Partial Translation Algebras for Trees

P AR TIAL TRANSLA TION ALGEBRAS FOR TREES J. BRODZKI, G.A. NIBLO, AND N.J. WRIGHT Abstract. In [1] we in troduced the notion of a partial translation C ∗ -algebra for a discr ete metric space. Here we demonstrate that sev eral im p ortant clas- sical C ∗ -algebras and extensions aris e naturally by considering partial trans- lation al gebras ass o ciated with subspaces of trees. 1. Introduction The uniform Ro e algebra C ∗ u ( X ) is a C ∗ -algebra asso cia ted with any discrete metric spac e X which enco des a na lytically the co arse ge o metry o f the space [4, Ch. 4]. F or example, the space X has Y u’s prop erty A [8] if and only if the uniform Ro e a lgebra of X is n uclear [7]. A rich sourc e of examples of int ere s ting metric spaces is the c la ss of discrete g r oups equipp ed with a left inv ar iant metric. F or such a gr oup G , the uniform Ro e algebra contains the (right) r educed C ∗ -algebra of the group C ∗ ρ ( G ). The uniform Ro e algebra is v a stly lar g er than C ∗ ρ ( G ), indeed, unless G if finite, it is not sepa rable. F o r this reason, it is us eful for g eneral metric spac e s to consider a n analo gue of the reduce d g roup C ∗ -algebra . In [1] we int ro duced the notion of a pa rtial transla tio n algebr a for a discrete metr ic space to play this role. In this pa pe r we demons trate the p ow er of this approa ch by exhibiting several well known algebras in the new fra mework. In the context o f subspaces of Z the partial tra nslation alg ebra enco des the additive structure. W e provide several ex - amples of this phenomenon. In pa rticular the T o eplitz ex tension ar ises her e by considering the algebr a a sso ciated to the inclusion of the natural n umbers in the int eger s (Theore m 1). Repla cing the na tur al num b ers by Z \ { 0 } we obtain a trivial extension o f C ( S 1 ) by the compacts which is therefore inequiv a lent to the T o eplitz extension (Theorem 2 ). It is also natural to ask what the ass o ciated alg ebra tells us ab out the additive structure o f the primes, and here we make a co nnection with the cla s sical de Polignac conjecture (Theorem 3 ). By generalis ing these ideas to consider the inclusion of the 3 -v alent tree a nd the ro oted 3-v alent tree in to the Cayley gra ph of the free gr oup on tw o gener a tors w e are a ble to r ecov er the extension used by Cun tz in his computation in [5] of the K -theory o f the Cuntz algebra O 2 (Theorem 4). It is s traightforw ar d to g eneralise this metho d for the alg ebras O n , by considering the C ayley graph of the free g roup F n on n genera tors, a nd we give an outline o f this. Finally we use the g eometry to co nstruct an explicit em b edding of C ∗ ρ ( F 2 ) into the Cuntz algebra O 2 (Theorem 5). W e do t his b y exhibiting explicit injective quasi-isometr ies of the regular 4- v alent tree in to the regular 3-v alent tree which are well b ehav ed with resp ect to the natura l partial tra nslations on these tr e es. 1 2 J. BRODZKI, G.A. NIBLO, AND N.J . WRIGHT 2. P ar tial transla tions Let G be a discre te group equipp ed with a left in v ariant metr ic d . This means that for every element l ∈ G the map on G defined using the left multiplication by l is a n isometry with resp ect to the metric d . On the o ther hand, the r ight m ultiplication by an elemen t r ∈ G mov es ev ery elemen t g ∈ G by t he same amount: d ( g , g r ) = d ( e, r ) . By a nalogy with metric geo metry , w e will call suc h maps tr anslations . These tw o actions to gether are resp ons ible fo r the s ymmetries and the homo geneity of the group G regar ded a s a metric space with r esp ect to the left inv ariant metr ic d . It is c le ar that one cannot hope to ha ve the same amoun t of informa tion for a general discrete metric space X . How e ver, in [1], we introduced a w ay of measuring homogeneity . A star ting point of our in vestigations w as the observ ation tha t, up to a bo unded a mo unt of distor tion, a metric subspa ce of a discrete gr oup retains a degree of symmetry . More ov er, this induced structure can b e co dified. Solid arrows describ e the partial trans- lation · a acting on the subset { b n a | n ∈ Z } of the free group F 2 ; dashed arrows denot e left multiplication b y b · . By a p artial tr anslation we mean a bijection t defined on a subs e t S ⊆ X , taking v alues in a subset of X , such that d ( s, t ( s )) is b ounded for all s ∈ S . Th is notio n was intro duced by Ro e [4 , Def. 10.21] in his discussion of the co arse group oid of Sk a nda lis, T u and Y u [7]. Equiv a - lent ly , a partial transla tion may b e descr ibed in terms of its graph; from that p oint of view it may b e defined as a subspace o f X × X which lies within a b o unded dista nce o f the dia gonal and such that the c o ordinate pro jections are in- jective. In the case when X is a discrete group, right multiplication by an element g ∈ X deter - mines a partia l transla tion t g : y 7→ y g , which is defined for every y ∈ X . Definition 1. Let X be a uniformly discr ete bo unded geometr y s pace, and let T b e a