in [EP13] was to generalize two interestingclasses of C*-algebras. The first class is the C*-algebras that Nekrashevych associated toself-similar groups in [Nek04] and [Nek09]. In [EP13], it was shown that these algebrasarise from self-similar graph actions (
G, E, ϕ ) for which E is a finite graph with only onevertex. In the work of Nekrashevych and in other work on self-similar groups, the map ϕ takes the form of a restriction , ( g, x ) g | x .The second class generalized in [EP13] are the C*-algebras constructed by Katsura[Kat08] from pairs of integer matrices A and B , denoted O A,B . The pair (
A, B ) gives riseto a self-similar graph action ( Z , E A , ϕ ), where E A is the graph whose incidence matrix is A and the action of Z and the cocycle ϕ are determined by the entries of A and B . It isshown in [EP13], Example 3.4, that O A,B ∼ = O Z ,E A . From [Kat08], it is a fact that everyKirchberg algebra in the UCT class arises as O A,B for some A and B , so the algebras weconsider here constitute a large class.In [EP13], O G,E is realized as the C*-algebra of the tight groupoid of a certain inversesemigroup S G,E , using the theory of such algebras from [Exe08]. This groupoid is shownto be Hausdorff in the case where (
G, E, ϕ ) is pseudo-free . In Theorem 3.3, we show that(
G, E, ϕ ) is pseudo-free if and only if inverse semigroup S G,E is strongly E*-unitary . We thenuse the results of [MS11] to prove that when ( G, E, ϕ ) is pseudo-free, O G,E is isomorphicto a partial crossed product of a commutative C*-algebra by the universal group U ( S G,E )of S G,E .In our final section, we use our results to realize the Cuntz algebra O n as partial crossedproducts C ( Y ) ⋊ H for Y homeomorphic to the Cantor set and H the universal group of any S G,E satisfying certain conditions.
Let X be a finite set. We will denote by X ∗ the set of all words x x · · · x n where n ≥ x , x , . . . , x n ∈ X , together with the symbol ∅ , which is called the empty word . For α = α α , · · · α n ∈ X ∗ , we let | α | = n and call this the length of α – we take | ∅ | = 0. Wemay concatenate two words in X ∗ : if α, β ∈ X ∗ with α = α α · · · α n and β = β β · · · β m then their concatenation is given by αβ = α α · · · α n β β · · · β m .
2e also take, for any α ∈ X ∗ , α ∅ = ∅ α = α. This operation gives X ∗ the structure of a semigroup with identity (or monoid ) and is calledthe free monoid on X . We may also consider the space Σ X of infinite words in X . For anelement x ∈ Σ X and α ∈ X ∗ , the concatenation αx is again in Σ X . For α ∈ X ∗ , let α Σ X = { αx ∈ Σ X | x ∈ Σ X } . Sets of this type are called cylinder sets , and they generate the product topology on Σ X .In this topology, cylinder sets are both open and closed and Σ X is homeomorphic to theCantor set.A directed graph is a quadruple E = ( E , E , r, d ) where E and E are sets and r, d arefunctions from E to E . The set E is called the set of vertices of E and E is called theset of edges of E . A vertex x ∈ E is said to be a source if r − ( x ) = ∅ , and it is said to bea sink if d − ( x ) = ∅ . A directed graph E is called finite if E and E are finite sets. For n ≥ E n = { x x · · · x n | x i ∈ E , d ( x i ) = r ( x i +1 ) for 1 ≤ i ≤ n − } and take E ∗ = ∞ [ n =0 E n . The set E ∗ is called the set of finite paths in E . We may concatenate two paths α and β and obtain αβ if r ( β ) = d ( α ), taking the convention that d ( x ) = r ( x ) = x for all vertices x , and that r ( α ) α = α = αd ( α ). We will also consider the set of infinite paths Σ E = { e e · · · | e i ∈ E , s ( e i ) = r ( e i +1 ) for all i ≥ } given the product topology. With an abuse of notation, for α ∈ E ∗ we will let α Σ E = { αx | x ∈ E ∗ , r ( x ) = d ( α ) } and call these cylinder sets as well – sets of this type generate the topology on Σ E . If E has one vertex, we may refer to paths in E as “words”A semigroup S is called an inverse semigroup if for each s ∈ S there exists a uniqueelement s ∗ ∈ S such that s ∗ ss ∗ = s ∗ and ss ∗ s = s . An element e ∈ S is called an idempotent if e = e – the set of all such elements will be denoted E ( S ). It is true that, for all s, t ∈ S and e, f ∈ E ( S ) we have that ( st ) ∗ = t ∗ s ∗ , ( s ∗ ) ∗ = s , e ∗ = e , ef = f e and ef ∈ E ( S ). Forall s ∈ S , the elements ss ∗ and s ∗ s are idempotents.Recall that a groupoid is a set G together with a subset G (2) ⊂ G × G , called the set ofcomposable pairs, a product map G (2) → G with ( γ, η ) γη , and an inverse map from G to G with γ γ − such that 3. ( γ − ) − = γ for all γ ∈ G ,2. If ( γ, η ) , ( η, ν ) ∈ G (2) , then ( γη, ν ) , ( γ, ην ) ∈ G (2) and ( γη ) ν = γ ( ην ),3. ( γ, γ − ) , ( γ − , γ ) ∈ G (2) , and γ − γη = η , ξγγ − for all η, ξ with ( γ, η ) , ( η, ξ ) ∈ G (2) .Elements of the form γγ − are called units , and the set of all such elements is denoted G (0) and is called the unit space of G . The maps r : G → G (0) and d : G → G (0) defined by r ( γ ) = γγ − , d ( γ ) = γ − γ are called the range and source maps