Approximating the group algebra of the lamplighter by infinite matrix products
aa r X i v : . [ m a t h . R A ] M a y APPROXIMATING THE GROUP ALGEBRA OF THE LAMPLIGHTER BYINFINITE MATRIX PRODUCTS
PERE ARA AND JOAN CLARAMUNT
Abstract.
In this paper, we introduce a new technique in the study of the ∗ -regular closure of somespecific group algebras KG inside U ( G ) , the ∗ -algebra of unbounded operators affiliated to the groupvon Neumann algebra N ( G ) . The main tool we use for this study is a general approximation result fora class of crossed product algebras of the form C K ( X ) ⋊ T Z , where X is a totally disconnected compactmetrizable space, T is a homeomorphism of X , and C K ( X ) stands for the algebra of locally constantfunctions on X with values on an arbitrary field K . The connection between this class of algebrasand a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, westudy an explicit approximation for the lamplighter group algebra. This is used in another paper bythe authors to obtain a whole family of ℓ -Betti numbers arising from the lamplighter group, most ofthem transcendental. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Background and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ -regular rings and rank functions . . . . . . . . . . . . 52.2 Division closure and rational closure . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 ℓ -Betti numbers for group algebras and the Atiyah Conjecture . . . . . . . . . . . . 7 . . . . . . . . . . . . .
84 The ∗ -regular closure R A and its ideal structure . . . . . . . . . . . . . . . . . . . R A and R ∞ assuming existence of a periodic point . . . . . . . 114.2 The ∗ -regular closure R B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Z -crossed product algebras . . . . . . . . . . . . . . . .
256 The lamplighter group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A n for the lamplighter group algebra . . . . . . . . . . 316.2 The algebra of special terms for the lamplighter group algebra . . . . . . . . . . . . 326.3 Analysis of the algebra A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction
For a discrete group G and a subfield K of C closed under complex conjugation, the group algebra KG can be naturally seen as a ∗ -subalgebra of the von Neumann algebra N ( G ) of G . The followingproblem, known as the Strong Atiyah Conjecture (SAC), was solved in the negative by Grigorchukand Żuk [20] (see also [21] and [12]):
Date : Wednesday 27 th May, 2020.2010
Mathematics Subject Classification.
Primary 16E50; Secondary 16S35, 37A05, 16D70.
Key words and phrases. ∗ -regular closure, ℓ -Betti number, Atiyah conjecture, rank function, lamplighter.Both authors were partially supported by DGI-MINECO-FEDER through the grants MTM2014-53644-P andMTM2017-83487-P, and by the Generalitat de Catalunya through the grant 2017-SGR-1725. The second named authorwas also partially supported by DGI-MINECO-FEDER through the grant BES-2015-071439 and by the research fundingBrazilian agency CAPES. trong Atiyah Conjecture: If T is an m × n matrix over KG , then dim N ( G ) ( ker T ) ∈ X H | H | Z , where H ranges over the family of all the finite subgroups of G , and ker T denotes the kernel of theoperator ℓ (Γ) n → ℓ (Γ) m given by left multiplication by T (see e.g. [31, page 369]).Here dim N ( G ) stands for the von Neumann dimension, and the real number dim N ( G ) ( ker T ) is calledthe ℓ -Betti number of T . The first counterexample to the SAC was found in [20] and is given by the lamplighter group Γ = Z ≀ Z = (cid:16) M Z Z (cid:17) ⋊ Z , where Z acts on L Z Z by translation. More precisely, denoting by t the generator corresponding to Z and by a i the generator corresponding to the i th copy of Z in G , it was shown in [20] that theself-adjoint element T = s + s ∗ , where s = (1 + a ) t , satisfies that dim N (Γ) ( ker T ) = 1 / . Thisgives a counterexample for the SAC, because all the finite subgroups of Γ have order a power of ,but does not solve the original question posed by Atiyah which asks whether all the ℓ -Betti numbers dim N ( G ) ( ker T ) are rational whenever T ∈ Z G . This question was solved, also in the negative byAustin [7] in a non-constructive way, and then other authors gave further counterexamples, see [18],[19] and [34]. It is important to remark that Grabowski gave in [19] an example of an irrational(indeed transcendental) ℓ -Betti number associated to the lamplighter group. Using, among otherthings, the techniques developed in the present paper, we build in [5] a large family of real numbers,most of them transcendental, which arise as ℓ -Betti numbers of the lamplighter group.The von Neumann algebra N ( G ) can be naturally embedded in the ∗ -algebra U ( G ) of unboundedoperators affiliated to N ( G ) . This algebra can be realized algebraically as the classical ring of quotientsof N ( G ) , meaning that every element in U ( G ) can be represented as a “fraction” ab − where a, b ∈ N ( G ) and b has trivial kernel. The study of ℓ -Betti numbers can be done by using the notion of aSylvester matrix rank function. Such a function assigns to each matrix (of finite size) A over a ring R a real number rk( A ) satisfying some axioms, and it is a generalization of the usual rank in matricesover a field. (See Section 2 for the precise definitions of these concepts.)The canonical trace tr on N ( G ) induces a canonical Sylvester matrix rank function rk on U ( G ) bythe rule rk( A ) = tr( P ) , where A ∈ M n ( U ( G )) and P ∈ M n ( N ( G )) is its support projection. We havethe formula rk( A ) = n − dim N ( G ) ( ker A ) connecting the rank of A with the von Neumann dimensionof the kernel of A (thought as an operator acting on ℓ ( G ) n ).The ∗ -algebra U ( G ) is a ∗ -regular ring (see [8] and [37]) and therefore there is a smallest ∗ -regularsubalgebra R KG of U ( G ) containing KG ([6] and [30]). The algebra R KG is called the ∗ -regularclosure of KG in U ( G ) . The structure of R KG has been recently investigated in connection with thevalidity of the Strong Atiyah Conjecture and the Lück Approximation Conjecture in several papers,including [6], [23], [24] and [25]. An interesting result in this respect is given by Jaikin-Zapirain in[24, Lemma 6.2], which implies that for any ∗ -subring S of a ∗ -regular ring U such that U agreeswith the ∗ -regular closure of S in U , and every Sylvester matrix rank function rk on U , the subgroup G ( S ) of ( R , +) generated by { rk( A ) | A ∈ M n ( S ) } coincides with φ ( K ( U )) , where φ is the stateof K ( U ) induced by rk . Applying this to the canonical rank function rk restricted to R KG oneimmediately recovers the known fact that G ( KG ) = Z if and only if R KG is a division ring (in whichcase, obviously, it must agree with the division closure of KG in U ( G ) , see [38, Lemma 3]). Anotherfact one can easily show from the Jaikin-Zapirain result is that if the SAC holds for G , and there isa bound on the orders of the finite subgroups of G , then R KG must be an Artinian semisimple ring(see [30, Theorem 1.5] for a stronger related result).In this paper we begin a systematic study of the ∗ -regular algebra R K Γ of the lamplighter group Γ , and indeed of a much more general class of ∗ -regular closures, as follows. We consider groups ofthe form G = H ⋊ ρ Z , the semidirect product of a countable discrete torsion abelian group H by anaction ρ of Z on H . The first step in our strategy consists in using the Fourier transform to expressthe group algebra KG in the form C K ( X ) ⋊ ˆ ρ Z , where X = b H is the Pontryagin dual of H , C K ( X ) s the algebra of continuous functions f : X → K , where K is endowed with the discrete topology ,and ˆ ρ : Z y b H is the dual action. This process works smoothly for any field K of characteristic containing all the n th roots of , where n ranges over the orders of the elements of H . In characteristic p > we have to impose in addition the condition that p is coprime with all the orders of elements of H . (See Section 5 for details.)Since H is a discrete countable torsion abelian group, the space X = b H is a totally disconnectedcompact metrizable space. We can thus generalize the above setting by studying ∗ -algebras A = C K ( X ) ⋊ T Z , where T is a homeomorphism of a totally disconnected infinite compact metrizablespace X and K is any field endowed with a positive definite involution. This is precisely the situationstudied in [3], where it was shown that given a full T -invariant ergodic measure µ on X , there is a unique Sylvester matrix rank function rk on A such that rk( χ U ) = µ ( U ) for each clopen subset U of X . Here χ U ∈ C K ( X ) denotes the characteristic function of U . It is worth to mention here that, usingthe methods of [40] and [29], it is possible to obtain a Sylvester matrix rank function on A satisfyingthe above condition. Although the method of [3] is quite different from the methods employed inthose articles, these rank functions must agree by the uniqueness result mentioned above. Moreoverby [3] the completion R rk of A with respect to the metric topology induced by rk is a ∗ -regular ring,which is ∗ -isomorphic to the von Neumann continuous regular factor M K . We show in Proposition5.10 that for a group algebra KG , where G = H ⋊ ρ Z is as above, the canonical rank function rk KG corresponds under Fourier transform to the rank function rk b µ , where b µ is the Haar measure on b H . Ifin addition rk KG is extremal in the compact convex set of Sylvester matrix rank functions, then b µ isergodic and we are exactly in the setting of [3]. Moreover we show in Theorem 5.12 that, with theabove hypothesis, there is a rank-preserving ∗ -isomorphism R KG ∼ = R A , where R A is the ∗ -regularclosure of A = C K ( b H ) ⋊ ˆ ρ Z in R rk . It is well-known that the Haar measure on the group L Z Z is afull invariant ergodic measure and so the lamplighter group Γ falls under the umbrella of our theory.In the following, we fix a totally disconnected infinite compact metrizable space X and a homeo-morphism T of X . We also fix a full T -invariant ergodic measure µ on X , and write A := C K ( X ) ⋊ T Z ,where K is a field with positive definite involution. The basic idea to obtain approximating algebrasfor A was developed in the setting of C ∗ -algebras by Putnam et al, mainly in the case where T is aminimal homeomorphism, see e.g. [22], [35], [36]. We do not require in this paper that T is minimal.Given a non-empty clopen subset E of X and a (finite) partition P of X \ E into clopen subsets,let A ( E, P ) be the unital ∗ -subalgebra of A generated by the partial isometries χ Z t , where Z ∈ P .Then A ( E, P ) can be embedded in a (possibly infinite) direct product R of finite matrix algebrasover K . Now fixing a point y ∈ X and considering sequences ( E n , P n ) such that E n ⊇ E n +1 for all n , T n ≥ E n = { y } , each partition P n +1 ∪ { E n +1 } refines P n ∪ { E n } , and such that S n ≥ ( P n ∪ { E n } ) generate the topology of X , we obtain a sequence of approximating ∗ -algebras A n = A ( E n , P n ) , eachof them embedded in a direct product of finite matrix algebras R n . We set A ∞ = S n ≥ A n and R ∞ = S n ≥ R n .With this notation at hand, we can summarize our main results in the following theorem (seePropositions 4.3 and 4.6, and Theorem 4.11). Theorem 1.1.