family of disjoint par tial translations o n X . Ea ch of the partia l translations t ∈ T induces a partial isometr y τ on ℓ 2 ( X ) defined by τ ( δ x ) = δ t ( x ) for x in the domain of t , and τ ( δ x ) = 0 for x not in the domain. The p artial tr anslation algebr a C ∗ ( T ) is the C ∗ sub-algebra of the uniform Ro e algebra, C ∗ u ( X ), ge ne r ated by the partia l tr anslations in T , regar ded as partial isometries in ℓ 2 ( X ) (see [ ? , p. 67 ] for a discussio n of C ∗ u ( X )). Note tha t for a ny pa rtial transla tion t the inv erse t − 1 is a lso defined as a partial translation on X a nd that as an element of ℓ 2 ( X ), it induces the adjoint τ ∗ of τ . The notion of a partial translation algebra was in tro duced to play the role of the reduced group C ∗ -algebra for a discr ete metric space, and in [1, Thm 27] we prove that in a countable discrete gro up there is a c a nonical family of partia l tra nslations T such that the alge br a C ∗ ( T ) is isomor phic to C ∗ ρ ( G ). In genera l, witho ut additional co nstraints on the par tial translations , we do no t exp ect to be able to recover (or use) g eometric information. Howev er, as we show ed in [1], in many cas e s, a nd in particula r in the case of a subspace of a discrete group, P AR TIAL TRANSLA TION ALGEBRAS F OR TREES 3 we can choose our partial trans lations to satisfy str ong (partial) homogeneity con- ditions which control the structure o f the corr esp onding par tial transla tion a lgebras and r elate this to the geo metr y of the space. The analytic prop erties of this al- gebra capture some interesting metric prop erties o f the space X . One ex ample of this r elation is the statement that when X is sufficiently ho mogeneous (i.e., in the languag e of [1] it admits a free a nd globally controlled atlas for some partial translation structure), then the following statements a re all equiv alent [1, Thm 29 ]: (1) The spac e X has prop er ty A ; (2) The uniform Ro e a lgebra C ∗ u ( X ) is nuclear; (3) The alg ebra C ∗ u ( X ) is exact. The co nditions of this s tatement a re sa tisfied, fo r example, when X admits an injectiv e uniform e mbedding into a co un table discrete g roup. In this pap er we will conside r subspaces o f trees , which of course embed in free groups and therefore inher it well b ehav ed par tial transla tions. 3. Transla tion subsp aces of Z Coburn’s theo r em, [3], states that the T o eplitz a lgebra is the middle ter m of an extension: 0 → K → T → C ( S 1 ) → 0 , where K d enotes the co mpacts. This extensio n arise s na turally by viewing the generator of the T o eplitz algebr a a s the unilateral shift on ℓ 2 ( N ) and identifying the generator of C ( S 1 ) with the biliteral shift on ℓ 2 ( Z ) , induce d b y the ident ificatio n of C ( S 1 ) with the r educed C ∗ algebra of Z . The p o int of intro ducing it her e is that, as we shall see, it yields the fir st non-trivia l ex ample of a par tia l translatio n algebra, arising from the inclusion of N in Z . 3.1. The translation algebra C ∗ ( N ) . In this section we establish the following: Theorem 1. The tra nslation algebr a C ∗ ( N ) arising fr om t he inclusion of the natu- r al nu mb ers in the int e gers is isomorphic to the T o eplitz algebr a, and mor e over the inclusion induc es t he T o eplitz extension. Regarding Z as a n infinite cyclic g roup, it is equipped with a canonical family of partial translations inherited from the right actio n of the group on itself. The partial transla tion algebra of Z induced b y this is , by definition, the reduced g r oup C ∗ -algebra C ∗ ρ ( Z ). This is gene r ated by a single element, σ 1 , the bilater al shift o n ℓ 2 ( Z ) induced by the par tial trans lation (actually a tra nslation) n 7→ n + 1. The subspace N (which for our pur p o ses will include 0) inherits a family o f partial translations by restricting, and cores tr icting the tra nslations of Z to N . That is, the set of partial translations o n N consists of all maps t n defined o n N of the for m t n ( j ) = j + n where n ≥ 0, alo ng w ith all maps of the form t − 1 n : { n , n + 1 , n + 2 , . . . } → N , t : j 7→ j − n, where n > 0. The co rresp onding partial tr anslation algebra is by definition the C ∗ -algebra C ∗ ( N ) ⊂ B ( ℓ 2 ( N )) gener ated b y the partial isometrie s τ i and τ − i corres p o nding to the partial transla tions s : N → N \ { 0 , . . . , i − 1 } , s ( j ) = j + i and s − 1 : N \ { 0 , . . . , i − 1 } → N , s − 1 ( j ) = j − 1. 4 J. BRODZKI, G.A. NIBLO, AND N.J . WRIGHT Lemma 1. The p artial t r anslation algebr a C ∗ ( N ) is gener ate d by τ 1 (and its ad- joint) wher e τ 1 acts on ℓ 2 ( N ) as a unilater al shift. I t c ontains the algebr a of c omp act op er ators on ℓ 2 ( N ) . Pr o of. It is clear that for each n the ope rator τ n 1 is induced by the partial trans la tion t n , while ( τ ∗ 1 ) n is induced by t − 1 n proving the first part of the lemma. The op era to r τ ∗ 1 τ 1 − τ 1 τ ∗ 1 is the pro jection o nt o the subspace spanned b y 0 , and co njuga ting b y the op er ators τ n i we o btain all the matrix elements, so C ∗ ( N ) contains the alg ebra of compact op erator s.  