respectively. We note that ( γ, η ) ∈ G (2) if and only if r ( η ) = d ( γ ). A topological groupoid is a groupoid which is a topological space for which theinverse and product maps are continuous (where G (2) is given the product topology inheritedfrom G ×G ). An isomorphism of topological groupoids is a homeomorphism which preservesthe groupoid operations. A topological groupoid G is called ´etale if it is locally compact,second countable, and the maps r and d are local homeomorphisms. We note that theseimply that G (0) is open in G and that for all x ∈ G (0) the spaces G x := r − ( x ) , G x := d − ( x )are discrete.An open set S ⊂ G of a topological groupoid is called a slice if the restrictions of r and d to S are both injective. In an ´etale groupoid G , the collection of slices forms a basis forthe topology of G , cf. [Exe08], Proposition 3.5.´Etale groupoids can be constructed from actions on topological spaces. Recall that an action of an inverse semigroup S on a space X is a pair( { D e } e ∈ E ( S ) , { θ s } s ∈S )such that each D e ⊂ X is an open set, the union of the D e coincides with X , and the maps θ s : D s ∗ s → D ss ∗ are continuous bijections which satisfy θ s ◦ θ t = θ st , where the composition is on the largestdomain possible. Given an action θ of an inverse semigroup S on a space X , one may forman ´etale groupoid G ( S , X, θ ), called the groupoid of germs . As a set, G ( S , X, θ ) = { [ s, x ] | x ∈ D s ∗ s } where [ s, x ] are equivalence classes of elements of S × X such that [ s, x ] = [ t, y ] iff x = y and there exists some idempotent e ∈ E ( S ) such that x ∈ D e and se = te . One may always4ssume that s ∗ se = e and t ∗ te = e , and we note that this implies that θ s and θ t agree onthe neighborhood D e of x . The unit space of G ( S , X, θ ) is identified with X , and[ s, x ] − = [ s ∗ , θ s ( x )] , r ([ s, x ]) = θ s ( x ) , d ([ s, x ]) = x, [ t, θ s ( x )][ s, x ] = [ ts, x ] . A related construction arises from a group G acting on a topological space by partialhomeomorphisms. Let X be a set and let G be a group. Recall that a partial action of G on X is a pair ( { D g } g ∈ G , { θ g } g ∈ G )consisting of a collection { D g } g ∈ G of subsets of X , and a collection { θ g } g ∈ G of functions θ g : D g − → D g , such that D = X , θ is the identity map, and θ g ◦ θ h ⊂ θ gh . Here, θ g ◦ θ h may be ambiguous,because the range of θ h may not be contained in the domain θ g . This function is then definedon the largest domain possible, that is, θ − h ( D h ∩ D g − ). On this domain it is defined to be θ g ◦ θ h . The notation θ g ◦ θ h ⊂ θ gh means that the function θ gh extends the function θ g ◦ θ h .With the above, one can show that each θ g is a bijection on its domain and that θ − g = θ g − .A partial topological dynamical system is a quadruple( X, G, { D g } g ∈ G , { θ g } g ∈ G )such that X is a topological space, G is a group, ( { D g } g ∈ G , { θ g } g ∈ G ) is a partial action of G on the set X such that each D g is an open subset of X and each θ g is a homeomorphism.Given a partial topological dynamical system θ = ( X, G, { D g } g ∈ G , { θ g } g ∈ G ) with X alocally compact Hausdorff space and G discrete, using a construction of Abadie [Aba04] wemay form the ´etale groupoid G ⋉ θ X := { ( g, x ) ∈ G × X | x ∈ X g − } (1)with ( G ⋉ θ X ) (2) = { (( g, x ) , ( h, y )) | θ h ( y ) = x } , r ( g, x ) = θ g ( x ) , d ( g, x ) = x and ( g, x ) − =( g − , θ g ( x )). If y = θ g ( x ), then we have ( g, x )( h, y ) = ( gh, y ).There is a procedure for producing a C*-algebra from an ´etale groupoid G , given byRenault in [Ren80]. One considers the linear space C c ( G ) of compactly supported complexfunctions on G equipped with product and involution given by( f g )( γ ) = X r ( η )= s ( γ ) f ( γη ) g ( η − ) f ∗ ( γ ) = f ( γ − ) . C c ( G ) the structure of a complex ∗ -algebra. The C*-algebra of G is then definedto be the completion of this ∗ -algebra in a certain norm, and is denoted C ∗ ( G ). It is a factthat if G and H are isomorphic ´etale groupoids, then their C*-algebras are isomorphic.A partial C*-dynamical system is a quadruple( A, G, { D g } g ∈ G , { θ g } g ∈ G )such that A is a C*-algebra, G is a group, ( { D g } g ∈ G , { θ g } g ∈ G ) is a partial action of G onthe set A such that each D g is a closed two-sided ideal of A and each θ g is a ∗ -isomorphism.If we are given a partial C*-dynamical system as above such that G is a discrete group,then we can form the partial crossed product by first considering the ∗ -algebra of formalfinite linear combinations X g ∈ G a g δ g such that a g ∈ D g for each g ∈ G . Here the symbols δ g have no meaning other thanplaceholders. Addition and scalar multiplication are defined in the obvious way, andmultiplication is determined by the rule( aδ g )( bδ h ) = θ g ( θ g − ( a ) b ) δ gh while the ∗ is determined by ( aδ g ) ∗ = θ g − ( a ∗ ) δ g − This multiplication is not always associative, but for the cases we consider here it is.A partial topological dynamical system (