Let A = C K ( X ) ⋊ T Z , T and µ as stated above. Then the nested sequence of approx-imating ∗ -subalgebras A n of A satisfies the following properties:(i) For n ∈ N , each algebra A n can be embedded in a ∗ -regular ring R n which is a (possiblyinfinite) direct product of finite matrix algebras over K .(ii) There are natural inclusion maps R n ֒ → R n +1 so that the following diagram is commutative: A n (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) A n +1 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / A n +2 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / A ∞ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / A (cid:127) _ (cid:15) (cid:15) R n (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) R n +1 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / R n +2 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / R ∞ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / R A (cid:127) _ (cid:15) (cid:15) R n (cid:31) (cid:127) / / R n +1 (cid:31) (cid:127) / / R n +2 (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / R ∞ (cid:31) (cid:127) / / R rk , here, R rk is the rank-completion of A , R A is the ∗ -regular closure of A in R rk , and for each n ∈ N ∪ {∞} , R n is the ∗ -regular closure of A n in R n .(iii) Assume that y is a periodic point for T of period l . Then there is an ideal I of A ∞ which isalso an ideal of A such that A ∞ /I ∼ = M l ( K ) and A /I ∼ = M l ( K [ t, t − ]) . (iv) The extension e I of I to R ∞ satisfies that R ∞ / e I ∼ = M l ( K ) . Moreover the ideal e I can be furtherextended to an ideal I of R A such that R A /I ∼ = M l ( K ( t )) . In particular, I is a maximal idealof the ∗ -regular closure R A . Parts (i) and (iii) come from [3] and are listed here for completeness. Observe that (iii) means thatthe ∗ -subalgebra A ∞ is large in A . Similarly (iv) indicates that the ∗ -regular subalgebra R ∞ is largein the ∗ -regular algebra R A .Using this result we can infer the following corollary, which provides useful information on the idealstructure of R A : Corollary 1.2.
With the above hypothesis and notation, suppose that T has some periodic point.Then there is an injective map from the set of finite orbits for the action of T to the set of maximalideals of R A . In particular, R A is not a simple ring. See Lemma 4.12 for the proof of Corollary 1.2. The algebra R A might be simple if T has no periodicpoints, see [5].Although the above theorem reduces in principle the study of the ∗ -regular closure R A to thedetermination of the ∗ -regular subalgebras R n , it is a challenging problem to elucidate the structureof these algebras. For a given ∗ -subalgebra B := A ( E, P ) associated to a pair ( E, P ) as above,we start the investigation of the ∗ -regular closure R B of B in R . For this, we are guided by thestudy in [6] of a particular case, which corresponds to the choices E = [0] and P = { [1] } , where A = C K ( { , } Z ) ⋊ T Z ∼ = K Γ is the lamplighter group algebra and [ ǫ ] = { ( x n ) ∈ { , } Z | x = ǫ } for ǫ = 0 , . In that paper, the structure of the ∗ -regular closure R of A := A ( E , P ) in itscorresponding direct product of finite matrix algebras R was completely determined, see [6, Theorem6.13]. For a general algebra B = A ( E, P ) , we find in the present paper a ∗ -subalgebra E of R B whichprovides a generalization of the construction in [6] (see Subsection 6.3 for the exact relation betweenboth constructions). We expect that E will generally be only a proper ∗ -subalgebra of R B , but thereare indications that the algebra E is very large, see below.The algebra B has a description as a partial crossed product over Z [14], and we can write B = L i ∈ Z B i t i , where B is a ∗ -regular subalgebra of C K ( X ) and each B i = e i B is an ideal of B generated by an idempotent e i (see [3] and Subsection 4.2). We observe that we can restrict theinjective representation map π : B ֒ → R to a representation π + : B + = L i ≥ B i t i → R by lowertriangular matrices. Considering now a skew partial formal power series algebra B [[ t ; T ]] := n X i ≥ b i t i | b i ∈ B i for all i ≥ o , we can extend π + to an injective representation π + : B [[ t ; T ]] → R by lower triangular matrices.Similarly we obtain a representation π − : B [[ t − , T − ]] → R by upper triangular matrices. Thealgebra D + is defined as the division closure of B + in B [[ t ; T ]] ; that is, D + is the smallest subalgebraof B [[ t ; T ]] containing B + and closed under inversion. Similarly we obtain a corresponding subalgebra D − of B [[ t − , T − ]] . It turns out that π + ( D + ) ⊆ R B and π − ( D − ) ⊆ R B , and that π + ( D + ) ∗ = π − ( D − ) (see Proposition 4.16). Since R B is closed under the involution, the ∗ -algebra D generatedby π + ( D + ) must also be contained in R B . The situation is summarized in the following diagram: + (cid:31) (cid:127) / / (cid:15) o ❅❅❅❅❅❅❅❅ O O ∗ (cid:15) (cid:15) D + (cid:16) p ❆❆❆❆❆❆❆❆ O O ∗ (cid:15) (cid:15) B .(cid:14) > > ⑤⑤⑤⑤⑤⑤⑤ (cid:16) p ❇❇❇❇❇❇❇❇ B (cid:31) (cid:127) / / D (cid:31) (cid:127) / / R B B − (cid:31) (cid:127) / / /(cid:15) > > ⑦⑦⑦⑦⑦⑦⑦⑦ D − .(cid:14) > > ⑥⑥⑥⑥⑥⑥⑥⑥ Finally by using the somewhat technical notion of special terms (see Definition 4.17) we are ableto enlarge the algebra D by considering a certain ∗ -regular subalgebra Ψ( Q ) p E of p E R p E , where p E := π ( χ E ) . The ∗ -algebra E is then defined as the subalgebra of R generated by D and Ψ( Q ) p E (Definition 4.25). It is shown in Proposition 4.26 that we have inclusions of ∗ -algebras B ⊆ D ⊆ E ⊆ R B .In the last section of the paper, we give a specific approximation sequence { A n } for the lamplightergroup algebra A := C K ( X ) ⋊ T Z ∼ = K Γ , where here X = { , } Z and T is the shift T ( x ) i = x i +1 for x ∈ X . We use as a sequence of approximations the algebras A n := A ( E n , P n ) associated topartitions given by cylinder sets [ ǫ − n . . . ǫ · · · ǫ n ] = { x = ( x i ) ∈ X | x − n = ǫ − n , ..., x = ǫ , ..., x n = ǫ n } , where ǫ − n , . . . , ǫ n ∈ { , } . For n = 0 , we recover the algebra A studied in [6]. It is worth men-tioning here that, for n ≥ , the algebra R n := R A n contains a copy of the algebra K rat h X i ofnon-commutative rational series in infinitely many indeterminates (see Proposition 6.7). This haspotential applications for the computation of ℓ -Betti numbers for the lamplighter, as explained in[5].The paper is organized as follows. Section 2 contains preliminary definitions and results, andSection 3 contains details on the basic construction used in the paper, which is fully developed in[3]. We undertake in Section 4 the general study of the ∗ -regular closure R A of a crossed productalgebra A = C K ( X ) ⋊ T Z , where X is a totally disconnected compact metrizable space, and T is ahomeomorphism of X . We first obtain the results about the approximation of R A by a sequence of ∗ -regular algebras R n , including the study about the difference between R A and R ∞ = S n ≥ R n incase y is a periodic point of T . In Subsection 4.2, we start our general study of the ∗ -regular algebra R B associated to an algebra B = A ( E, P ) , by identifying several interesting subalgebras therein. InSection 5, we explain the connection between the crossed product algebras C K ( X ) ⋊ T Z and groupalgebras KG , where G = H ⋊ ρ Z is a semidirect product with H a countable torsion abelian group.The connection uses essentially the Fourier transform, but, since we impose only minimal conditionson our base field K , we need to work out some additional details. Finally Section 6 contains our studyof the group algebra K Γ of the lamplighter group Γ . Using a concrete sequence { ( E n , P n ) | n ∈ Z + } of partitions of X = { , } Z , we are able to concretely compute several of the objects introducedin Section 4 in the general setting. In particular we show in Proposition 6.7 that, for n ≥ , the ∗ -regular algebras R n contain a copy of the algebra of non-commutative rational series in infinitelymany indeterminates. We conclude the paper by determining the exact relation of our theory withthe algebra studied in [6, Section 6].2. Background and preliminaries
Here we collect background definitions, concepts, and results needed during the course of the paper.2.1.
Von Neumann regular rings, ∗ -regular rings and rank functions. A ring R is called regular , or von Neumann regular , if for every element x ∈ R there exists y ∈ R such that x = xyx .In this case, the element e = xy is an idempotent and generates the same right ideal as x . In fact,a characterization for regular rings is that every finitely generated right ideal of R is generated by asingle idempotent [17, Theorem 1.1]. ∗ -regular ring is a regular ring R endowed with an involution ∗ which is proper , meaning thatthe equation x ∗ x = 0 implies x = 0 . The involution is called positive definite in case, for each n ≥ ,the equation P ni =1 x ∗ i x i = 0 implies x i = 0 for all ≤ i ≤ n . If R is a ∗ -regular ring with a positivedefinite involution, then M n ( R ) is also a ∗ -regular ring when endowed with the ∗ -transpose involution.In a ∗ -regular ring R every principal right/left ideal is generated by unique projections. Specifically,for every x ∈ R there exist unique projections e, f ∈ R (usually denoted by LP( x ) and RP( x ) , andtermed the left and right projections of x , respectively) satisfying that xR = eR and Rx = Rf . Inthis setting, there also exists a unique element x ∈ f Re such that xx = e and xx = f . The element x is called the relative inverse of x . We refer the reader to [1, 8] for further information on ∗ -regularrings.For any subset S ⊆ R of a unital ∗ -regular ring R , there exists a smallest unital ∗ -regular subringof R containing S ([6, Proposition 6.2], see also [30, Proposition 3.1]). This ∗ -regular ring is calledthe ∗ -regular closure of S in R , and is denoted by R ( S, R ) . In fact, R ( S, R ) = [ n ≥ R n ( S, R ) , where R ( S, R ) is the unital ∗ -subring of R generated by the set S , and R n +1 ( S, R ) is generated by R n ( S, R ) and the relative inverses in R of the elements of R n ( S, R ) . It was observed in [24] that R n +1 ( S, R ) can be described as the subring of R generated by the elements of R n ( S, R ) and therelative inverses of the elements of the form x ∗ x for x ∈ R n ( S, R ) .A pseudo-rank function on a unital regular ring R is a map rk : R → [0 , that satisfies the followingproperties:a) rk(0) = 0 , rk(1) = 1 ;b) rk( xy ) ≤ min { rk( x ) , rk( y ) } for every x, y ∈ R ;c) if e, f ∈ R are orthogonal idempotents, then rk( e + f ) = rk( e ) + rk( f ) .If rk satisfies the additional propertyd) rk( x ) = 0 only if x = 0 ,then rk is called a rank function on R . For general properties of pseudo-rank functions over regularrings one can consult [17, Chapter 16].Every (pseudo-)rank function rk on a regular ring R defines a (pseudo-)metric d on R by the rule d ( x, y ) = rk( x − y ) . Since the ring operations are continuous with respect to this (pseudo-)metric,one can consider the completion R of R with respect to d . It is also a regular ring, and rk can beuniquely extended continuously to a rank function rk on R such that R is also complete with respectto the metric induced by rk .Regular rings are also of great interest since every (pseudo-)rank function rk on R can be uniquelyextended to a (pseudo-)rank function on matrices over R (see e.g. [17, Corollary 16.10]). This is nolonger true if we do not assume R to be regular. The definition that seems to fit in the general settingis the notion of a Sylvester matrix rank function. Definition 2.1.