Since o ne can ca ncel pair s τ ∗ 1 τ 1 , a g eneral ele men t o f C ∗ ( N ) of the form τ i 1 1 ( τ ∗ 1 ) j 1 τ i 2 1 ( τ ∗ 1 ) j 2 . . . τ i k 1 ( τ ∗ 1 ) j k can b e reduced to τ i 1 ( τ ∗ 1 ) j . Suppo se that n = i − j is p ositive. Then it is easy to see that t i t − 1 j is a s ubtrans- lation of t n , that is, its doma in is a subs et of the domain of t n and where b oth a re defined they a re equa l. Mo reov er the domains differ only by a finite set. Hence as op erator s τ i 1 ( τ ∗ 1 ) j and τ n 1 differ by a finite rank op era tor. Similar ly if i − j = n is negative then t i t − 1 j is a subtranslation of t n , and the ope r ators τ i 1 ( τ ∗ 1 ) j and ( τ ∗ 1 ) n again differ by a finite r ank op er ator. Thu s elements of the for m τ n 1 and ( τ ∗ 1 ) n along with finite rank oper ators span a dense subspace of C ∗ ( N ). It is easy to see that τ i 1 and ( τ ∗ 1 ) j never differ b y a compact op erator (ther e is no cancellation be t ween them) while τ i 1 and τ j 1 differ by a compact op er a tor only if i = j . Hence the map τ n 1 7→ σ n 1 , ( τ ∗ 1 ) l 7→ ( σ ∗ 1 ) n , and k 7→ 0 for every co mpact op erator k , extends to a well defined linear map from a dense subspace o f C ∗ ( N ) to C ∗ ρ ( Z ). This ma p is mor eov er a *-a lgebra homomo rphism, and extends by contin uity to a * -homomor phism from C ∗ ( N ) to C ∗ ρ ( Z ) w ith kernel consisting of compact o p erators , g iving us an extension 0 → K ( ℓ 2 ( N )) → C ∗ ( N ) → C ∗ ρ ( Z ) → 0 . W e now make the following identifications. The Hilb ert space ℓ 2 ( N ) is natura lly ident ified with the Ha rdy space H 2 , by taking e n to z n for n ∈ N . Similarly ℓ 2 ( Z ) is na tur ally identified with L 2 ( S 1 ), again the map takes e n to z n (how e ver this is now for all n ∈ Z ). With these identifications τ 1 is identified with T z , the T o e plitz op erator ass o ciated with the identit y map z : S 1 → S 1 , while the gene r ator σ 1 of C ∗ ρ ( Z ) is identified with M z , the op erator of p o int wise multiplication by the function z . Since C ∗ ( N ) and T are g enerated b y τ 1 and T z resp ectively , the ab ov e ident ificatio n o f Hilb ert spa ces gives an isomorphis m C ∗ ( N ) ∼ = T . Simila rly we hav e C ∗ ρ ( Z ) ∼ = C ( S 1 ). The isomor phisms C ∗ ( N ) ∼ = T and C ∗ ρ Z ∼ = C ( S 1 ) identify this with the T o eplitz extension, 0 → K ( H 2 ) → T → C ( S 1 ) → 0 . 3.2. The translatio n algebra C ∗ ( Z \ { 0 } ) . F ro m now o n we will abuse notation, denoting b oth a par tial trans lation and the op erator that it defines with the sa me symbol, using context to determine the meaning . Hence if s is a par tia l transla tion we may write s ∗ to denote the adjoint of the o p erator co rresp onding to s . W e next consider the effect of removing a sing le p oint from the gro up Z . P AR TIAL TRANSLA TION ALGEBRAS F OR TREES 5 Theorem 2. The p artial tr anslation algebr a C ∗ ( Z \ { 0 } ) is a trivial exten sion of C ( S 1 ) by the c omp act op er ators which is ther efor e not e quivalent (in the sense of K -homolo gy) t o the T o eplitz extension. Pr o of. Let X = Z \ { 0 } . The partial transla tio ns o n X that we obtain b y restric ting and cor e stricting the tr anslations of Z hav e the for m s n : Z \ {− n, 0 } → Z \ { 0 , n } , s n : j 7→ j + n. Note that s 0 is the identit y and s − n = s ∗ n for all n ∈ Z . It app ears a priori that we need all of these pa rtial translatio ns to genera te the algebra C ∗ ( Z \ { 0 } ), since it is not true in this ca se that s n = ( s 1 ) n . In fact we will see that it s uffices to have s 0 = 1 , s 1 and s 2 . Consider s 1 s ∗ 1 . This acts a s the iden tity on i for all i 6 = 1, while it is undefined at 1 . T hus, a s a n element of the algebra , s 1 s ∗ 1 = 1 − p 1 where p 1 denotes the r a nk 1 pro jectio n onto the basis element e 1 in ℓ 2 ( X ). Hence the algebra C ∗ ( X ) cont ains the rank 1 pro jection p 1 . Now for any i, j ∈ X , the matrix element e i,j is given by s i − 1 p 1 ( s ∗ j − 1 ), hence the algebra contains all matr ix elements, and hence all compact op erator s. Now co nsider comp ositio ns of the for m s i 1 s i 2 . . . s i k . Wh ere this is defined it translates b y l = i 1 + · · · + i k . In other words it is a subtranslatio n of s l . The domain on which this is defined is Z \ { 0 , − i k , − ( i k − 1 + i k ) , . . . , − l } in particular it differs fr om the domain of s l by o nly finitely many p oints. As befo r e s i − s j is co mpact only if i = j a nd hence we deduce that we hav e a n extension of the for m 0 → K ( ℓ 2 ( X )) → C ∗ ( X ) → C ∗ ρ ( Z ) → 0 , where the map C ∗ ( X ) → C ∗ ρ ( Z ) is giv en b y extending linearly the ma p taking s l to [ l ] a nd v anishing o n the c ompacts. W e ca n identify the a lgebra C ∗ ( X ) mor e explicitly as follows. Consider the partial transla tion s 1 . This takes Z \ {− 1 , 0 } to Z \ { 0 , 1 } . W e can extend it to a globally defined tra nslation t on X by defining t ( j ) = s 1 ( j ) for j ∈ X , j 6 = − 1 and t ( − 1) = 1 . Note that this is a compact per turbation of s 1 and hence lies in the