X, G, { D g } g ∈ G , { θ g } g ∈ G ) with X locally com-pact and Hausdorff gives rise to a partial C*-dynamical system ( C ( X ) , G, { C ( D g ) } g ∈ G , { e θ g } g ∈ G )with e θ g : C ( D g − ) → C ( D g ) e θ g ( f ) (cid:12)(cid:12)(cid:12) x = f ( θ − g ( x ))and C ( D g ) is understood to denote all the functions f ∈ C ( X ) which vanish outside of D g .It is a fact that in this situation, the resulting partial action crossed product is isomorphicto the C*-algebra of the groupoid G ⋉ θ X . An important special case is the situation where X and each D g is compact – in this case C ( X ) becomes C ( X ) and each C ( D g ) becomes C ( D g ). One may consider instead the linear space of finitely supported functions f from G into A such that f ( g ) ∈ D g , and in this case, the element aδ g is identified with the function which takes the value a on g and 0 elsewhere. Inverse Semigroups from Graph Actions
Let E = ( E , E , r, d ) be a finite directed graph. An automorphism of E is a bijective map h : E ∪ E → E ∪ E such that h ( E i ) ⊂ E i for i = 0 , h ◦ d = d ◦ h and h ◦ r = r ◦ h .An action of a group G on a graph E is a group homomorphism from G to the group ofautomorphisms of E .Suppose that G acts on a set X . A one-cocycle for the action of G on X is a map ϕ : G × X → G such that ϕ ( gh, x ) = ϕ ( g, hx ) ϕ ( h, x )for all g, h ∈ G and x ∈ X . Setting g = h = 1 into the above we get that ϕ (1 , x ) = 1 foreach x .We now assume that E is a finite directed graph with no sources or sinks, G is a countablediscrete group, and that we have a homomorphism from G to the group of automorphismsof E . We denote the image of a group element g under this homomorphism simply as g .We also assume that we have a one-cocycle ϕ : G × E → G for the restriction of our actionto E , which also satisfies ϕ ( g, e ) x = gx, for all g ∈ G, e ∈ E , x ∈ E In [EP13], they show that the action of G and the cocycle ϕ extend to E ∗ in a natural way.This induced action preserves lengths. Furthermore, for every g, h ∈ G , for every x ∈ E and for every α and β in E ∗ such that d ( α ) = r ( β ) we have(E1) ( gh ) α = g ( hα )(E2) ϕ ( gh, α ) = ϕ ( g, hα ) ϕ ( h, α )(E3) ϕ ( g, x ) = g (E4) r ( gα ) = gr ( α ) (E5) d ( gα ) = gd ( α )(E6) ϕ ( g, α ) x = gx (E7) g ( αβ ) = ( gα ) ϕ ( g, α ) β (E8) ϕ ( g, αβ ) = ϕ ( ϕ ( g, α ) , β ).The triple ( G, E, ϕ ) is called a self-similar graph action .Given a self-similar graph action (
G, E, ϕ ), we construct an action of G and a cocycleon the graph obtained from E by collapsing all the vertices to a single vertex. To be moreprecise, consider the directed graph e E := ( { ∅ } , E , r, d ), that is, the directed graph withone vertex whose edge set is equal to the edge set of E . Then the set of paths e E ∗ is just thefree monoid on E , with identity equal to the empty word ∅ = r ( e ) = d ( e ) for all e ∈ E .7t is clear that our given action of G on E induces an action of G on e E , and that herewe have g ∅ = ∅ for all g ∈ G . Furthermore, the cocycle ϕ is defined on G × E , so it isalso a cocycle for the induced action on e E . For clarity, we denote the induced cocycle on G × e E ∗ by e ϕ . By the above list, we have for every w, v ∈ e E ∗ and g, h ∈ G :(SS1) 1 w = w (SS2) ( gh ) w = g ( hw )(SS3) g ∅ = ∅ (SS4) g ( vw ) = ( gv ) e ϕ ( g, v ) w (SS5) e ϕ ( g, ∅ ) = g (SS6) e ϕ ( g, vw ) = e ϕ ( e ϕ ( g, v ) , w )(SS7) e ϕ (1 , w ) = 1(SS8) e ϕ ( gh, w ) = e ϕ ( g, hw ) e ϕ ( h, w )These properties mean that the pair ( G, e E ) is a self-similar action in the sense of Lawson([Law08], Section 3), which he proves is equivalent to ( G, e E ) being a self-similar group inthe sense of Nekrashevych [Nek05] (except that the action on E ∗ may not be faithful). Forthis reason, we call ( G, e E, e ϕ ) the induced self-similar group of ( G, E, ϕ ). We note that anyself-similar graph action (
G, E, ϕ ) such that E is finite and has only one vertex will satisfy(SS1)–(SS8), and so from now on we will call any such triple a self-similar group .We say that ( G, E, ϕ ) is pseudo-free if whenever we have g ∈ G and e ∈ E suchthat ge = e and ϕ ( g, e ) = 1 then we have that g = 1. This is equivalent to saying thatwhenever we have g ∈ G and w ∈ E ∗ such that gw = w and ϕ ( g, w ) = 1 then we have that g = 1 ([EP13], Proposition 5.2). Because this property is phrased in terms of the actionand cocycle on the edge set, it is clear that pseudo-freeness of ( G, E, ϕ ) is equivalent topseudo-freeness of ( G, e E, e ϕ ).Given ( G, E, ϕ ), we let (as in [EP13]), S G,E = { ( α, g, β ) ∈ E ∗ × G × E ∗ | d ( α ) = gd ( β ) } . This set becomes an inverse semigroup when given the operation( α, g, β )( γ, h, ν ) = ( αgγ ′ , ϕ ( g, γ ′ ) h, ν ) , if γ = βγ ′ , ( α, gϕ ( h − , β ′ ) − , νh − β ′ ) , if β = γβ ′ , α, g, β ) ∗ = ( β, g − , α ) . Recall that an inverse semigroup with zero S is called E*-unitary if whenever one has s ∈ S and e ∈ E ( S ), then se ∈ E ( S ) \ { } implies that s ∈ E ( S ). In [EP13], Proposition 5.4, itis shown that S G,E is E*-unitary if and only if (