Let R be a unital ring, and set M ( R ) = S n ≥ M n ( R ) . A Sylvester matrix rankfunction on R is a map rk : M ( R ) → R + satisfying the following conditions:a) rk( M ) = 0 if M is a zero matrix, and rk(1) = 1 ;b) rk( M M ) ≤ min { rk( M ) , rk( M ) } for any matrices M and M of appropriate sizes;c) rk (cid:18) M M (cid:19) = rk( M ) + rk( M ) for any matrices M and M ;d) rk (cid:18) M M M (cid:19) ≥ rk( M ) + rk( M ) for any matrices M , M and M of appropriate sizes.Sylvester matrix rank functions were first introduced by Malcolmson in [32] in order to study ho-momorphisms to division rings. For more theory and properties about Sylvester matrix rank functionswe refer the reader to [24], [28], [29] and [39, Part I, Chapter 7]. e denote by P ( R ) the compact convex set of Sylvester matrix rank functions on R . By [17,Proposition 16.20] this space coincides with the space of pseudo-rank functions on R when R is aregular ring.As in the case of pseudo-rank functions on a regular ring, a Sylvester matrix rank function rk ona unital ring R gives rise to a pseudo-metric by the rule d ( x, y ) = rk( x − y ) . We call the Sylvestermatrix rank function faithful if its kernel, defined as the set of all element x ∈ R with zero rank, isexactly { } . In this case d becomes a metric on R . Again, the ring operations are continuous withrespect to d , so one can consider the completion R of R with respect to d , and rk can be uniquelyextended continuously to a Sylvester matrix rank function rk on R .A very useful result connecting the ∗ -regular closure R ( S, R ) and possible values of a Sylvestermatrix rank function defined on R is given in the following proposition, which can be thought of asan analogue of the classical Cramer’s rule. Proposition 2.2 (Corollary 6.2 of [24]) . Let S be a unital ∗ -subring of a ∗ -regular ring R , and let R = R ( S, R ) be the ∗ -regular closure of S in R .Then for any matrices r , ..., r k ∈ M n × m ( R ) , there exists a matrix M ∈ M a × b ( S ) and matrices A , ..., A k ∈ M n × b ( S ) such that, for any other square-matrices t , ..., t k ∈ M n ( S ) and any Sylvestermatrix rank function rk defined on R , rk( t r + · · · + t k r k ) = rk (cid:18) Mt A + · · · + t k A k (cid:19) − rk( M ) . In particular, any Sylvester matrix rank function on R is completely determined by its values onmatrices over S . Division closure and rational closure.
In this subsection we recall the well-known conceptsof the division closure and the rational closure. We also introduce the notion of universal localizationin a special case. See [11] for a detailed treatment.Let R be a unital subring of a ring T . The division closure D ( R, T ) of R in T is the smallestsubring of T containing R and closed under inverses in T . So we have R ⊆ D ( R, T ) ⊆ T and d − ∈ D ( R, T ) whenever d ∈ D ( R, T ) is invertible in T . The rational closure of R in T is the smallestsubring R at ( R, T ) of T containing R and closed under inverses of square matrices. We thus have R ⊆ R at ( R, T ) ⊆ T , and whenever a matrix A ∈ M n ( R at ( R, T )) is invertible in M n ( T ) , then all theentries of A − belong to R at ( R, T ) .We summarize in the next lemma some properties concerning the rational, division and ∗ -regularclosures. Its proof is straightforward, so we omit it. Lemma 2.3.
Let R be a unital subring of T . Then the following properties hold:i) D ( D ( R, T ) , T ) = D ( R, T ) = D ( R, D ( R, T )) .ii) R at ( R at ( R, T ) , T ) = R at ( R, T ) = R at ( R, R at ( R, T )) .iii) D ( R, T ) ⊆ R at ( R, T ) .Moreover, if T is ∗ -regular, then the ∗ -regular closure of R in T , which we denote by R ( R, T ) , containsthe rational closure R at ( R, T ) . We will need in Section 6 the notion of universal localization of a ring, but only with respect toelements of the ring. Let Σ be a subset of a unital ring R . Then the universal localization of R withrespect to Σ is the ring Σ − R obtained by universally adjoining to R inverses of elements of Σ . Thereis a canonical ring homomorphism ι Σ : R → Σ − R satisfying the following universal property: foreach ring homomorphism ϕ : R → S such that ϕ ( s ) is invertible for all s ∈ Σ , there exists a uniquehomomorphism ˜ ϕ : Σ − R → S such that ϕ = ˜ ϕ ◦ ι Σ .2.3. ℓ -Betti numbers for group algebras and the Atiyah Conjecture. Here we recall somebasic facts on ℓ -Betti numbers associated to a group. We refer the reader to [31] for more informationon this subject.For a discrete countable group G and a subfield K of C closed under complex conjugation we considerthe group ∗ -algebra KG acting on the Hilbert space ℓ ( G ) by left multiplication. We denote by N ( G ) he weak-completion of C G in B ( ℓ ( G )) , which is commonly known as the group von Neumann algebraof G . An equivalent algebraic definition can be given: it consists exactly of those bounded operators T : ℓ ( G ) → ℓ ( G ) that are G -equivariant, i.e. T ( ξg ) = T ( ξ ) g for ξ ∈ ℓ ( G ) and g ∈ G .The algebra N ( G ) is endowed with a normal, positive and faithful trace, defined as tr N ( G ) ( T ) := h T ( ξ e ) , ξ e i ℓ ( G ) for T ∈ N ( G ) , where { ξ g } g ∈ G is the canonical orthonormal basis of ℓ ( G ) . Note that for an element T = P γ ∈ G a γ γ ∈ C G , its trace is simply the coefficient a e .All the above constructions can be extended to k × k matrices: the ∗ -algebra M k ( KG ) acts faith-fully on ℓ ( G ) k by left multiplication. We denote by N k ( G ) the weak-completion of M k ( C G ) inside B ( ℓ ( G ) k ) , which coincides with M k ( N ( G )) . The previous trace can be extended to an (unnormalized)trace in N k ( G ) by setting, for a matrix T = ( T ij ) ∈ N k ( G ) , Tr N k ( G ) ( T ) := k X i =1 tr N ( G ) ( T ii ) . Every matrix operator A ∈ M k ( KG ) gives rise to an ℓ -Betti number , in the following way. Consider A as an operator A : ℓ ( G ) k → ℓ ( G ) k acting on the left, and take p A ∈ N k ( G ) to be the projectiononto the kernel of A . One can then define the von Neumann dimension dim N ( G ) ( ker A ) of ker A asthe trace of the projection p A . Definition 2.4.
A real positive number r is called an ℓ -Betti number arising from G with coefficientsin K if for some integer k ≥ , there exists a matrix operator A ∈ M k ( KG ) such that dim N ( G ) ( ker A ) := Tr N k ( G ) ( p A ) = r. We denote the set of all ℓ -Betti numbers arising from G with coefficients in K by C ( G, K ) . It shouldbe noted that this set is always a subsemigroup of ( R + , +) . The subgroup of ( R , +) generated by C ( G, K ) will be denoted throughout the article by G ( G, K ) .It is also possible to define the von Neumann dimension by means of a rank function , as follows. Let U ( G ) be the classical ring of quotients of N ( G ) . It is a ∗ -regular ring possessing a Sylvester matrixrank function rk defined by rk( A ) := Tr N k ( G ) (LP( A )) = Tr N k ( G ) (RP( A )) for any matrix A ∈ M k ( U ( G )) , where LP( A ) and RP( A ) are the left and right projections of A inside M k ( U ( G )) , respectively. In particular, we obtain by restriction a Sylvester matrix rank function rk KG on KG , and we have the equality(2.1) dim N ( G ) ( ker A ) = k − rk KG ( A ) for each matrix operator A ∈ M k ( KG ) .3. A dynamical approximation for crossed product algebras
Let us recall the general construction used in [3] for providing an essentially unique Sylvester matrixrank function on an algebraic crossed product algebra A := C K ( X ) ⋊ T Z , where T : X → X is ahomeomorphism of a totally disconnected, compact metrizable space X , which we also assume tobe infinite (e.g. one can take X to be the Cantor space). Here K is an arbitrary field and C K ( X ) is the algebra of continuous functions f : X → K where K is endowed with the discrete topology;equivalently, is the algebra of locally constant functions f : X → K . For the construction, a T -invariant, ergodic and full probability measure µ on X is also needed. We refer the reader to [3,Section 3]. This construction is used throughout the paper.For ∅ 6 = E ⊆ X any clopen subset and P any (finite) partition of the complement X \ E into clopensubsets, define B to be the unital ∗ -subalgebra of A generated by the family of partial isometries { χ Z t } Z ∈ P . By [3, Lemma 3.4], the ∗ -algebra B = C K ( X ) ∩ B is linearly spanned by the unit andthe projections of the form χ C , being C a non-empty clopen subset of X of the form(3.1) T − r ( Z − r ) ∩ · · · ∩ Z ∩ · · · ∩ T s − ( Z s − ) , here Z − r , ..., Z , ..., Z s − ∈ P and r, s ≥ . Here χ U denotes the characteristic function of the clopensubset U ⊆ X . We have the decomposition B = M i ∈ Z B ( χ X \ E t ) i = M i ∈ Z B i t i with B i = χ X \ ( E ∪···∪ T i − ( E )) B and B − i = χ X \ ( T − ( E ) ∪···∪ T − i ( E )) B for i > . By [3, Lemma 3.8], for i > the algebra B i is linearly spanned by the terms χ C , where C is of the form (3.1) with s ≥ i , andsimilarly B − i is linearly spanned by the terms χ C , where C is of the form (3.1) with r ≥ i .There exists a quasi-partition of X (i.e. a countable family of non-empty, pairwise disjoint clopensubsets whose union has full measure) given by the T -translates of clopen subsets W of the form(3.2) W = E ∩ T − ( Z ) ∩ · · · ∩ T − k +1 ( Z k − ) ∩ T − k ( E ) for