alg ebra. Mor eov er, t n is a compact per turbation o f s n for all n , thus C ∗ ( X ) is generated by t a lo ng with all co mpact op era tors. Now co nsider the a lgebra gener ated by t alone. Let φ : X → Z be the bijectiv e coarse e q uiv alence defined by φ ( j ) = j for j > 0 and φ ( j ) = j + 1 for j < 0. If U denotes the co rresp onding unitary from ℓ 2 ( X ) to ℓ 2 ( Z ) then U tU ∗ is rig ht translation by 1, i.e. it is the element [1] in C ∗ ρ ( Z ). Hence using U to identify these tw o Hilb ert spaces, the alg ebra C ∗ ( t ) is identified with C ∗ ρ ( Z ), the alg ebra o f compacts K ( ℓ 2 ( X )) is identified with K ( ℓ 2 ( Z )) and hence (since these together generate C ∗ ( X ) we deduce that C ∗ ( X ) is identified with the sum C ∗ ρ ( Z )+ K ( ℓ 2 ( Z )). This identifies the extension as 0 → K ( ℓ 2 ( Z )) → C ∗ ρ ( Z ) + K ( ℓ 2 ( Z )) → C ∗ ρ ( Z ) → 0 . As a K -homo logy cycle for C ∗ ρ ( Z ) this extension is ( ℓ 2 ( Z ) , ρ, 1) where ρ denotes the rig ht regula r r e presentation. This cycle is degenerate, hence it is a trivial element in K -ho mo logy . How ever the K- homology class represented by the T o e plitz extension is non-trivial, thus the tw o extensions are not equiv a lent.  6 J. BRODZKI, G.A. NIBLO, AND N.J . WRIGHT 3.3. Coarsel y disconnected subspaces. W e will now consider subspace s of Z such as { i 2 : i ∈ N } , having arbitr arily large gaps. These are said to b e coarsely disconnected. Prop ositi o n 1. L et X b e a subset of Z which is c o arsely e quivalent t o { i 2 : i ∈ N } . Then C ∗ ( X ) is iso morphic to the u nitise d c omp acts f K ( l 2 ( X )) . Pr o of. Subspaces of Z whic h ar e coa rsely equiv alent to { i 2 : i ∈ N } can be c harac- terised as follows. Le t X + denote the non-neg ative par t of X and let X − denote the negative part o f X . Then X + is either finite or cons is ts o f an incr easing sequence a i of p oints with a i +1 − a i tending to infinit y a s i → ∞ . Similar ly X − is either finite or consists of an decreas ing sequence b i of p oints with b i − b i +1 tending to infinit y as i → ∞ , and X + , X − cannot b oth b e finite. Now fix so me n 6 = 0, a nd consider t n the translation b y n on X . The doma in of t n consists o f those x ∈ X such that x + n lies in X . Since if X − is infinite, the g a ps b n − b n +1 tend to infinit y , it follows that if the domain is non-empty then there is a lea st s uch x . Simila rly there is a g reatest s uch x , a nd hence the domain of t n is finite. Th us as an op era tor on l 2 ( X ) it fo llows that t n is compact. F or n = 0 the tra nslation by n is the ide ntit y on X , and hence we deduce that C ∗ ( X ) is a s ubalgebra o f the unitisation o f the compact o p erators f K ( l 2 ( X )). W e will now sho w that C ∗ ( X ) con tains all matrix elements. Pick some n > 0 for which the domain of t n is non-empty , let x b e the lea s t element of the do main., and let y b e the gr e atest element. Then y − x = mn for some m > 0, a nd the translation ( t n ) m takes x to y and is undefined otherwise . Hence as an op erator ( t n ) m is the rank 1 o p er ator taking e x to e y . Now, for a ny a, b in X , the comp os ition t b − y ( t n ) m t x − a is the rank 1 op era tor taking e a to e b . The closed s pan of these op erator s is K ( l 2 ( X )), hence we co nclude that C ∗ ( X ) = f K ( l 2 ( X )). As in the previo us examples there is a n extension: in this case we s e e tha t taking the quotient by the co mpact op erator s we obtain a map to C , since only the ident ity on X has infinite supp or t, and the quotient her e ca n be regarded as the group C ∗ -algebra of the trivial gr oup. Thus we have the extensio n 0 → K ( l 2 ( X )) → f K ( l 2 ( X )) → C ∗ ρ ( { 0 } ) → 0 .  3.4. The translation alge bra of the prim es. The primes inherit a partial tr ans- lation algebra C ∗ ( P ) fro m the int ege r s and for ea ch po sitive integer n there is sp ecific element t n ∈ C ∗ ( P ) repres ent ed by translation b y n . Since there is only one even prime the element t ∗ 1 t 1 is a rank 1 pro jectio n, and since the translations act tra ns itively the partial translatio n algebr a contains every compact op erator . Clearly the element t 0 is the identit y , so the partial translation algebra of the primes actually contains the unitised algebra f K ( l 2 ( P )) which is an extension of the for m: 0 → K ( l 2 ( X )) → f K ( l 2 ( X )) → C ∗ ρ ( { 0 } ) → 0 . Now the twin prime conjectur e is equiv a lent to the statement that the op era tor t 2 is no t co mpact. Indeed de Polignac’s genera lisation of the twin primes conjecture, which as serts the ex is tence of infinitely many pr ime pairs separated by distance P AR TIAL TRANSLA TION ALGEBRAS F OR TREES 7 n for eac h even n , ([6]) is equiv alent to the statemen t that t n is not compact for any even n . Note that if de Polignac’s co njecture holds for some even n then the algebra C ∗ ( P ) is strictly larger than f K ( l 2 ( X )). The algebr a f K ( l 2 ( X )) has a unique idea l, na mely K ( l 2 ( X )). I f C ∗ ( P ) is isomor- phic to f K ( l 2 ( X )) it also must contain a unique ideal and this must b e K ( l 2 ( X )) since this is an ideal in C ∗ ( P ). It fo llows that taking the q uotient by this ideal we obtain in b oth cases a c o py of C . Hence C ∗ ( P ) = f K ( l 2 ( X )). Theorem 3. The algeb r a C ∗ ( P ) is not isomorphic to f K ( l 2 ( X )) if and only if de Polignac’ s c onje ctu r e holds for some even n , if and only if the quotient of C ∗ ( P ) by the c omp act op er ators is st rictly lar ger than C . 