G, E, ϕ ) is pseudo-free. In the remainder of This was originally termed residually free in a preprint of [EP13]. S G,E .A prehomomorphism from an inverse semigroup with zero S to a group H is a function θ : S \ { } → H such that whenever we have s, t ∈ S \ { } such that st = 0, then θ ( st ) = θ ( s ) θ ( t ). Aprehomomorphism θ defined on S is called idempotent pure if θ − (1) = E ( S ). Every inversesemigroup S admits a prehomomorphism into a group U ( S ) called the universal group of S . The group U ( S ) is generated by the set S \ { } subject to the relations s · t = st if st = 0. An inverse semigroup S is called strongly E*-unitary if there exists an idempotentpure prehomomorphism from S to a group H . This is equivalent to saying that the naturalmap σ from S \ { } to U ( S ) (which is a prehomomorphism) is idempotent pure. It is clearthat if S is strongly E*-unitary then it is E*-unitary, because if se is a nonzero idempotentthen 1 = σ ( se ) = σ ( s ).In the special case of a self-similar group ( G, E, ϕ ), S G,E has extra structure. In thiscase, the set of elements of the form ( α, g, ∅ ) ∈ S G,E form a subsemigroup of S G,E . Thissemigroup is isomorphic to what is called the
Zappa-Sz´ep product of the free semigroup E ∗ by the group G , denoted E ∗ ⊲⊳ G . See [Law08], Section 3 for a discussion of the constructionof this semigroup. As a set, E ∗ ⊲⊳ G is E ∗ × G and the semigroup operation is( α, g )( β, h ) = ( αgβ, ϕ ( g, β ) h ) . One sees that this agrees with the operation from S G,E restricted to elements of the form( α, g, ∅ ). Lemma 3.1. (Lawson-Wallis) Let (
G, E, ϕ ) be a self-similar group. Then the inversesemigroup with zero S G,E is strongly E*-unitary if and only if E ∗ ⊲⊳ G is cancellative. Proof.
By [Law08], E ∗ ⊲⊳ G is a left Rees monoid. By [LW13] Theorem 5.5, a left Reesmonoid can be embedded into a group if and only if it is cancellative. By [Law99] Theorem8, E ∗ ⊲⊳ G can be embedded into a group if and only if S G,E is strongly E*-unitary. Theresult follows.
Lemma 3.2.
Let (
G, E, ϕ ) be a self-similar group. Then the semigroup E ∗ ⊲⊳ G is can-cellative if and only if ( G, E, ϕ ) is pseudo-free.
Proof.
We first prove the “only if” part. Suppose that E ∗ ⊲⊳ G is cancellative, and supposethat we have g ∈ G and e ∈ E such that ge = e and ϕ ( g, e ) = 1. Then we calculate( ∅ , g )( e,
1) = ( ∅ ge, ϕ ( g, e )1) = ( e, ∅ , e )( e,
1) = ( e, . Since we assume cancellation, this implies that ( ∅ , g ) = ( ∅ , g = 1. Hence( G, E, ϕ ) is pseudo-free.We now prove that “if” part. Suppose that (
G, E, ϕ ) is pseudo-free. It is straightforwardto show that E ∗ ⊲⊳ G is always left cancellative. Suppose then that we have v, v ′ , w ∈ X ∗ and g, g ′ , h ∈ G such that ( v, h )( w, g ) = ( v ′ , h ′ )( w, g ) . This implies ( v ( hw ) , ϕ ( h, w ) g ) = ( v ′ ( h ′ w ) , ϕ ( h ′ , w ) g ) . Equating the second coordinates gives us that ϕ ( h, w ) = ϕ ( h ′ , w ). Also, since the action of G preserves length, equating the first coordinates implies that v = v ′ and hw = h ′ w . Byproperties of ϕ we have ϕ ( h − h ′ , w ) = ϕ ( h − , h ′ w ) ϕ ( h ′ , w )= ϕ ( h − , hw ) ϕ ( h, w )= ( ϕ ( h, w )) − ϕ ( h, w )= 1 . Also, by the above we have h − h ′ w = w . Since ( G, E, ϕ ) is pseudo-free, this implies that h = h ′ . Thus we have proven that ( v, h ) = ( v ′ , h ′ ) and so E ∗ ⊲⊳ G is cancellative.We now prove the main result of this section, which addresses when S G,E is strongly E*-unitary for an arbitrary self-similar graph action (
G, E, ϕ ) using the two previous lemmasand the induced self-similar group ( G, e E, e ϕ ). Theorem 3.3.
Let (
G, E, ϕ ) be a self-similar graph action. Then the inverse semigroup S G,E is strongly E*-unitary if and only if (
G, E, ϕ ) is pseudo-free.
Proof.
We prove the “if” part first. Suppose that (
G, E, ϕ ) is pseudo-free (and therefore so is( G, e E, e ϕ )). Hence by Lemmas 3.1 and 3.2, S G, e E is strongly E*-unitary. Define Q : E ∗ → e E ∗ by Q ( α ) = α if α / ∈ E ∅ if α ∈ E (2)and define a function ι : S G,E → S G, e E by ι ( α, g, β ) = ( Q ( α ) , g, Q ( β )) . ι ( st ) = ι ( s ) ι ( t ) as long as st = 0. Then if σ : S G, e E → U ( S G, e E ) is the standard mapfrom S G, e E to its universal group, the map σ ◦ ι : S G,E → U ( S G, e E )is a prehomomorphism. Since S G, e E is strongly E*-unitary, σ is idempotent pure. Take( α, g, β ) ∈ S G,E . If σ ◦ ι ( α, g, β ) = 1, then this implies that ( Q ( α ) , g, Q ( β )) is a nonzeroidempotent, that is, Q ( α ) = Q ( β ) and g = 1. Hence either α = β or both α and β arepaths of length zero (ie vertices). It is not possible for α and β to be different paths oflength zero, because ( α, g, β ) ∈ S G,E and g = 1 implies that d ( α ) = d ( β ). Hence σ ◦ ι is anidempotent pure prehomomorphism and thus S G,E is strongly E*-unitary.Now we prove the “only if” part. Suppose that S G,E is strongly E*-unitary. Then,in particular, it is E*-unitary. By [EP13], Proposition 5.4, this implies that (
G, E, ϕ ) ispseudo-free.