some k ≥ and Z i ∈ P , whenever these are non-empty. In fact, if we write | W | := k (the length of W ) and V := { W = ∅ as above } , then for a fixed W ∈ V the characteristic functions { χ W , ..., χ T | W |− ( W ) } belong to B ,and moreover the set of elements e ij ( W ) = ( χ X \ E t ) i χ W ( t − χ X \ E ) j , ≤ i, j ≤ | W | − , forms a setof | W | × | W | matrix units in B (i.e. they satisfy e ij ( W ) e ts ( W ) = δ j,t e is ( W ) for all allowable indices i, j, t, s ). In addition, the element h W := e ( W ) + · · · + e | W |− , | W |− ( W ) is central in B and, by [3,Proposition 3.11], we have(3.3) h W B ∼ = M | W | ( K ) . From now on, and slightly abusing language, we will identify h W B with M | W | ( K ) whenever convenient.In this way one constructs an injective ∗ -representation π : B ֒ → Q W ∈ V M | W | ( K ) =: R defined by π ( a ) = ( h W · a ) W [3, Proposition 3.13].The ∗ -algebra B corresponding to ( E, P ) as above will be denoted by A ( E, P ) .Take now { E n } n ≥ to be a decreasing sequence of clopen sets of X together with a family { P n } n ≥ ,being each P n a (finite) partition of X \ E n into clopen subsets, such that:a) the intersection of all the E n consists of a single point y ∈ X ;b) P n +1 ∪ { E n +1 } is a partition of X finer than P n ∪ { E n } ;c) S n ≥ ( P n ∪ { E n } ) generates the topology of X .By writing V n for the (non-empty) sets (3.2) corresponding to the pair ( E n , P n ) and setting A n := A ( E n , P n ) , we get injective ∗ -representations π n of A n into R n := Q W ∈ V n M | W | ( K ) , in such a waythat the diagrams(3.4) A n (cid:31) (cid:127) ι n / / (cid:127) _ π n (cid:15) (cid:15) A n +1 (cid:127) _ π n +1 (cid:15) (cid:15) (cid:31) (cid:127) ι n +1 / / A n +2 (cid:127) _ π n +2 (cid:15) (cid:15) (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / A ∞ (cid:127) _ π ∞ (cid:15) (cid:15) R n (cid:31) (cid:127) j n / / R n +1 (cid:31) (cid:127) j n +1 / / R n +2 (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / R ∞ commute. Here ι n is the natural embedding ι n ( χ Z t ) = P Z ′ χ Z ′ t where the sum is taken with respect toall Z ′ ∈ P n +1 satisfying Z ′ ⊆ Z , the maps j n : R n ֒ → R n +1 are the embeddings given in [3, Proposition4.2], and A ∞ , R ∞ are the inductive limits of the direct systems ( A n , ι n ) , ( R n , j n ) , respectively. Infact, the algebra A ∞ can be explicitly described in terms of the crossed product, as follows. For U ⊆ X an open set, denote by C c,K ( U ) the ideal of C K ( X ) generated by the characteristic functions χ V , where V ranges over the clopen subsets V ⊆ X contained in U . By [3, Lemma 4.3], A ∞ coincideswith the ∗ -subalgebra of A = C K ( X ) ⋊ T Z generated by C K ( X ) and C c,K ( X \{ y } ) t . This will be usedin the next section when describing the ∗ -regular closure of A .One can define a Sylvester matrix rank function on each R n by the rule rk n ( M ) = X W ∈ V n µ ( W ) Rk( M W ) for M = ( M W ) W ∈ R n , being Rk the usual rank of matrices. These rank functions are compatible with respect to the em-beddings j n , so they give rise to a well-defined rank function rk ∞ on R ∞ . Using this rank function t is possible to define a Sylvester matrix rank function over A , unique with respect to a certaincompatibility property concerning the measure µ , as the following theorem states. Theorem 3.1. [3, Theorem 4.7 and Proposition 4.8] If R rk denotes the rank-completion of R ∞ withrespect to its rank function rk ∞ , then we can embed A ֒ → R rk in such a way that it coincides with theembedding A ∞ ֒ → R ∞ when restricted to A ∞ , and the element t is sent to lim n π n ( χ X \ E n t ) .Moreover, the Sylvester matrix rank function rk A induced by restriction of rk ∞ (the extension of rk ∞ to R rk ) on A is extremal, and unique with respect to the following property: rk A ( χ U ) = µ ( U ) for every clopen subset U ⊆ X. Finally, the rank-completion of A with respect to rk A gives back R rk . The ∗ -regular closure R A and its ideal structure In this section we undertake the study of the ∗ -regular closure of the crossed product algebra A = C K ( X ) ⋊ T Z inside R rk (see Theorem 3.1) when ( K, − ) is a field with positive definite involution.In this situation, the direct product R n = Q W ∈ V n M | W | ( K ) has a structure of ∗ -regular K -algebra,where each matrix algebra M | W | ( K ) is endowed with the ∗ -transpose involution. Since this structureis compatible with the transition homomorphisms, we obtain a stucture of ∗ -regular algebra on R ∞ ,and thus on its completion R rk . Moreover, A sits naturally as a ∗ -subalgebra of R rk . See [3, Theorem4.9] for more details.Let R A := R ( A , R rk ) denote the ∗ -regular closure of A inside R rk . The structure of this ∗ -algebrais closely related to the possible values of the rank function rk A ; in other words, it is related with theset(4.1) C ( A ) := rk A (cid:16) [ n ≥ M n ( A ) (cid:17) ⊆ R + . As in Definition 2.4, this set has the structure of a semigroup, inherited from ( R + , +) . We will denoteby G ( A ) the subgroup of ( R , +) generated by C ( A ) . The set C ( A ) can be thought of as the set of complementary ℓ -Betti numbers arising from A (cf. Definition 2.4 and (2.1)).The exact relation between G ( A ) and R A is given in the following proposition, which is motivatedby Proposition 2.2. Proposition 4.1.
The group G ( A ) coincides with the subgroup of ( R , +) generated by the set rk R A ( R A ) = { rk R A ( r ) | r ∈ R A } , where rk R A is the restriction of rk ∞ to R A (see Theorem 3.1). Equivalently, it coincides with theimage of the state φ : K ( R A ) → R , [ p ] − [ q ] rk R A ( p ) − rk R A ( q ) . Proof.
Write S for the subgroup generated by rk R A ( R A ) . By Proposition 2.2, we have the inclusion S ⊆ G ( A ) .For the other inclusion note first that, since R A is a ∗ -regular ring with positive definite involution,each matrix algebra M n ( R A ) is ∗ -regular too. So for each A ∈ M n ( R A ) there exists a projection P ∈ M n ( R A ) such that rk R A ( A ) = rk R A ( P ) . We conclude that C ( A ) is contained in the set of positive realnumbers of the form rk R A ( P ) , where P ranges over matrix projections with coefficients in R A . Noweach such projection P is equivalent to a diagonal one [17, Proposition 2.10], that is, one of the formdiag ( p , ..., p r ) for some projections p , ..., p r ∈ R A , so that rk R A ( P ) = rk R A ( p )+ · · · +rk R A ( p r ) ∈ S ,and G ( A ) ⊆ S .Now the last part of the proposition follows easily, since φ ( K ( R A )) = S . (cid:3) Before continuing, it is worth to mention that we can completely determine the rank-completionof R A : it is the well-known von Neumann continuous factor M K , defined as the completion of lim −→ n M n ( K ) with respect to its unique rank function (see [4, 13] for details). Here the direct limit istaken with respect to the block diagonal embeddings x diag ( x, x ) . roposition 4.2. With the above notation, we have that R A rk RA ∼ = M K as ∗ -algebras.Proof. Here M K has the involution induced from the ∗ -transpose involution on each matrix algebra M n ( K ) . Since A ⊆ R A ⊆ R rk and A rk A = R rk ∼ = M K due to [3, Theorems 4.7 and 4.9], the resultfollows. (cid:3) Our strategy is to make use of our sequence { A n } n ≥ of approximating algebras from Section 3 toapproximate R A in a suitable way. As already mentioned, in our present setting the algebras R n , R ∞ and R rk become ∗ -regular algebras, and all the connecting maps in the commutative diagram (3.4)become ∗ -homomorphisms. We denote by R n = R ( A n , R n ) the ∗ -regular closure of A n inside R n .Similarly, R ∞ = R ( A ∞ , R ∞ ) . Proposition 4.3.
We have inclusions R n ⊆ R n +1 , and moreover S n ≥ R n = R ∞ . Therefore thediagram (3.4) extends to a commutative diagram (4.2) A n (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) A n +1 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / A n +2 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / A ∞ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / A (cid:127) _ (cid:15) (cid:15) R n (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) R n +1 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / R n +2 (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / R ∞ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / R A (cid:127) _ (cid:15) (cid:15) R n (cid:31) (cid:127) / / R n +1 (cid:31) (cid:127) / / R n +2 (cid:31) (cid:127) / / · · · (cid:31) (cid:127) / / R ∞ (cid:31) (cid:127) / / R rk . Proof.
Since A n ⊆ A n +1 ∩ R n ⊆ R n +1 ∩ R n ⊆ R n , and R n +1 ∩ R n is ∗ -regular, we have R n ⊆ R n +1 ∩ R n ⊆ R n +1 . In particular, this shows the commutativity of the left sides of the diagram.The proof for the right sides is similar: A ∞ ⊆ A ∩ R ∞ ⊆ R A ∩ R ∞ ⊆ R ∞ , and since R A ∩ R ∞ is ∗ -regular, we have again R ∞ ⊆ R A ∩ R ∞ ⊆ R A .To prove the equality S n ≥ R n = R ∞ , note that each A n ⊆ R n ⊆ S n ≥ R n ⊆ R ∞ , hence A ∞ ⊆ S n ≥ R n ⊆ R ∞ . It is easy to check, using that R n ⊆ R n +1 , that S n ≥ R n is ∗ -regular, so bydefinition R ∞ ⊆ S n ≥ R n . The other inclusion is trivial because each R n ⊆ R ∞ . (cid:3) The following lemma gives some examples of elements that appear inside R A . Lemma 4.4.