4. Transla tion subsp a ces of F n 4.1. T ranslation algebras and the Cun tz e xtension. In this sectio n w e will consider the trans la tion a lgebras arising fro m subspaces of the regular 4-v alent tree, the Cayley graph o f the free group of rank 2, F 2 . W e will indicate briefly how the arguments c arry o ver to the g eneral case. Let X denote the set of p os itive words in F 2 including the iden tity . That is X consists of e alo ng with all w or ds in the generator s a and b . Consider the partial transla tions o n X defined by a a nd b (acting by right tr ans- lation). Then a is globally defined, while the image of a is the set of all p ositive words ending in a . Similar ly b is globally defined with image co nsisting of p ositive words ending in b . The partial tra nslations a ∗ and b ∗ are left inv erses of a and b , resp ectively . Viewed as an oper ator on ℓ 2 ( X ), the element a ∗ acts as a bijection from the words ending in a to all words, a nd a cts as zer o on e a nd all words ending with b . Similarly for b , hence we have the following a lg ebra relations: a ∗ a = b ∗ b = 1 , a ∗ b = b ∗ a = 0 , aa ∗ + bb ∗ = 1 − p e where p e denotes the rank 1 pro jection onto e in ℓ 2 ( X ). A general transla tio n is given by rig ht multiplication by a reduced word x 1 . . . x k with x i ∈ { a, b, a − 1 , b − 1 } . I t is easy to see that as a partial tra nslation ( x 1 . . . x k ) acts as the comp osition y k y n − 1 . . . y 1 where y i equals a or b resp ectively if x i is a or b , while y i = a ∗ , b ∗ resp ectively if x i is a − 1 or b − 1 . Note the r eversal o f the order of co mpo sition, due to the fact that we ar e using r ight multiplications, e.g. the op er ator ab is r ight multiplication by the gr oup element ba . W e th us see that the algebra C ∗ ( X ) is the C ∗ -algebra g enerated by a and b . (W e do not need to explicitly include the identit y s ince a ∗ a = 1.) W e now compare C ∗ ( X ) with the algebra C ∗ ( Y ) of a slightly bigger subset of F 2 . Le t Y denote the set of elements in F 2 of the form a n w ( a, b ), where n ∈ Z and w ( a, b ) is a word in a and b . Tha t is, Y consists o f reduced words in a, a − 1 and b , where a − 1 can only o ccur b efore the first b . Using Y fixe s a defect in X : each vertex of X has three neighbours with the exc eption of the identit y , which only has t wo. In Y ho wev er every vertex has three neighbour s , i.e. Y is a thre e regula r tree. W e will now co nsider the alg ebra C ∗ ( Y ). Again the algebr a is generated by tw o elements a and b , with a ∗ a = b ∗ b = 1. Let A denote the set o f words ending with either a or a − 1 , alo ng with the iden tit y e . (Note that the only words ending with a − 1 are words o f the for m a − n .) Let B denote the set of w or ds ending with b . Note that B is the co mplement of A in Y . The partia l translation a g ives a bijection 8 J. BRODZKI, G.A. NIBLO, AND N.J . WRIGHT from Y to A while, b gives a bijection from Y to B . The partial transla tions a ∗ and b ∗ are left in verses to a and b , hence, a s an o p erator, aa ∗ is the identit y on A and v anishes o n B , i.e. it is the pro jection of ℓ 2 ( Y ) ont o ℓ 2 ( A ). Conv ersely bb ∗ is the pro jection of ℓ 2 ( Y ) o nto ℓ 2 ( B ). Thu s we conclude that C ∗ ( Y ) is a s ubalgebra of B ( l 2 ( Y )) genera ted by tw o isometries a and b , with the pr op erty that aa ∗ + bb ∗ = 1 . This is the defining prop erty of the Cuntz a lgebra O 2 , thus we have C ∗ ( Y ) ∼ = O 2 . 1 W e will now r elate C ∗ ( Y ) to C ∗ ( X ). It will b e imp orta nt to remem b er at e ach stage whether a w ord in a, b, a ∗ , b ∗ is to b e considered as an op er ator on ℓ 2 ( X ) or on ℓ 2 ( Y ). Let x = x k x k − 1 . . . x 1 be a word in a, b, a ∗ , b ∗ , considere d a s an op era tor on ℓ 2 ( X ), and let y = y k y k − 1 . . . y 1 denote the corresp onding op era to r on ℓ 2 ( Y ). W e claim that this gives a well defined map from C ∗ ( X ) to C ∗ ( Y ). W e will say that a w or d is quasi-r e duc e d if it do es not contain a ∗ a o r b ∗ b (since these w ill a c t a s the identit y), and do es not contain a ∗ b or b ∗ a (sinc e these will act as zero ). The quasi-r educed words (as op era tors on ℓ 2 ( X )) span a dense subalgebr a of C ∗ ( X ). W e claim tha t the quasi-reduce d words are linea rly indep endent. Note that a quasi-reduce d word is neces sarily of the for m w ( a, b ) w ′ ( a ∗ , b ∗ ), wher e w ( a, b ) (res p. w ′ ( a ∗ , b ∗ ) denotes a positive word in a, b (res p. a ∗ , b ∗ ). Say that a word is o f t yp e l if w ′ is a w ord o f length l . Note that a qua si-reduced word of t yp e l acts as the zero op era tor on words in X of length 0 , 1 , . . . l − 