Example 3.4. (The Odometer)
Take a natural number n ≥ n edges and one vertex R n . That is, R n = { , , , . . . n − } , and R n = { ∅ } . We writethe group Z of integers multiplicatively, with generator z , so that Z = { z m | m ∈ Z } . For x ∈ R n , let zx = x + 1 if x = n −
10 if x = n − . This formula defines a self-similar group ( Z , R n , ϕ ) with cocycle ϕ ( z, x ) = e if x = n − z if x = n − . We claim that ( Z , R n , ϕ ) is pseudo-free. Suppose that ν ∈ R ∗ n , z m ∈ Z , z m ν = ν and ϕ ( z m , ν ) = 1. We suppose that m >
0. Since ϕ ( z m , ν ) = 1, we must have that | ν | > log n ( m ). Furthermore, since z m ν = ν , we must have that | ν | is a multiple of log n ( m ),which is impossible. The case of m < m = 0, andso ( Z , R n , ϕ ) is pseudo-free.By Theorem 3.3, S Z ,R n is strongly E*-unitary, and so the identity map σ : S Z ,R n \ { } → U ( S Z ,R n ) is idempotent pure. We describe the universal group U ( S Z ,R n ). One can alwaysassume that the universal group is generated by the image of σ . Recall that S Z ,R n = { ( α, z n , β ) | n ∈ Z , α, β ∈ R ∗ n } . Let a i := σ (( i, z , ∅ )) , i ∈ R n , Z := σ (( ∅ , z , ∅ )) . S Z ,R n we find quickly that U ( S Z ,R n ) = h a , a , . . . a n − , Z | Za i = a i +1 for 0 ≤ i < n − , Za n − = a Z i . We can use the relations to reduce this to U ( S Z ,R n ) = (cid:10) a , Z | Z = a − Z n a (cid:11) obtaining that U ( S Z ,R n ) is isomorphic to the Baumslag-Solitar group BS (1 , n ). We notethat the relationship between this self-similar action and these groups has been noted in[BRRW14], Example 3.5. There it is seen directly that the Zappa-Sz´ep product R ∗ n ⊲⊳ Z embeds as a suitable positive cone in BS (1 , n ). ( G, E, ϕ ) as a Partial Crossed Prod-uct To a self-similar graph action (
G, E, ϕ ) one associates a C*-algebra, denoted in [EP13] as O G,E . Definition 4.1.
Let (
G, E, ϕ ) be a self-similar graph action. Then O G,E is the universalC*-algebra for the set { p x | x ∈ E } ∪ { s e | e ∈ E } ∪ { u g | g ∈ G } subject to the following:(CK1) { p x | x ∈ E } is a set of mutually orthogonal projections,(CK2) { s e | e ∈ E } is a set of partial isometries,(CK3) s ∗ e s e = p d ( e ) for each e ∈ E ,(CK4) p x = X e ∈ r − ( x ) s e s ∗ e for each x ∈ E with 0 < r − ( x ) < ∞ ,(EP1) g u g is a unitary representation of G ,(EP2) u g s e = s ge u ϕ ( g,e ) for all g ∈ G and e ∈ E ,(EP3) u g p x = p gx u g for all g ∈ G and x ∈ E .The first four relations are the Cuntz-Krieger relations for the graph E . Our maintheorem describes when this C*-algebra is isomorphic to a partial crossed product, and isa combination of Theorem 3.3 with the following previously known results. Theorem 4.2. ([EP13], Theorem 9.5) Let (
G, E, ϕ ) be a self-similar graph action suchthat E is a finite graph with no sources. Then O G,E is isomorphic to the C*-algebra of thegroupoid G tight ( S G,E ). 12 heorem 4.3. ([MS11], Theorem 5.3) Let S be a countable E*-unitary inverse semigroupwith universal group U ( S ) and tight spectrum b E tight . Then there is a natural partial actionof U ( S ) on b E tight such that the groupoids G tight ( S ) and U ( S ) ⋉ b E tight are isomorphic.We now define what is meant by “ b E tight ” and “ G tight ( S )” in the above, and then describethe situation in our case.Each inverse semigroup S possesses a natural order structure. Two elements s and t satisfy s t if and only if s = ss ∗ t . This is equivalent to saying that s = te for someidempotent e . If e and f are idempotents, then e f if and only if ef = e . Recall that a filter F in a partially ordered set X is a proper subset that is downward directed (that is,for each x, y ∈ F there is an element z ∈ F such that z x, y ) and upwards closed (thatis, if x ∈ F and x y then y ∈ F ). If F is a proper subset which is downward directed,then it is called a filter base and F = { x ∈ X | f x for some f ∈ F } is a filter. Also recall that an ultrafilter is a filter which is not properly contained in anotherfilter. Filters in E ( S ) are closed under multiplication and, if S has a zero element, thenfilters in E ( S ) do not contain the zero element. If ξ ⊂ E ( S ) is a filter and e ∈ ξ , then it isstraightforward that both eξ and ξe are filter bases and ξ = eξ = ξe .Suppose that S is countable and consider { , } E ( S ) , the power set of E ( S ). This is acompact Hausdorff space homeomorphic to the Cantor set when given the product topology.Let b E denote the closed subspace of filters in E ( S ) – this is called the spectrum of S . Let b E ∞ denote the space of ultrafilters, and let b E tight denote the closure of b E ∞ in b E – this iscalled the tight spectrum of S .Any inverse semigroup acts naturally on its spectra. Fix an inverse semigroup S withset of idempotents E . For each e ∈ E , let D e = { ξ ∈ b E tight | e ∈ ξ } . Then define θ s : D s ∗ s → D ss ∗ by θ s ( ξ ) = sξs ∗ . These sets and maps define an action of S on b E tight . The groupoid of germs associated tothis action is called the tight groupoid of S and is denoted G tight ( S ). For details, see [Exe08].We now turn to constructing a partial action of U ( S ) on the tight spectrum of S fromthe canonical action of S . Let σ : S \ { } → U ( S ) be the standard identity map from S \ { } to U ( S ). The basic idea from [MS11] is that for a given group element g ∈ U ( S )one “bundles together” the partial homeomoprhisms corresponding to all its preimagesunder σ . It turns out that to guarantee that such functions agree on their domains, oneneeds that S is strongly E*-unitary.To be more precise, let ( { D e } e ∈ E ( S ) , { θ s } s ∈S ) be the canonical action of S on b E tight .13efine a partial action ( { F g } g ∈ U ( S ) , { e θ g } g ∈ U ( S ) } ) of the group U ( S ) on b E tight , by setting F g = [ s ∈ σ − ( g ) D ss ∗ (3)such that, for all ξ in some D s ∗ s ⊂ F g − , we have e θ g ( ξ ) = θ s ( ξ ). To see why this is well-defined, suppose that we have s, t ∈ σ − ( g ) for some g ∈ U ( S ), and that we have a filter ξ ∈ D s ∗ s ∩ D t ∗ t – this means that s ∗ s, t ∗ t ∈ ξ , and so s ∗ st ∗ t = 0. Hence st ∗ = 0. A shortcalculation gives σ ( st ∗ ) = σ ( s ) σ ( t ) − = gg − = 1 , and so st ∗ = ts ∗ is an idempotent.Since t ∗ t ∈ ξ and sξs ∗ is a filter, we must have that st ∗ ts ∗ = 0. This means that st ∗ st ∗ = 0, and so similar to the above t ∗ s = 0 and is an idempotent. Furthermore, s ∗ t = s ∗ ts ∗ t = s ∗ st ∗ t ∈ ξ . Now, suppose that e ∈ sξs ∗ . Then there exists f ∈ ξ such that sf s ∗ e , which is to say that esf s ∗ = sf s ∗ . Now we have e ( ts ∗ sf s ∗ st ∗ ) = ts ∗ ( esf s ∗ ) st ∗ = ts ∗ sf s ∗ st ∗ ∈ tξt ∗ and so e is greater than an element of tξt ∗ , whence e ∈ tξt ∗ . This argument is symmetricin s and t , so we must have that sξs ∗ = tξt ∗ , and thus the functions θ s and θ t agree on F g − from (3), and so e θ g is well-defined. We note that by (3), the only g ∈ U ( S ) for which F g isnonempty will be those in the image of σ .It is straightforward to show that map from G ( S , b E tight , θ ) to U ( S ) ⋉ e θ b E tight given by[ s, ξ ] ( σ ( s ) , ξ )is a well-defined isomorphism of topological groupoids.Now, let ( G, E, ϕ ) be a self-similar graph action such that E is a finite graph withno sinks or sources. Then the tight spectrum of S G,E is homeomorphic to the space ofinfinite paths, see [EP13], Section 8. The action of G on E ∗ extends to an action on Σ E byhomeomorphisms determined by the following: for each g ∈ G and x ∈ Σ E we have( gx ) i = ϕ ( g, x x · · · x i − ) x i . As above, each ( α, g, β ) ∈ S G,E acts via a partial homeomorphism on its tight spectrum Σ E given by ( α, g, β ) : β Σ E → α Σ E βx α ( gx ) . The above together with Theorem 3.3 and previously known Theorems 4.2 and 4.3directly imply the following. 14 heorem 4.4.
Let (
G, E, ϕ ) be a self-similar graph action such that E is a finite graphwith no sinks or sources. Suppose further that ( G, E, ϕ ) is pseudo-free. Then O G,E isisomorphic to a partial crossed product C (Σ E ) ⋊ e θ U ( S G,E ). The action e θ is the action on C (Σ E ) derived from the action θ described above. Example 4.5. (The 2-Odometer)
We consider the self-similar group ( Z , R , ϕ ), that is,Example 3.4 with n = 2. Recall that S Z ,R = { ( α, z n , β ) | n ∈ Z , α, β ∈ R ∗ } . For convenience and clarity of computations, we will make the identifications S α := ( α, z , ∅ )and U := ( ∅ , , ∅ ), so that S Z ,R = { S α U n S ∗ β | n ∈ Z , α, β ∈ { , } ∗ } . The universal group of S Z ,R is the Baumslag-Solitar group BS (1 , σ ( S ) = aσ ( S ) = b. Doing this, we obtain H := U ( S Z ,R ) = (cid:10) a, b | aba − b − = ba − (cid:11) with σ ( U ) = ba − .Because this action is pseudo-free, σ is idempotent-pure and Theorem 4.3 applies. Hencethe C*-algebra of the tight groupoid of S Z ,R is isomorphic to the partial crossed product ofthe tight spectrum of S Z ,R by H . We will describe this partial action ( { D g } g ∈ H , { θ g } g ∈ H ).We begin by looking closer at the group H . For a word α in a and b , let e α be the wordin 0 and 1 obtained from α by replacing a with 0 and b with 1; that is, α = σ ( S e α ). Supppsethat α and β are words in a and b and that | α | = | β | . Then we claim that the groupelement αβ − is equal to σ ( U k ) for some k ∈ Z . One may view e α and e β as binary numbers(with the powers of 2 increasing from left to right rather than right to left). For a word ν in a and b , let n ν denote the integer corresponding to the binary number determined by e ν .Now, notice that U n β − n α S e α S ∗ e β = U n β U − n α S e α S ∗ e β = U n β S | α | S ∗ e β = S e β S ∗ e β ∈ E ( S Z ,R )15here, in the above, 0 | α | denotes the word consisting of | α | σ ( U n β − n α S e α S ∗ e β ) = 1,and so αβ − = σ ( U n α − n β ) as desired. In fact, from this it is easy to see that σ ( U k ) ∈ H can always be written in the form αβ − for words α, β in a and b with | α | = | β | . Hence,the image of σ consists only of group elements in H which can be written in the form αβ − for words α and β (not necessarily of the same length). Since D g is only nonempty if g isin the image of σ , the only group elements for which D g will be nonempty are those whichcan be written in the form αβ − .Next we further clarify the different forms that group elements can take in H . Claim: If αβ − = νω − in H , then | α | − | β | = | ν | − | ω | . If | α | − | β | >
0, then the initialsegment of α of length | α | − | β | is equal to the initial segment of ν . Similarly, if | α | − | β | < β of length | β | − | α | is equal to the initial segment of ω . Proof.