Take p ( x ) = λ + λ x + · · · + λ k x k ∈ K [ x ] a polynomial with λ = 0 . Then p ( t ) ∈ A isinvertible in R A . Moreover, R A contains a copy of the rational function field K ( t ) .Proof. Inside R rk we identify the element t ∈ A with lim n π n ( χ X \ E n t ) (cf. Theorem 3.1). Hence p ( t ) = lim n p ( π n ( χ X \ E n t )) . Note that π n ( χ X \ E n t ) = ( h W · χ X \ E n t ) W . We compute h W · χ X \ E n t = e ( W ) + · · · + e | W |− , | W |− ( W ) =: u W , so p ( π n ( χ X \ E n t )) = ( λ Id W + λ u W + · · · + λ k u kW ) W . These are all lower triangular matrices inside each matrix algebra M | W | ( K ) , and in fact invertiblesince λ = 0 . Hence p ( π n ( χ X \ E n t )) is invertible inside R n ⊆ R rk for each n ≥ , and so is its limit lim n p ( π n ( χ X \ E n t )) = p ( t ) . Since t is already invertible in R A , it follows that K ( t ) ⊆ R A . (cid:3) Difference between R A and R ∞ assuming existence of a periodic point. We determinein this subsection the exact relationship between R ∞ and R A in case y is a periodic point. To start,we recall the following proposition from [3], which determines how big is the subalgebra A ∞ insidethe algebra A in this case of interest. Proposition 4.5. [3, Proposition 4.5]
Let us assume the above notation. Suppose that y is a periodicpoint for T with period l . Let I be the ideal of A generated by C c,K ( X \{ y, . . . , T l − ( y ) } ) . Then:(i) I is also an ideal of A ∞ , and we have ∗ -algebra isomorphisms A /I ∼ = M l ( K [ s, s − ]) , A ∞ /I ∼ = M l ( K ) . ii) There exists some M ≥ such that for each n ≥ M there is exactly one W n ∈ V n of length l and containing y , and such that the isomorphism h W n A n ∼ = M l ( K ) given in (3.3) coincideswith the restriction of the projection map q : A ∞ → A ∞ /I on h W n A n . That is, the diagram A ∞ q / / A ∞ /I ∼ = (cid:15) (cid:15) h W n A n ∼ = / / ?(cid:31) O O M l ( K ) commutes, where the right isomorphism comes from ( i ) . Moreover, h W ∈ I for all W ∈ V n , W = W n .(iii) A n / ( I ∩ A n ) ∼ = A ∞ /I ∼ = M l ( K ) and (1 − h W n ) A n = I ∩ A n for every n ≥ M . We continue with a proposition concerning the structure of R ∞ . Proposition 4.6.
Let us assume the same notation as in Proposition 4.5. Let e I be the ideal of R ∞ generated by I . Then:(i) e I = S n ≥ M (1 − h W n ) R n , and there is a ∗ -isomorphism R ∞ / e I ∼ = M l ( K ) . (ii) If R denotes the ∗ -subalgebra of R A generated by e I, h W M A M and K [ t, t − ] , then e I is also anideal of R , A is contained in R , and there is a ∗ -isomorphism R / e I ∼ = M l ( K [ t l , t − l ]) . Remark 4.7.
Since the ideal e I is already ∗ -regular and the quotient R / e I is close to be ∗ -regular, the ∗ -subalgebra R is not the ∗ -regular closure R A , but is a good approximation to it. We will see laterhow to enlarge R in order to obtain the whole R A . First, we prove Proposition 4.6. Proof of Proposition 4.6. ( i ) For n ≥ M , let I n = I ∩ A n and let e I n be the ideal of R n generated by I n .Claim 1: e I = S n ≥ M e I n .Proof: Clearly each e I n ⊆ e I , so S n ≥ M e I n ⊆ e I . For the other inclusion, recall that I ⊆ A ∞ byProposition 4.5. If we take a to be an element of e I , then a = P mj =1 r j b j s j for some r j , s j ∈ R ∞ and b j ∈ I ⊆ A ∞ . There exists then an index n ≥ M such that r j , s j ∈ R n and b j ∈ I ∩ A n = I n for all ≤ j ≤ m . Therefore a = P mj =1 r j b j s j ∈ R n I n R n = e I n , and we obtain the inclusion e I ⊆ S n ≥ M e I n . (cid:3) Claim 2: e I n = (1 − h W n ) R n .Proof: Since I n = (1 − h W n ) A n due to Proposition 4.5, and taking into account that (1 − h W n ) iscentral in R n , we compute e I n = R n I n R n = R n (1 − h W n ) A n R n = R n (1 − h W n ) R n = (1 − h W n ) R n , as required. (cid:3) Using Claims and , we get e I = S n ≥ M e I n = S n ≥ M (1 − h W n ) R n . In particular, we see that the ideal e I is ∗ -regular.Claim 3: For m ≥ n ≥ M , we have isomorphisms R n / e I n ∼ = R m / e I m ∼ = M l ( K ) , via e ij ( W n ) + e I n e ij ( W m ) + e I m e ij . Proof: To see this, note first that h W n R n ∼ = M l ( K ) through e ij ( W n ) e ij , since A n ⊆ R n ⊆ R n and h W n A n = h W n R n ∼ = M l ( K ) . Now each h W n is a central idempotent in R n , so we havedecompositions R n = (1 − h W n ) R n ⊕ h W n R n = e I n ⊕ h W n R n . Hence R n / e I n ∼ = h W n R n ∼ = M l ( K ) ∼ = h W m R m ∼ = R m / e I m through the cited maps. (cid:3) ix n ≥ M and consider the composition R n ֒ → R ∞ → R ∞ / e I . Since e I n ⊆ e I , we get a ∗ -homomorphism R n / e I n → R ∞ / e I, r + e I n r + e I. From Claim it follows easily that, for n, m ≥ M , the diagram R n / e I n ∼ = (cid:15) (cid:15) / / R ∞ / e I R m / e I m : : ✉✉✉✉✉✉✉✉✉ is commutative. This proves surjectivity of R n / e I n → R ∞ / e I . For injectivity, it is enough to showthat the element h W n does not lie inside e I . But if h W n ∈ (1 − h W n ) R n for some n ≥ n then h W n = h W n h W n = 0 , a contradiction. From this we obtain the desired ∗ -isomorphism R ∞ / e I ∼ = R n / e I n ∼ = M l ( K ) . ( ii ) We show first that e I is stable under multiplication by elements of K [ t, t − ] .Claim 4: t e I = e I .Proof: Let’s prove the inclusion t e I ⊆ e I , so take a ∈ (1 − h W n ) R n for some n ≥ M . Since a = (1 − h W n ) a , it is enough to show that t (1 − h W n ) ∈ e I . But t (1 − h W n ) = tχ X \ ( W n ∪ T ( W n ) ∪···∪ T l − ( W n )) = χ X \ ( T ( W n ) ∪ T ( W n ) ∪···∪ T l ( W n )) t and χ X \ ( T ( W n ) ∪ T ( W n ) ∪···∪ T l ( W n )) ∈ C c,K ( X \{ y } ) , so by the description of A ∞ given in Section 3we deduce that t (1 − h W n ) ∈ A ∞ , hence t (1 − h W n ) ∈ e I . To show the other inclusion it is enough toshow that t − e I ⊆ e I , which in turn will follow once we show that t − (1 − h W n ) ∈ e I . This is obviousbecause − h W n ∈ C c,K ( X \{ y } ) . (cid:3) As a consequence, we have p ( t ) e I, e Ip ( t ) ⊆ e I for any Laurent polynomial p ( t ) in t , so e I is an ideal of R .Claim 5: e I is a proper ideal of R .Proof: This follows from the fact that rk A (1 − h W n ) < for all n ≥ M . (cid:3) Claim 6: A ⊆ R .Proof: We show by induction that h W n A n ⊆ R for all n ≥ M . For n = M , this follows fromthe definition of R . Now assume that h W n A n ⊆ R for some n ≥ M . Under the quotient map A ∞ → A ∞ /I ∼ = M l ( K ) the matrix units e ij ( W n ) correspond to the matrix units e ij (Proposition4.5), hence the differences e ij ( W n +1 ) − e ij ( W n ) belong to I ⊆ e I ⊆ R for all ≤ i, j ≤ l − . Since e ij ( W n ) ∈ h W n A n ⊆ R , we deduce that e ij ( W n +1 ) ∈ R , and so the whole algebra h W n +1 A n +1 liesinside R . Therefore h W n A n ⊆ R for all n ≥ M . Hence A n = (1 − h W n ) A n ⊕ h W n A n ⊆ e I + h W n A n ⊆ R for all n ≥ M , so A ∞ ⊆ R . In particular C K ( X ) ⊆ R , and since K [ t, t − ] ⊆ R already, we obtain A ⊆ R , as claimed. (cid:3) Claim 7: We have a ∗ -isomorphism R / e I ∼ = M l ( K [ t l , t − l ]) .Proof: Since − h W M ∈ e I , the family { e ij ( W M ) + e I | ≤ i, j ≤ l − } is a complete system ofmatrix units for R / e I , so there is an isomorphism R / e I ∼ = M l ( T ) , being T the centralizer of thefamily { e ij ( W M ) + e I | ≤ i, j ≤ l − } in R / e I . The isomorphism is given explicitly by s l − X i,j =0 s ij e ij , with s ij = l − X k =0 e ki ( W M ) · s · e jk ( W M ) ∈ T, which is also a ∗ -isomorphism. We thus only need to prove that T = K [ t l , t − l ] . The inclusion K [ t l , t − l ] ⊆ T is clear since, using that e ij ( W ) = χ T i ( W ) t i − j for ≤ i, j ≤ | W | − , we get t l e ij ( W M ) − e ij ( W M ) t l = ( χ T i + l ( W M ) − χ T i ( W M ) ) t i − j + l hich belongs to I ⊆ e I due to the fact that χ T i + l ( W M ) − χ T i ( W M ) ∈ C c,K ( X \{ y, . . . , T l − ( y ) } ) since y is a periodic point of period l which belongs to W M (see Proposition 4.5). Therefore R / e I ∼ = M l ( T ) ⊇ M l ( K [ t l , t − l ]) . In order to prove the equality it is enough to check that theelement t + e I ∈ R / e I belongs to M l ( K [ t l , t − l ]) under the previous isomorphism. Note that we canwrite t + e I = th W M + e I = l − X i =0 e i +1 ,i ( W M ) + t l e ,l − ( W M ) + e I which is mapped to the element P l − i =0 e i +1 ,i + t l e ,l − under the previous isomorphism, and so itbelongs to M l ( K [ t l , t − l ]) . We have thus obtained the desired ∗ -isomorphism R / e I ∼ = M l ( K [ t l , t − l ]) . (cid:3) This concludes the proof of the proposition. (cid:3)
We now want to describe the ∗ -regular closure R A of A in R rk in terms of the ∗ -algebra R introducedin Proposition 4.6(ii). For that we need a couple of technical lemmas, together with a definition. Definition 4.8.
Let R be a non-unital ring. We say that a family E ⊆ R of idempotents is a leftlocal unit for R if for every r , ..., r n ∈ R there exists an idempotent e ∈ E such that er i = r i for all ≤ i ≤ n. The concept of right local unit is defined analogously. A local unit will be a right and left local unit.Note that, in the case that R is a ring endowed with an involution ∗ and E is a left local unit for R , then E ∗ = { e ∗ | e ∈ E } is a right local unit for R .Recall that e I is the ideal of R ∞ generated by I . We write S for the ∗ -subalgebra of R A generatedby e I, h W M A M and K ( t ) (compare with R from Proposition 4.6). It may be the case that e I is not anideal of S anymore; nevertheless, we have the following result. Lemma 4.9.