1. Thus for a linea r combination of q uasi-reduced words, the part of type 0 determines the a ction on e . Having remov ed the type 0 part, the w ords of t yp e 1 then determine the action o n words in X of length 1, etc. Hence considering the ac tio n on words o f length 0 , 1 , 2 , . . . we deduce that a linear combination which acts a s zero, must b e zer o; that is, the quasi-reduce d words are linearly indep endent. Now we return to the ab ov e map x = x k x k − 1 . . . x 1 ∈ B ( ℓ 2 ( X )) 7→ y = y k y k − 1 . . . y 1 ∈ B ( ℓ 2 ( Y )) . Since the quas i-reduced words ar e linearly indepe ndent , this gives a well defined map from their linea r span to C ∗ ( Y ). This is a *-alg ebra homomor phism, and hence contractive, so it extends to a well-defined * -homomor phis m fro m C ∗ ( X ) to C ∗ ( Y ). Since a, b genera te C ∗ ( Y ), this ho momorphism is surjective. Clearly the kernel includes p e since aa ∗ + bb ∗ 7→ 1. Hence in fact the kernel includes K ( ℓ 2 ( X )), since one can eas ily construct a ll matrix elements by pre- and po st-comp osing p e with tra nslations. Lemma 2. L et x = x k x k − 1 . . . x 1 b e a wor d in a, b, a ∗ , b ∗ , c onsider e d as an op er ator on ℓ 2 ( X ) . Consider the c orr esp onding op er ator y = y k y k − 1 . . . y 1 on ℓ 2 ( Y ) , and let x ′ b e the t runc ation P y P wh er e P is the pr oje ction of ℓ 2 ( Y ) onto ℓ 2 ( X ) . Then x ′ is a c omp act p erturb ation of x . Pr o of. Note that as o p erators o n ℓ 2 ( Y ), P y i and y i P differ only on a s ingle basis vector. Hence P y − y P is a compact op erato r. Thus x ′ = P y P = P y k y k − 1 . . . y 1 P is a compact per turbation of ( P y k P )( P y k − 1 P ) . . . ( P y 1 P ). It no w s uffices to note 1 Recall that the Cuntz algebra is constructed concretely as the al gebra generated by tw o suc h isometries. Cun tz show ed that this is the unique algebra with these prop erties. P AR TIAL TRANSLA TION ALGEBRAS F OR TREES 9 that P y i P = x i , i.e. for a, b etc. vie wed a s translations o f Y , trunca ting to X g ives the co r resp onding translatio n on X .  Since we hav e a rig ht-in verse, which is als o a left-inv erse mo dulo compact op era- tors, it follows that the kernel is precise ly the co mpact op erator s. W e thus pro duce an extension of O 2 0 → K ( ℓ 2 ( X )) → C ∗ ( X ) → C ∗ ( Y ) ∼ = O 2 → 0 . This is an a na logue for the Cuntz algebr a of the T o eplitz extens io n. Note that the argument showing that there is a map from C ∗ ( X ) to C ∗ ( Y ) in fact shows that C ∗ ( X ) has a universal pro p erty: If A is any C ∗ -algebra genera ted by tw o elements s, t satis fying s ∗ s = t ∗ t = 1 a nd s ∗ t = t ∗ s = 0 then there is a surjection from C ∗ ( X ) to A , taking a to s a nd b to t . These relations s uffice to show that there is a homomorphism from the a lgebra spanned by the quas i-reduced words to A , and this extends by contin uity to a sur jective *-homomor phism. Another example o f an alge br a satisfying these r elations is the algebr a E 2 of Cun tz, [5]. By definition this is the suba lgebra of O 3 generated by the first t wo isometries V 1 , V 2 , and since V 1 V ∗ 1 + V 2 V ∗ 2 + V 3 V ∗ 3 = 1, it fo llows that V ∗ 1 V 2 = V ∗ 2 V 1 = 0. Thus there is a sur jection from C ∗ ( X ) to E 2 . In [5], Cuntz show ed that ther e is an extension 0 → J 2 → E 2 → O 2 → 0 where J 2 is an ideal in E 2 isomorphic to the algebra o f compact operato rs. Here the quotient map takes the generato rs of E 2 to the gener a tors o f O 2 . Thus the quotient map C ∗ ( X ) → C ∗ ( Y ) ∼ = O 2 factors through the ma p C ∗ ( X ) → E 2 . The kernel of the latter is thus an ide a l in the kernel of the former, which is K ( ℓ 2 ( X )). Since this is simple, and the kernel is not the whole of K ( ℓ 2 ( X )), we deduce that in fact the surjection fro m C ∗ ( X ) to E 2 is in fact a n iso morphism. Thus we have prov ed the following theorem. Theorem 4. Ther e is a c anonic al isomo rphism b etwe en the Cuntz extension 0 → J 2 → E 2 → O 2 → 0 and the ex tension 0 → K ( ℓ 2 ( X )) → C ∗ ( X ) → C ∗ ( Y ) → 0 wher e X and Y ar e subsets of F 2 as ab ove. 4.2. The algebras O n . F ollowing the construction in the previous section w e re - place the free group F 2 with F n . Again we define the subtree X to b e spanned by all p ositive words, a nd choo sing a generator we extend this to a regular n + 1-v alent tree which we ca ll Y . The inclusio ns of X and Y into the Cayley gr aph endow them with partial translatio n algebras C ∗ ( X ) , C ∗ ( Y ) resp ectively a nd w e obta in isomorphisms C ∗ ( X ) ∼ = E n and C ∗ ( Y ) ∼ = O n , with the algebras defined b y Cuntz in [5]. By the s a me arg ument a s ab ov e the inclusion of X into Y ca n b e s hown to induce the extension 0 → J n → E n → O n → 0 . 