First suppose that | α | = | β | . Then by the discussion above, αβ − = σ ( U n ) for some n . Hence we must have that σ ( U n S e ω S ∗ e ν ) = αβ − ων − = 1 . Since σ is idempotent pure, U n S e ω S ∗ e ν must be an idempotent, and so | ω | = | ν | .Now suppose that | α | > | β | , Then αβ − = α α β − with | α | = | β | . Again, by thediscussion above, α β − = σ ( U n ) for some n . Hence σ ( S e α U n S e ω S ∗ e ν ) = α α β − ων − = 1and again S e α U n S e ω S ∗ e ν is an idempotent. This can only happen if α is an initial segmentof ν and | α | + | ω | = | ν | . Hence | α | − | β | = | ν | − | ω | . The case of | β | > | α | is similar. (cid:3) As above, the tight spectrum of S Z , { , } is homeomorphic to Σ { , } , the space of infinitewords in 0 and 1. There is a homeomorphism λ : Σ { , } → Σ { , } which takes an infinitesequence x , looks for the first entry which is not equal to 1, switches it to 1, switches theprevious entries to 0, and leaves the rest of the entries unchanged. If all entries are equalto 1, λ switches them all to 0. One sees that this is the extension of the action of Z inExample 3.4 to infinite sequences. Our maps θ g will involve this homeomorphism.We can now describe the partial action ( { D g } g ∈ H , { θ g } g ∈ H ). If g is not of the form αβ − for some words α, β in a and b , then D g = ∅ . Otherwise, we have three cases.1. If g = αβ − with | α | = | β | , then D g = D g − = Σ { , } , and θ g ( x ) = λ n α − n β ( x ) .
2. If g = αβ − with | α | > | β | , then αβ − = α g α β − with | α | = | β | . Further, the word α g does not depend on the particular representation of αβ − by the above claim. Inthis case we have D g − = Σ { , } and D g = α g Σ { , } . The map θ g is given by θ g ( x ) = α g λ n α − n β ( x )16here above we are concatenating the infinite sequence λ n α − n β ( x ) with α g .3. The third case, where g = αβ − with | α | > | β | , is completely determined by thesecond case above. Here αβ − = αβ − β − g with | α | = | β | , θ g : β g Σ { , } → Σ { , } , and θ g ( β g x ) = λ n α − n β ( x )The partial action θ induces a partial action e θ on C (Σ E ). By Theorem 4.4 and Example3.4, we have that O Z ,R ∼ = C (Σ R ) ⋊ e θ BS (1 , O Z ,R n ∼ = C (Σ R n ) ⋊ e θ BS (1 , n )We finish this example by pointing out that this gives a realization of the C*-algebra Q associated in [LL12] to the 2-adic integers as a partial crossed product, because Q ∼ = O Z ,R .See [BRRW14], Example 6.5 for more details. O n Suppose that θ = ( A, G, { D g } g ∈ G , { θ g } g ∈ G ) is a partial C*-dynamical system and that G isa subgroup of H . Then one can extend this partial C*-dynamical system to H , creating b θ = ( A, H, { b D g } h ∈ H , { b θ g } h ∈ H ) by setting b D h = D h and b θ h = θ h if h ∈ G and b D h and b θ h to be the zero ideal and zero map if h / ∈ G . In this situation, A ⋊ θ G ∼ = A ⋊ b θ H . In thisway we see that a result of the form “ B is isomorphic to a partial crossed product by H ”doesn’t contain as much information as one would like, because perhaps a subgroup of H would suffice. In our cases so far, we have shown that the O G,E are isomorphic to crossedproducts by certain groups, and that these groups are actually generated by the elementswhose corresponding ideals are nonempty.It is well-known that the Cuntz algebra O n can be realized as a partial crossed product.The first such construction appears in [QR97], where it is shown that O n is isomorphicto a partial crossed product of the Cantor set by F n , the free group on n elements. Wereproduce this construction below in Example 5.1. In [Hop07], O n is realized as a partialcrossed product of the Cantor set by the Baumslag-Solitar group BS (1 , n ) ∼ = Z (cid:2) n (cid:3) ⋊ Z ,an amenable group. In both of these situations, the elements g for which the ideal D g is nonzero generate the group. In this section, we show that if ( G, E, ϕ ) is pseudo-freeself-similar graph action such that E has only one vertex and which satisfies a conditionwe call exhausting (Definition 5.3), then O | E | is isomorphic to a partial crossed productby the group U ( S G,E ) and the group elements g such that the ideal corresponding to g isnonzero generate U ( S G,E ). 17 xample 5.1. (The Cuntz Algebra)
This is a construction seen in [QR97]. Let A be afinite alphabet and consider the space Σ A of right-infinite words in elements of A . Giventhe product topology, Σ A is homeomorphic to the Cantor set. Let F A denote the free groupgenerated by A . We will describe a partial action of F A on Σ A . If g ∈ F A is not of the form αβ − for words α, β ∈ A ∗ , then D g = ∅ . Otherwise, for all α, β ∈ A ∗ we have D αβ − = α Σ A = { αx | x ∈ Σ A } and the map θ αβ − : β Σ A → α Σ A is defined by θ αβ − ( βx ) = αx. Here, Σ A is compact and each D g is clopen. One can show that the C*-algebra of thegroupoid F A ⋉ θ Σ A is isomorphic to O | A | . Definition 5.2.