Denote by I the ideal of S generated by e I , and consider E = { p ( t l ) − (1 − h W n ) p ( t l ) ∈ I | p ( t ) ∈ K [ t ] \{ } , n ≥ M } . Then E is a left local unit for I .Proof. Since e I is closed under the involution, it follows that I is a ∗ -ideal of S .Note that every element of I is a sum of elements of the form(4.3) p ( t ) q ( t ) − e i ,j ( W M ) · · · p s ( t ) q s ( t ) − e i s ,j s ( W M ) p s +1 ( t ) q s +1 ( t ) − (1 − h W n ) y, where p k , q k ∈ K [ t ] \{ } , ≤ i k , j k ≤ l − , n ≥ M and y ∈ S . Since e I is stable under multiplicationby K [ t, t − ] , the product p s +1 ( t )(1 − h W n ) belongs to e I , so we can assume that p s +1 ( t ) = 1 .Claim: Each element of the form (4.3) can be further written as a sum of elements of the form q ( t l ) − (1 − h W n ) e y for some q ∈ K [ t l ] \{ } , n ≥ M and e y ∈ S . Proof: Since the field extension K ( t ) /K ( t l ) has degree l , with basis { , t, ..., t l − } , we can write q s +1 ( t ) − as q s +1 ( t ) − = N X i =0 t i g i ( t l ) − for some N ≥ and polynomials g i ∈ K [ t l ] \{ } . Thus we can assume that q s +1 is a polynomial in t l .Recall that, modulo the ideal e I , the matrix units e ij ( W M ) commute with the element t l . As aconsequence the element b s := q s +1 ( t l ) e i s ,j s ( W M ) − e i s ,j s ( W M ) q s +1 ( t l ) belongs to e I , so there existsan integer n s ≥ M such that b s = (1 − h W ns ) b s . Therefore e i s ,j s ( W M ) q s +1 ( t l ) − − q s +1 ( t l ) − e i s ,j s ( W M ) = q s +1 ( t l ) − (1 − h W ns ) b s q s +1 ( t l ) − , o that p ( t ) q ( t ) − e i ,j ( W M ) · · · p s ( t ) q s ( t ) − e i s ,j s ( W M ) q s +1 ( t l ) − (1 − h W n ) y = p ( t ) q ( t ) − e i ,j ( W M ) · · · p s ( t ) q s ( t ) − q s +1 ( t l ) − e i s ,j s ( W M )(1 − h W n ) y + p ( t ) q ( t ) − e i ,j ( W M ) · · · p s ( t ) q s ( t ) − q s +1 ( t l ) − (1 − h W ns ) b s q s +1 ( t l ) − (1 − h W n ) y. Since e i s ,j s ( W M ) ∈ R M ⊆ R n and − h W n is central in R n , the first term becomes p ( t ) q ( t ) − e i ,j ( W M ) · · · p s ( t ) e q s ( t ) − (1 − h W n ) y ′ with e q s ( t ) = q s ( t ) q s +1 ( t l ) ∈ K [ t ] \{ } and y ′ = e i s ,j s ( W M ) y ∈ S , and the second term becomes p ( t ) q ( t ) − e i ,j ( W M ) · · · p s ( t ) e q s ( t ) − (1 − h W ns ) y ′′ with now y ′′ = b s q s +1 ( t l ) − (1 − h W n ) y ∈ S . Again, due to the fact that K [ t, t − ] e I ⊆ e I , we canassume that p s = 1 in each of these terms. Now the claim follows by induction on s . (cid:3) Let now x , ..., x n ∈ I . By the above claim, we can assume that each x i is a monomial of theform q i ( t l ) − (1 − h W ni ) y i with q i ∈ K [ t l ] \{ } , n i ≥ M and y i ∈ S . Consider the polynomial q := q · · · q n ∈ K [ t l ] \{ } . We see that, for each ≤ i ≤ n , the result of multiplying x i by q tothe left is always an element of the form e x i y i , where e x i ∈ e I . Therefore there exists N ≥ M suchthat (1 − h W N ) q ( t l ) x i = q ( t l ) x i for all ≤ i ≤ n . The lemma follows by taking the idempotent e := q ( t l ) − (1 − h W N ) q ( t l ) . (cid:3) As a consequence of Lemma 4.9, the ideal I must be a proper ideal of S , since for e ∈ E we have rk R A ( e ) < .At this moment we could argue as in the proof of Proposition 4.6 and compute the quotient S /I .It turns out that this quotient is ∗ -isomorphic to M l ( K ( t l )) , which is a ∗ -regular ring. The problemwe encounter now is that the ideal I may not be ∗ -regular. To fix this, we consider the non-unitalsubalgebra of R A generated by I and the relative inverses x of elements x ∈ I , denoted by I . It isin fact a ∗ -subalgebra because of the equality x ∗ = x ∗ .From now on, we let P be the set of all the left projections LP( e ) , for e ∈ E . So for each p ∈ P there is an idempotent e ∈ E such that p = LP( e ) ; in particular ep = p and pe = e . Note that P ⊆ I . Lemma 4.10.
The following statements hold:i) The set P is a local unit for I .ii) If S denotes the ∗ -subalgebra of R A generated by I , h W M A M and K ( t ) , then I is a properideal of S , and there is a ∗ -isomorphism S /I ∼ = M l ( K ( t l )) . Proof.
For i ) , let x , ..., x n ∈ I . We can assume that each x i is a monomial of one of the forms ( I ) r r · · · with r i ∈ I ; ( II ) r r · · · with r i ∈ I . Consider the sets J = { r ∈ I | r appears as a first term in one of the x i } ,J = { r ∗ ∈ I | r appears as a first term in one of the x i } , so that J = J ∪ J is a finite subset of I . By Lemma 4.9 there exists an idempotent e ∈ E suchthat er = r for all r ∈ J . Take p = LP( e ) ∈ P , so pe = e and ep = p . Now for an element r ∈ J , wecompute pr = per = er = r. Also for an element r ∈ I such that r ∗ ∈ J , we compute pr ∗ = per ∗ = er ∗ = r ∗ , so by taking ∗ wehave rp = r . Multiplying to the left by the relative inverse r we get ( rr ) p = rr , which is a projection.Hence rr = ( rr ) ∗ = ( rrp ) ∗ = prr , and pr = prrr = rrr = r. We deduce from these computations that px i = x i for all ≤ i ≤ n . Since P = P ∗ , part i ) follows. i ) . By i ) , it is immediate to check that I ⊆ S is proper. To prove that it is an ideal, it isenough to show that p ( t ) I , I p ( t ) ⊆ I for all p ( t ) ∈ K ( t ) and that e ij ( W M ) I , I e ij ( W M ) ⊆ I forall ≤ i, j ≤ l − . By taking ∗ , we only need to show that p ( t ) I , e ij ( W M ) I ⊆ I .Let p ( t ) ∈ K ( t ) and a ∈ I . We can assume that a is a monomial of the form either (I) or (II). Inthe first case, a = ra ′ for some r ∈ I and a ′ ∈ I ; then p ( t ) a = p ( t ) ra ′ ∈ I since p ( t ) r ∈ I . In thesecond case, a = ra ′ for r ∈ I and a ′ ∈ I . Consider p ∈ P such that pr = r and e ∈ E such that p = LP( e ) . Then p ( t ) e ∈ I , so that p ( t ) a = p ( t ) ra ′ = p ( t ) pra ′ = p ( t ) epra ′ = p ( t ) ea ∈ I , as required. Similar computations can be used to show that e ij ( W M ) I ⊆ I .The rest of the proof follows exactly the same arguments as in the proof of Proposition 4.6. (cid:3) We are now ready to determine the ∗ -regular closure R A . Theorem 4.11.
Following the previous assumptions and caveats, we define I m to be the non-unitalsubalgebra of R A generated by I m − and the relative inverses of elements of I m − , starting from our I . Let also S m be the ∗ -subalgebra of R A generated by I m , h W M A M and K ( t ) . Then:(1) I m admits the set P as a local unit;(2) I m is a proper ∗ -ideal of S m , and S m /I m ∼ = M l ( K ( t l )) for all m ≥ ;(3) I ∞ = S m ≥ I m is a proper ∗ -regular ideal of R A , and R A /I ∞ ∼ = M l ( K ( t l )) . Moreover, R A is generated as a ∗ -algebra by I ∞ , h W M A M and K ( t ) .Proof. We observe that each I m is also a ∗ -subalgebra of R A . (1) and (2) follows easily by induction,taking into account that the same arguments as in the proof of Lemma 4.10 apply here.For (3) , let S ∞ be the ∗ -subalgebra of R A generated by I ∞ , h W M A M and K ( t ) . Clearly I ∞ ⊆ S ∞ is proper since each I m ⊆ S m is so. To prove that I ∞ is an ideal of S ∞ it is enough to show that K ( t ) I ∞ ⊆ I ∞ and that e ij ( W M ) I ∞ ⊆ I ∞ for all ≤ i, j ≤ l − . For the first inclusion, take p ( t ) ∈ K ( t ) and a ∈ I ∞ . Then a ∈ I m for some m ≥ , so by (2) we have p ( t ) a ∈ I m ⊆ I ∞ . Thesecond inclusion is obtained analogously.By construction of our sequence { I m } m ≥ , it is straightforward to show that I ∞ is ∗ -regular too.Therefore I ∞ is a ∗ -regular ideal of S ∞ and, just as before, its quotient S ∞ /I ∞ is ∗ -isomorphic to M l ( K ( t l )) , which is ∗ -regular. It follows from [17, Lemma 1.3] that S ∞ is ∗ -regular. Since A ∞ ⊆ S ∞ by Claim 6 of Proposition 4.6 and t ∈ S ∞ , we get A ⊆ S ∞ ⊆ R A . We conclude that S ∞ = R A . (cid:3) In conclusion, in the case that there exists a periodic point y ∈ X of finite period l , we havebeen able to determine part of the ideal structure of the ∗ -regular closure R A : for each such point y ∈ X one can apply the above process to construct a maximal ideal I ∞ ( y ) of R A , thus provingthat, in particular, R A is not simple. In fact, the construction of the ideal I ∞ ( y ) not only dependson the periodic point y ∈ X , but on the whole orbit O ( y ) = { y, T ( y ) , ..., T l − ( y ) } . This defines acorrespondence O ( y ) I ∞ ( y ) between the whole set of orbits of periodic points in X and maximal ideals of R A . The next lemmashows that this correspondence is in fact injective. Lemma 4.12.