10 J. BRODZKI, G.A. NIBLO, AND N.J . WRIGHT 4.3. Emb edding C ∗ ρ F 2 in O 2 . In [2] it was shown tha t there is an injection of the reduced C ∗ -algebra C ∗ ρ F 2 int o O 2 . W e conclude by constructing such an inclusion explicitly us ing o ur identification of O 2 with C ∗ ( Y ). T o do so we will co nstruct explicit injective quasi-is ometries fro m F 2 int o Y . It is well known that for any n, m > 2 the n - regular tree is quasi-iso metr ic to the m -regular tree , a nd such quasi- isometries a re easy to cons tr uct. The purp o se of the co nstruction given here is that these quasi- isometries are defined in such a wa y that they b ehav e well with res p ect to the natural pa rtial tra nslations on F 2 and o n Y . An element g ∈ F 2 is uniquely determined by a g eo desic s egment ema nating from the identit y element, and this is enco ded by a sequence o f turns in the Cayley graph of F 2 . T o fo r malise wha t we mean by a turn we consider the extensio n of F 2 by the cyclic group of order 4 g enerated by a rotatio n o f the Cayley graph ar ound the bas e p oint as follows. Let α b e the automorphism of F 2 taking a to b and b to a − 1 . W e will use the notation x α for the image of x under α . Let H b e the gr oup of a utomorphisms of F 2 generated by α , and let G be the semi-direc t pro duct H ⋉ F 2 . This gr oup is generated by a, b, α with the re la tions α 4 = 1, αaα − 1 = b and αbα − 1 = a − 1 . In general we hav e αxα − 1 = x α , for x a word in F 2 . An element o f F 2 , viewed as a subgro up of G , can b e written uniquely in the fo rm a − n α i 0 aα i 1 a . . . α i d − 1 aα i d , where n , d ≥ 0, i j ∈ {− 1 , 0 , 1 } for j < d , i 0 + · · · + i d = 0 mo d 4 , a nd if n > 0 then i 0 6 = 0 . A reduced word in F 2 can b e dir ectly transcrib ed in this fo r m, and we note that the condition that the word is reduce d translates directly into the r estrictions that i 0 6 = 0 if n > 0, and that there are no factors of α 2 , exc e pt p oss ibly for the final term α i d . W e will now consider a couple of examples. The word aba is equa l to the normal form word a 0 α 0 aα 1 aα − 1 aα 0 . Geometrically we ca n interpret this as s aying that starting from an initia l hea ding of Eas t (the a directio n) we go fo rwards ( α 0 ) then turn left and mov e for ward o ne unit ( α 1 ) then turn right and move forward one unit ( α − 1 ). Our final heading is Ea st and this ma y b e rea d fro m the ter minal α 0 . The word a − 1 b 2 is equal to a − 1 α 1 aα 0 aα − 1 , which geometric ally w e interpret as moving backw ards by 1 while facing East, turning left and moving o ne unit ( α 1 ) then contin uing forwards for one unit ( α 0 ). Note that at the end w e are facing North, which may als o b e r ead fr om the terminal α − 1 . Backwards mov es a re specia l in the following sense: they can only o c c ur a s initial mo ves, a nd they do no t change the direction in which we are facing . W e will call the direction in whic h we are facing at any stage the he ading . W e define a heading function h : F 2 → Z / 4 Z by h ( a − n α i 0 a . . . aα i d ) = − i d . As i 0 + · · · + i d = 0 mo d 4 , we hav e h ( x ) = i 0 + · · · + i d − 1 . This justifies the observ ation ab ov e that o ur final heading can b e r e a d from the terminal exp one nt of α . Indeed we no te that for a word x in F 2 , if x = a − n for n ≥ 0 then h ( x ) = 0, other wise h ( x ) determines the final term in the word: if x = x 1 . . . x k then x k = α h ( x ) aα − h ( x ) . T o embed F 2 int o Y , we will need to enco de the turns α i j and the headings h ( x ) as words in a, b . Define u 0 = a, u 1 = b 2 , u − 1 = ab , and define v 0 = a 2 , v 1 = b 2 , v 2 = ab a nd v 3 = b a (the index is interpreted mo dulo 4). W e can now define an embedding of F 2 int o Y φ 0 : F 2 → Y , φ 0 : x = a − n α i 0 aα i 1 a . . . α i d 7→ a − n u i 0 u i 1 . . . u i d − 1 v h ( x ) . This e xpression for φ 0 ( x ) is not necessar ily a reduce d word, how ever a t mos t there is cancella tion o f o ne factor of a fro m u i 0 with one factor of a − 1 . O ne can r ead off P AR TIAL TRANSLA TION ALGEBRAS F OR TREES 11 i d , i d − 1 , . . . i 0 from the rig ht of the w ord, hence one can recov er the original w ord x . Thus φ 0 is injective. W e ca n ex tend φ 0 to a map fr om G to Y in the fo llowing simple wa y . A g eneral element o f G is of the form a − n α i 0 aα i 1 a . . . α i d , where n, d ≥ 0, i j ∈ {− 1 , 0 , 1 } for j < d , and if n > 0 then i 0 6 = 0 . Note that w e now drop the requirement that i 0 + · · · + i d = 0 mo d 4 . W e will call this the nor mal form for a n e lement of G . W e define Φ : x = a − n α i 0 aα i 1 a . . . α i d 7→ a − n u i 0 u i 1 . . . u i d − 1 v h ( x ) where as b efor e h ( x ) = − i d . Again we can rec over the word x fro m its image , thus Φ is injective. Mor eov er it is a bijection: for any element y of Y we c an r ead off a corres p o nding word x s uch that Φ( x ) = y . The restr ic tio n of Φ to F 2 in G is φ 0 . Moreover G decomp oses as four left cosets o f F 2 , and these a re pres erved b y the rig ht action of F 2 on G . Using the bijection Φ we can define a corr e sp onding action of F 2 on Y . This action is fre e and ha s four or bits which ar e identified with the cosets b y Φ. Given the set of orbit representatives I = { v 0 , v 1 , v 2 , v 3 } the spac e l 2 ( Y ) is thus identified as l 2 ( F 2 ) ⊗ l 2 ( I ), and the action of F 2 on the right gives r ise to the the repr esentation