Suppose that we have a partial action of a group G on a set X , say( { D g } g ∈ G , { θ g } g ∈ G ) and suppose that ϕ : G → H is an onto homomorphism of groups.Suppose further that whenever h ∈ H , g , g ∈ ϕ − ( h ), and x ∈ D g − ∩ D g − , then θ g ( x ) = θ g ( x ). Then the induced partial action of H on X is the pair( { E h } h ∈ H , { θ h } h ∈ H )where E h = [ g ∈ ϕ − ( h ) D g and, with slight abuse of notation, the θ h are as before.This is well-defined because these functions agree on any possible intersections of thesets above. We will need the following condition. Definition 5.3.
A self-similar graph action (
G, E, ϕ ) is called exhausting if for all g ∈ G there exists α ∈ E ∗ such that ϕ ( g, α ) = 1 G .There is a natural homomorphism φ : F E → U ( S G,E ) determined by φ ( e ) = σ ( S e ) forall e ∈ E . In the case that ( G, E, ϕ ) is exhausting, we have the following.
Lemma 5.4.
Let (
G, E, ϕ ) be a self-similar graph action such that E is a finite graphwith no sinks or sources. If ( G, E, ϕ ) is exhausting, then the natural group homomorphism φ : F E → U ( S G,E ) is surjective.
Proof.
Let g ∈ G and find α ∈ E ∗ such that ϕ ( g, α ) = 1 G . Let σ : S G,E \ { } → U ( S G,E ) bethe natural map. Then U g S α S ∗ gα = S gα S ∗ gα , and so σ ( U g S α S ∗ gα ) = 1. Since σ is multiplicativeon nonzero products, we have that σ ( U g ) = σ ( S gα S ∗ g ). We know that U ( S G,E ) is generatedby the image of S G,E , so this implies that U ( S G,E ) is generated by { σ ( S x ) } x ∈ E , and theresult follows. 18n the following, we refer to the action in Example 5.1 as the Quigg-Raeburn action . Theorem 5.5.
Suppose that (
G, E, ϕ ) is a self-similar group. Suppose also that (
G, E, ϕ )is pseudo-free and exhausting. Then the partial crossed product associated to the action θ of U ( S G,E ) on Σ E induced by the Quigg-Raeburn action is isomorphic to O | E | , and theelements g ∈ U ( S G,E ) with corresponding ideals not equal to 0 generate U ( S G,E ). Proof.
We first show that the induced action is well-defined. Let φ : F E → U ( S G,E ) denotethe surjective group homomorphism from Lemma 5.4, given on generators by φ ( x ) = σ ( S x ).If we take the D αβ − as in Example 5.1, then for g ∈ U ( S G,E ) we have E h = [ αβ − ∈ φ − ( h ) D αβ − = [ αβ − ∈ ϕ − ( h ) α Σ E . In general, given two cylinder sets α Σ E and η Σ E , either they are disjoint, or one is containedin the other. In the latter case, α = ηα ′ (without loss of generality). So, suppose that h ∈ U ( S G,E ) and αβ − , ηγ − ∈ φ − ( h ) such that β Σ E ∩ γ Σ E = ∅ . Without loss of generality,we will assume that β Σ E ∩ γ Σ E = β Σ E , and so β = γβ ′ . This implies that1 = σ ( S α S ∗ β S γ S ∗ η )= σ ( S α S ∗ β ′ S ∗ η )= σ ( S α S ∗ ηβ ′ )and since σ is idempotent pure, we must have that α = ηβ ′ . Hence αβ − = ηβ ′ ( γβ ′ ) − = ηγ − , and so θ αβ − and θ ηγ − agree on the intersection of their domains (because if their domainsintersect, they must in fact be equal group elements).Now we have that the induced action is well-defined. The mapΦ : F E ⋉ θ Σ E → U ( S G,E ) ⋉ θ Σ E Φ( αβ − , βx ) = ( φ ( αβ − ) , βx )is easily shown to be an isomorphism of topological groupoids. We omit the details. Example 5.6.
If ( Z , R n , ϕ ) is the odometer from Examples 3.4 and 4.5, by Theorem 5.5we obtain that that O n ∼ = C (Σ R n ) ⋊ BS (1 , n ), reproducing the result from [Hop07].We also note that while the groups and algebra in this example are the same as Example4.5, we obtain nonisomorphic crossed products because the domains of group elements canbe different. For example, in the action in Example 4.5 we have that D ba − = Σ R , whilein the induced action described above, D ba − = [ n ≥ b n a Σ R = Σ R \ { bbb · · · } . eferences [Aba04] Fernando Abadie. On partial actions and groupoids. Proc. Amer. Math. Soc. ,132(4), 2004.[BRRW14] Nathan Brownlowe, Jacqui Ramagge, David Robertson, and Michael F. Whit-taker. Zappa–Sz´ep products of semigroups and their C*-algebras.
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Departamento de Matem´atica, Campus Universit´ario Trindade CEP 88.040-900Florian´opolis SC, Brasil.
Charles Starling (corresponding author): [email protected]@gmail.com