Let x, y ∈ X be two periodic points of periods l , l , respectively (not necessarily equal).Suppose that x / ∈ O ( y ) . Then the maximal ideals I ∞ ( x ) and I ∞ ( y ) of R A are different.Proof. Since O ( x ) ∩ O ( y ) = ∅ , we can find a clopen subset U ⊆ X such that O ( x ) ∩ U = ∅ but O ( y ) ∩ U = ∅ . We can in fact assume that y ∈ U .Since O ( x ) ∩ U = ∅ , we have χ U ∈ C c,K ( X \{ x, ..., T l − ( x ) } ) ⊆ I ∞ ( x ) . Assume for contradictionthat I ∞ ( x ) = I ∞ ( y ) , so χ U ∈ I ∞ ( y ) . By (1) of Theorem 4.11 there exists p ∈ P satisfying pχ U = χ U . Hence we can find a non-zero polynomial p ( t ) ∈ K [ t ] and N ≥ M such that p = LP( e ) with = p ( t l ) − (1 − h W N ) p ( t l ) . Here the collection { W n } n ≥ M is taken with respect to the point y ∈ X (see Proposition 4.5). In particular eχ U = epχ U = pχ U = χ U , so h W N p ( t l ) χ U = 0 .Write p ( x ) = a + a x + · · · + a m x m for some a i ∈ K . If we consider the non-empty clopen set V := U ∩ T l ( U ) ∩ · · · ∩ T l · m ( U ) , we obtain χ V · h W N p ( t l ) χ U = χ V · ( a h W N χ U + a h W N χ T l ( U ) t l + · · · + a m h W N χ T l · m ( U ) t l · m )= a h W N χ V + a h W N χ V t l + · · · + a m h W N χ V t l · m = h W N χ V p ( t l ) . Since p ( t l ) is invertible inside R A , necessarily h W N χ V = 0 . This is a contradiction because y ∈ W N ∩ V . (cid:3) It is therefore reasonable to think that, in order to uncover the whole structure of R A in the caseof existence of a periodic point, it is crucial to understand the structure of the ideals I ∞ , and inparticular the structure of R ∞ = S n ≥ R n , which in turn can be studied by studying their pieces R n .Therefore, in the next section we will concentrate on uncovering part of the structure of the ∗ -regularclosure R n , for a fixed n .4.2. The ∗ -regular closure R B . We return to the general setting we had in Section 3, with theextra hypothesis that K is now a field with a positive definite involution − . We fix a clopen subset E of X and a partition P of X \ E into clopen subsets. Recall that B denotes the unital ∗ -subalgebra of A generated by the partial isometries { χ Z t } Z ∈ P , and we write B = L i ∈ Z B i t i with B = C K ( X ) ∩ B , B i = χ X \ ( E ∪ T ( E ) ∪···∪ T i − ( E )) B and B − i = χ X \ ( T − ( E ) ∪···∪ T − i ( E )) B for i > . We also write π for the map π : B → R given by π ( b ) = ( h W · b ) W , where R = Q W ∈ V M | W | ( K ) .We aim to follow the same steps as in [6, Section 6] to study the ∗ -regular closure R B := R ( B , R ) .However, the situation here is much more involved, and we are only able to determine a (large) ∗ -subalgebra of R B .The first step is to consider, from B , a skew partial power series ring B [[ t ; T ]] by taking infiniteformal sums X i ≥ b i ( χ X \ E t ) i = X i ≥ b i t i , where b i ∈ B i for all i ≥ . It is worth to point out here that, for i > , the coefficients b i are restricted to belong to the generallyproper ideal B i of B . Similarly we can consider B [[ t − ; T − ]] . Now, given a W ∈ V , only a finitenumber of terms in the infinite sum P i ≥ b i t i can be non-zero in the factor corresponding to W , sincethe product h W · ( χ X \ E t ) i is exactly zero for i ≥ | W | . We have a similar situation for B [[ t − ; T − ]] .In this way we obtain faithful representations π + : B [[ t ; T ]] → R , b ( h W · b ) W and π − : B [[ t − ; T − ]] → R , b ( h W · b ) W by lower (resp. upper) triangular matrices. We will be mainly interested in the first one π + .We have the following key property. Lemma 4.13.
Let x = P i ≥ b i t i ∈ B [[ t ; T ]] . Then x is invertible in B [[ t ; T ]] if and only if b isinvertible in B . Analogously for the elements of B [[ t − , T − ]] .Proof. Assume first that x = P i ≥ b i t i is invertible in B [[ t ; T ]] . There exists then y = P i ≥ b ′ i t i in B [[ t ; T ]] such that xy = yx = 1 . In particular b b ′ = 1 , and so b is invertible in B .Conversely, assume that b is invertible in B . We can then assume that b = 1 , so that x = 1 − y ,where the order of y in t is greater than or equal to . We then have x − = (1 − y ) − = 1 + y + y + · · · ∈ B [[ t ; T ]] , hence x is invertible in B [[ t ; T ]] . The same arguments work for elements of B [[ t − , T − ]] . (cid:3) Note that, in the notation used in Section 4, R B = R n in case B is one of the ∗ -subalgebras A n . e now introduce the following definitions. Definition 4.14. a) We denote by B + the algebra of elements of B supported in non-negative degrees in t , thatis B + = L i ≥ B i t i . Clearly B + ⊆ B [[ t ; T ]] . The division closure of B + in B [[ t ; T ]] will bedenoted by D + .b) We set B − = L i ≥ B − i t − i . Again B − ⊆ B [[ t − , T − ]] , and we denote by D − the divisionclosure of B − in B [[ t − ; T − ]] .In order to study the division closures D + and D − , we need the following known lemma. Lemma 4.15 (cf. [27]) . Let S be a unital ∗ -subalgebra of C K ( X ) generated by a family of characteristicfunctions of the form { χ C } C , where C are clopen subsets of X . Then S is a ∗ -regular ring, and everynon-zero element of S can be expressed in the form n X i =1 λ i χ K i , where λ i ∈ K \{ } for all ≤ i ≤ n , and { K i } ni =1 are mutually disjoint clopen subsets of X such that χ K i ∈ S for all ≤ i ≤ n .In particular, the ∗ -subalgebra B of B is ∗ -regular, and b ∈ B is invertible if and only if, whenwriting b in the above form, the family { K i } ni =1 constitutes a partition of X .Proof. If a = P ni =1 λ i χ K i is as in the statement, then a = P ni =1 λ − i χ K i is the relative inverse of a ,hence S is ∗ -regular.We show that each element of S can be written in the stated form. It is clear that each elementof S is a K -linear combination of functions of the form χ L i , where L i is a clopen subset of X and χ L i ∈ S , since every product χ C χ C · · · χ C t belongs to S and equals χ L , where L = C ∩ · · · ∩ C t is clopen. Therefore, every non-zero element a of S can be written as a = P ni =1 λ i χ L i , with { L i } ni =1 clopen subsets of X such that χ L i ∈ S . We now show that this sum can be chosen to be an orthogonalsum. This is done by induction on n .The result is clear for n = 1 , so assume that n ≥ , that a = P n +1 i =1 λ i χ L i with { L i } n +1 i =1 clopensubsets of X such that χ L i ∈ S , and that P ni =1 λ i χ L i = P mj =1 µ j χ K j where now { K j } mj =1 are mutuallydisjoint clopen subets of X such that χ K j ∈ S . We compute a = m X j =1 µ j χ K j + λ n +1 χ L n +1 = m X j =1 ( µ j + λ n +1 ) χ K j ∩ L n +1 + m X j =1 µ j χ K j \ L n +1 + λ n +1 χ L n +1 \ ( K ∪···∪ K m ) . Since the clopen sets { K j ∩ L n +1 } mj =1 ∪ { K j \ L n +1 } mj =1 ∪ { L n +1 \ ( K ∪ · · · ∪ K m ) } are clearly disjointand all their characteristic functions belong to S , this completes the induction step.Now, since B is generated by a family of characteristic functions of the above form, it is ∗ -regular.To conclude, assume that b = P ni =1 λ i χ K i ∈ B is invertible in B . Then necessarily its inversemust be its relative inverse b = P ni =1 λ − i χ K i , and so b b = P ni =1 χ K i . This tells us that { K i } ni =1 forms a partition of X . The converse is easily verified, with also b − = b . (cid:3) Proposition 4.16.
With the preceding notation, we have:(i) D + coincides with the rational closure of B + in B [[ t ; T ]] , and similarly D − coincides with therational closure of B − in B [[ t − ; T − ]] .(ii) π + ( D + ) is the division closure of π + ( B + ) in R , and similarly π − ( D − ) is the division closure of π − ( B − ) in R .(iii) π + ( D + ) ⊆ R B , and similarly π − ( D − ) ⊆ R B .(iv) π + ( D + ) ∗ = π − ( D − ) .Proof. ( i ) This is a standard observation (see e.g. [2, Observation 1.18]). ( ii ) Recall that π + is an injective homomorphism from B [[ t ; T ]] into R . We first show that π + ( B [[ t ; T ]]) is division closed in R . For this, let x = P i ≥ b i t i be an element in B [[ t ; T ]] such that + ( x ) is invertible in R . Observe that each component of π + ( x ) is an invertible matrix, with diagonalcoming exclusively from elements of B . It follows that π + ( b ) = π ( b ) must be invertible in R . Butsince B is regular (Lemma 4.15), there exists b in B such that b b b = b . Applying π and takinginto account that π ( b ) is invertible in R , we get that π ( b ) − = π ( b ) , and so b is in fact invertiblein B . It follows from Lemma 4.13 that x is invertible in B [[ t ; T ]] , as required.Now we use the following general fact: if R ⊆ S ⊆ T are unital embeddings of unital rings, and S is division closed in T , then the division closure of R in T equals the division closure of R in S , thatis D ( R, T ) = D ( R, S ) . Using this and the fact just proved that π + ( B [[ t ; T ]]) is division closed in R ,we deduce that D ( π + ( B + ) , R ) = D ( π + ( B + ) , π + ( B [[ t ; T ]])) = π + ( D ( B + , B [[ t ; T ]])) = π + ( D + ) , as desired. Analogous arguments give that D ( π − ( B − ) , R ) = π − ( D − ) . ( iii ) By ( ii ) , we have that π + ( D + ) = D ( π + ( B + ) , R ) which is contained in the division closureof π ( B ) in R , D ( π ( B ) , R ) . This last one is contained in R B by Lemma 2.3, hence π + ( D + ) ⊆ R B .Similarly π − ( D − ) ⊆ R B . ( iv ) First observe that π + ( B + ) ∗ = π − ( B − ) and π + ( B [[ t ; T ]]) ∗ = π − ( B [[ t − , T − ]]) . The reason isthat, for x = P i ≥ b i t i ∈ B [[ t ; T ]] (resp. ∈ B + ), we have π + ( x ) ∗ = π − (cid:16) X i ≥ t − i b ∗ i (cid:17) = π − (cid:16) X i ≥ T − i ( b ∗ i ) t − i (cid:17) . The element b ∗ i is computed in the ∗ -algebra B . Also, by the description of B as a partial crossedproduct (Proposition 3.7 of [3]), it follows that T − i ( b ∗ i ) ∈ B − i and so P i ≥ T − i ( b ∗ i ) t − i ∈ B [[ t − ; T − ]] (resp. ∈ B − ). Analogous arguments show the other inclusion(s).Now, π + ( D + ) ∗ = D ( π + ( B + ) ∗ , π + ( B [[ t ; T ]]) ∗ ) = D ( π − ( B − ) , π − ( B [[ t − ; T − ]])) = π − ( D − ) , as required. (cid:3) We have two subalgebras π + ( D + ) and π − ( D − ) = π + ( D + ) ∗ of R B . We will write D for the ∗ -subalgebra of R B generated by π + ( D + ) , which coincides with the subalgebra generated by π + ( D + ) and π − ( D − ) . Intuitively, we obtain D by first adjoining all possible inverses of elements of B + andthen taking adjoints in R . B + (cid:31) (cid:127) / / (cid:15) o ❅❅❅❅❅❅❅❅ O O ∗ (cid:15) (cid:15) D + (cid:16) p ❆❆❆❆❆❆❆❆ O O ∗ (cid:15) (cid:15) B .