ρ ⊗ 1 where ρ is the right r egular repres e nt ation of l 2 ( F 2 ). This the natural embedding of C ∗ ρ ( F 2 ) int o C ∗ ρ ( G ). F urthermor e the bij ection Φ induces a n isomorphis m Φ ∗ betw een the bo unded linea r op erator s on ℓ 2 ( G ) a nd those o n ℓ 2 ( Y ). Theorem 5. The image of C ∗ ρ ( F 2 ) ⊗ 1 under the map Φ ∗ is c ontaine d in C ∗ ( Y ) ∼ = O 2 . Pr o of. Note that C ∗ ρ ( F 2 ) ⊗ 1 is generated by the elements ρ ( a ) ⊗ 1 , ρ ( b ) ⊗ 1. W e will construct tw o elemen ts t a , t b ∈ C ∗ ( Y ) and show that these are the images under Φ ∗ of ρ ( a ) ⊗ 1 , ρ ( b ) ⊗ 1 r esp ectively . It will thus follow that the imag e o f C ∗ ρ ( F 2 ) ⊗ 1 is contained in C ∗ ( Y ). Recall tha t righ t multiplication by a, b, a − 1 , b − 1 in F 2 induce partial isometries a, b, a ∗ , b ∗ on ℓ 2 ( Y ). Let t a , t b be the partial isometries defined a s follows: t a = a 3 ( a ∗ ) 2 + ba ( a ∗ ) 2 b ∗ + aba ∗ b ∗ a ∗ b ∗ + b 2 ( b ∗ ) 2 a ∗ b ∗ + a 2 ba ( b ∗ ) 2 + a 2 b 2 b ∗ a ∗ , t b = b 2 a ( b ∗ ) 2 + aba ∗ b ∗ a ∗ + a 2 a ∗ ( b ∗ ) 2 a ∗ + ba ( b ∗ ) 3 a ∗ + b 3 aa ∗ b ∗ + b 4 ( a ∗ ) 2 . Viewed as partial translatio ns our op era tors are chosen so that for any e le men t y ∈ Y ⊂ F 2 , there is a unique ter m in t a which is defined a t y , and likewise for t b . Moreov er we will see that t a and t b are bijections from Y to itself. In ter ms of the algebra C ∗ ( Y ) this mea ns that t a and t b are unitar ies. The following ta bles , s how which term in t a , t b acts on any given word in y , and how the word is changed. W ord ends in Applicable ter m for t a Ending re pla ced by a 2 = v 0 a 3 ( a ∗ ) 2 a 3 = u 0 v 0 a 2 b = u 0 v 2 ba ( a ∗ ) 2 b ∗ ab = v 2 abab = u − 1 v 2 aba ∗ b ∗ a ∗ b ∗ ba = v 3 b 2 ab = u 1 v 2 b 2 ( b ∗ ) 2 a ∗ b ∗ b 2 = v 1 b 2 = v 1 a 2 ba ( b ∗ ) 2 aba 2 = u − 1 v 0 ba = v 3 a 2 b 2 b ∗ a ∗ b 2 a 2 = u 1 v 0 12 J. BRODZKI, G.A. NIBLO, AND N.J . WRIGHT W ord ends in Applicable ter m for t b Ending replaced by b 2 = v 1 b 2 a ( b ∗ ) 2 ab 2 = u 0 v 1 aba = u 0 v 3 aba ∗ b ∗ a ∗ ba = v 3 ab 2 a = u − 1 v 3 a 2 a ∗ ( b ∗ ) 2 a ∗ a 2 = v 0 b 3 a = u 1 v 3 ba ( b ∗ ) 3 a ∗ ab = v 2 ab = v 2 b 3 aa ∗ b ∗ ab 3 = u − 1 v 1 a 2 = v 0 b 4 ( a ∗ ) 2 b 4 = u 1 v 1 F or the benefit of the rea der w e c o nsider the following e xample. Let y = Φ( α 0 aα 1 ) = u 0 v 3 . Then t a y = u 0 u 1 v 0 = Φ( α 0 aα 1 aα 0 ). Thus the action of t a on y , is the same as the right actio n of a on y , via the identification Φ of Y with G . Similarly t b y = v 3 = Φ( α ), and we note that the rig ht action of b on aα produces aα 2 aα − 1 = α 2 a − 1 aα − 1 = α . Thus the action of t b on y agr ees with the right action of b o n y . W e now consider the gener al c a se. Right multiplication by the element a ta kes a word o f the for m a − n α i 0 aα i 1 a . . . α i d to a − n α i 0 aα i 1 a . . . α i d aα 0 . This is in no rmal form unles s i d = 2 in which cas e we have cancellatio n as aα 2 a = α 2 a − 1 a = α 2 , th us a − n α i 0 aα i 1 a . . . α i d − 1 aα i d aα 0 = a − n α i 0 aα i 1 a . . . α i d − 1 +2 . In terms of the actio n of F 2 on Y we th us find that multiplication b y a ha s the effect of ta king a word of the form y v 0 to y u 0 v 0 , taking y v 1 to y u − 1 v 0 , taking y v 3 to y u 1 v 0 and tak ing y u i v 2 to y v 2 − i . Thus the tra nslation t a is pr e cisely the r ight action of a o n Y . Similarly r ight multiplication by the element b takes a word o f the form a − n α i 0 aα i 1 a . . . α i d to a − n α i 0 aα i 1 a . . . α i d +1 aα − 1 . This is in normal form unless i d + 1 = 2 in which case we hav e the c a ncellation a − n α i 0 aα i 1 a . . . α i d − 1 aα i d +1 aα − 1 = a − n α i 0 aα i 1 a . . . α i d − 1 +1 . Hence in ter ms o f the action of F 2 on Y we find that m ultiplica tion b y b ha s the effect of taking a word o f the fo rm y v 0 to y u 1 v 1 , tak ing y v 1 to y u 0 v 1 , tak ing y v 2 to y u − 1 v 1 and taking yu i v 3 to y v 3 − i . Aga in one can c heck that this is precisely the effect of t b . Thus the translation t b is the rig h t a ction of b o n Y . W e conclude that the suba lgebra C ∗ ( t a , t b ) of C ∗ ( Y ) is generated by tw o unitar ies which ac t on l 2 ( Y ) ∼ = l 2 ( F 2 ) ⊗ l 2 ( I ) a s ρ ( a ) ⊗ 1, ρ ( b ) ⊗ 1, wher e ρ is the r ight reg ula r representation of F 2 on l 2 ( F 2 ). This completes the pr o of.  W e c o nclude this section by rema rking that the a b ove theor em generalise s to show that C ∗ ρ G ( G ) embeds int o C ∗ ( Y ) where ρ G denotes the r ight r egular represen- tation of G on l 2 ( G ). T aking t a , t b as in the pro of of the theorem, we a dditionally define t α = ab ( a ∗ ) 2 + a 2 ( b ∗ ) 2 + b 2 a ∗ b ∗ + bab ∗ a ∗ . Then t α takes a word of the form y v i to y v i − 1 . This is pr e cisely the action of α by right multiplication o n Y , hence the subalgebr a C ∗ ( t a , t b , t α ) of C ∗ ( Y ) is cano nically ident ified as C ∗ ρ G ( G ). W e thus have an e mbedding of C ∗ ρ G ( G ) into O 2 = C ∗ ( Y ). P AR TIAL TRANSLA TION ALGEBRAS F OR TREES 13 References [1] J. Brodzki, G. A. 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