(cid:14) > > ⑤⑤⑤⑤⑤⑤⑤ (cid:16) p ❇❇❇❇❇❇❇❇ B (cid:31) (cid:127) / / D (cid:31) (cid:127) / / R B B − (cid:31) (cid:127) / / /(cid:15) > > ⑦⑦⑦⑦⑦⑦⑦⑦ D − .(cid:14) > > ⑥⑥⑥⑥⑥⑥⑥⑥ Note that D is indeed contained in the division closure D ( B , R ) of B in R .Our plan now is to adjoin to D the relative inverses of (some) elements of D , in a controlled way.This is done as follows. Definition 4.17 (Special terms inside B ) . a) A monomial b i t i ∈ B + (for i > ) is said to be special if the coefficient b i ∈ B i is exactly of theform χ S , with(4.4) S = T i − ( Z ′ i − ) ∩ T i − ( Z ′ i − ) ∩ · · · ∩ Z ′ , that is s = i and r = 0 in (3.1), and moreover we ask that E ∩ T − i ( S ) ∩ T − i − ( E ) = ∅ . The setof clopen subsets S ⊆ X of the form (4.4) with E ∩ T − i ( S ) ∩ T − i − ( E ) = ∅ will be denoted by W i . ) Similarly, a monomial b − j t − j ∈ B − (for j > ) is said to be special if the coefficient b − j ∈ B − j is exactly of the form χ S ′ , with(4.5) S ′ = T − ( Z ′− ) ∩ T − ( Z ′− ) ∩ · · · ∩ T − j ( Z ′− j ) , that is r = j and s = 0 in (3.1), and moreover we ask that E ∩ S ′ ∩ T − j − ( E ) = ∅ . The set ofclopen subsets S ′ ⊆ X of the form (4.5) with E ∩ S ′ ∩ T − j − ( E ) = ∅ will be denoted by W − j .c) If E ∩ T − ( E ) = ∅ , there is one term in B which we call special , namely the element χ S ∪ T − ( S ) ,where S = E ∪ (cid:16) [ Z ∈ P Z ∩ T − ( E ) = ∅ Z (cid:17) and S = E ∪ (cid:16) [ Z ∈ P T − ( Z ) ∩ E = ∅ Z (cid:17) . In case E ∩ T − ( E ) = ∅ , we set W = { S ∪ T − ( S ) } . If E ∩ T − ( E ) = ∅ , then there is nospecial term of degree and so W = ∅ .It is clear that, for i ≥ , the set W i is in bijection with the set of all W ∈ V having length i + 1 ,through the map S W ( S ) := E ∩ T − i ( S ) ∩ T − i − ( E ) . The inverse map will be written as W S ( W ) , so that S ( W ( S )) = S and W ( S ( W )) = W . Analo-gously, for j ≥ , the same set of all W ∈ V having length j + 1 is in bijection with W − j via S ′ W ( S ′ ) := E ∩ S ′ ∩ T − j − ( E ) . Again, the inverse map will be denoted by W S ′ ( W ) . When E ∩ T − ( E ) = ∅ , the set W containsonly one element, namely the clopen S ∪ T − ( S ) , and therefore is in bijective correspondence withthe set consisting of the only clopen W ∈ V with length , namely W = E ∩ T − ( E ) . We will use thenotation W S ( W ) := S ∪ T − ( S ) in this case. When E ∩ T − ( E ) = ∅ there is no W ∈ V havinglength and correspondingly W = ∅ .Note that, by construction, the element b = χ S ∪ T − ( S ) serves as a unit among the special terms,in the sense that b · b i t i = b i t i = b i t i · b and b · b − j t − j = b − j t − j = b − j t − j · b , for b i t i , b − j t − j special terms of degrees i, j ≥ , respectively.The special terms are exactly detected by the representation π : B → R , as follows. Lemma 4.18.
With the previous notation,i) For i > , let b i t i = χ S t i be a special term, with S as in (4.4) . Then h W · b i t i = e i, ( W ) , where W = W ( S ) . Moreover, if W ′ = W is of length k ≥ , then the component of e k − , ( W ′ ) in h W ′ · b i t i is .ii) For j > , let b − j t − j = χ S ′ t − j be a special term, with S ′ as in (4.5) . Then h W · b − j t − j = e ,j ( W ) ,where W = W ( S ′ ) . Moreover, if W ′ = W is of length k ≥ , then the component of e ,k − ( W ′ ) in h W ′ · b − j t − j is .iii) Suppose that E ∩ T − ( E ) = ∅ . Let b = χ S ∪ T − ( S ) be the special term of degree . Then h W · b = e , ( W ) , where W = E ∩ T − ( E ) . Moreover, if W = E ∩ T − ( E ) , then the componentsof e , ( W ) and e | W |− , | W |− ( W ) in h W · b are exactly .Proof. We will only prove i ) , being the other ones analogous. Take W = W ( S ) = E ∩ T − i ( S ) ∩ T − i − ( E ) , and note that T l ( W ) ∩ S = ∅ for ≤ l ≤ i − . For l = i , it gives T i ( W ) ∩ S = T i ( W ) .Hence h W · b i t i = i X l =0 χ T l ( W ) ∩ S t i = χ T i ( W ) t i = ( χ X \ E t ) i χ W = e i, ( W ) . For the second part, it is enough to show that the product e k − ,k − ( W ′ ) (cid:0) h W ′ · b i t i (cid:1) e ( W ′ ) is zero.This is a straightforward computation: e k − ,k − ( W ′ ) (cid:16) h W ′ · b i t i (cid:17) e ( W ′ ) = χ T k − ( W ′ ) (cid:16) k − X l =0 χ T l ( W ′ ) ∩ S t i (cid:17) χ W ′ = χ T k − ( W ′ ) ∩ T i ( W ′ ) ∩ S t i . ut W = E ∩ T − i ( S ) ∩ T − i − ( E ) , so T k − ( W ′ ) ∩ T i ( W ′ ) ∩ S ⊆ T k − ( W ′ ) ∩ T i ( W ′ ∩ W ) which isempty for W ′ = W . The result follows. (cid:3) Consider the subset S [[ t ; T ]] of B [[ t ; T ]] consisting of those elements X i ≥ b i ( χ X \ E t ) i = X i ≥ b i t i such that each b i ∈ B i belongs to span { χ S | S ∈ W i } . It is always a linear subspace of B [[ t ; T ]] , butit might not be a subalgebra. We will, however, see in Section 6.2 that in the special case of A beingthe lamplighter group algebra, S [[ t ; T ]] is indeed an algebra, and even an integral domain.As we shall see, S [[ t ; T ]] certainly is a subalgebra of B [[ t ; T ]] when it is endowed with the multi-plicative structure given by the Hadamard product ⊙ , defined by the rule (cid:16) X i ≥ b i t i (cid:17) ⊙ (cid:16) X j ≥ b ′ j t j (cid:17) := X i ≥ ( b i b ′ i ) t i . Observation 4.19.
Each b i belongs to the linear span of all the χ S with S ∈ W i , hence they can bewritten as b i = P S ∈ W i λ S χ S for λ S ∈ K . Let b i , b ′ i be two given elements of this form: b i = X S ∈ W i λ S χ S , b ′ i = X S ′ ∈ W i µ S ′ χ S ′ . Since the sets S ∈ W i are of the form T i − ( Z ′ i − ) ∩ T i − ( Z ′ i − ) ∩ · · · ∩ Z ′ for i > or S ∪ T − ( S ) in case i = 0 , we see that for S, S ′ ∈ W i , S ∩ S ′ = ∅ if they are different. Therefore the Hadamardproduct of S [[ t ; T ]] can be written as (cid:16) X i ≥ b i t i (cid:17) ⊙ (cid:16) X j ≥ b ′ j t j (cid:17) = X i ≥ ( b i b ′ i ) t i = X i ≥ (cid:16) X S ∈ W i λ S µ S χ S (cid:17) t i . We can also define an involution on S [[ t ; T ]] by X i ≥ (cid:16) X S ∈ W i λ S χ S (cid:17) t i := X i ≥ (cid:16) X S ∈ W i λ S χ S (cid:17) t i . These operations turn S [[ t ; T ]] into a commutative ∗ -algebra ( S [[ t ; T ]] , ⊙ , − ) . Indeed it is a ∗ -algebraisomorphic to Q V K . In the next proposition we show that it can be identified with the center of thealgebra R = Q W ∈ V M | W | ( K ) , and also with a certain corner of R . We first fix some notation: wewill denote the projections π ( χ C ) ∈ R by p C for any clopen C ⊆ X . So for example p E = π ( χ E ) =( e ( W )) W ∈ R , and p T − ( E ) = π ( χ T − ( E ) ) = ( e | W |− , | W |− ( W )) W ∈ R . Proposition 4.20.
We have an isomorphism of ∗ -algebras S [[ t ; T ]] Ψ ∼ = Z ( R ) , the center of R . Inparticular, we have a ∗ -isomorphism S [[ t ; T ]] ∼ = p E R p E given by d Ψ( d ) p E , d ∈ S [[ t ; T ]] .Proof. Write an element P i ≥ b i t i ∈ S [[ t ; T ]] as X i ≥ b i t i = X i ≥ (cid:16) X S ∈ W i λ S χ S (cid:17) t i = X W ∈ V ( λ S ( W ) χ S ( W ) ) t | W |− . Note that Z ( R ) = Z ( Q W ∈ V M | W | ( K )) = Q W ∈ V K . We define a map Ψ : S [[ t ; T ]] → Z ( R ) by Ψ (cid:16) X W ∈ V ( λ S ( W ) χ S ( W ) ) t | W |− (cid:17) = ( λ S ( W ) · h W ) W . It is straightforward to check that it is indeed an isomorphism of ∗ -algebras. Since Z ( R ) ∼ = p E R p E through z p E z , the result follows. (cid:3) Our next step is to prove the following formulas, which will be useful later.
Lemma 4.21.
For
A, B ∈ S [[ t ; T ]] , the following formulas hold inside R : p E · π + ( A ) ∗ · p T − ( E ) · π + ( B ) · p E = Ψ( A ⊙ B ) p E ,p T − ( E ) · π + ( A ) · p E · π + ( B ) ∗ · p T − ( E ) = Ψ( A ⊙ B ) p T − ( E ) . roof. We will only prove the first formula, being the second one analogous. Note that it is enoughto prove that the W -component of the left-hand side and the right-hand side of the formula agree.More precisely, we have to check the equality (cid:0) p E · π + ( A ) ∗ · p T − ( E ) · π + ( B ) · p E (cid:1) W = (cid:0) Ψ( A ⊙ B ) p E (cid:1) W for any fixed W ∈ V . Write A = P i ≥ (cid:16) P S ∈ W i λ S χ S (cid:17) t i and B = P j ≥ (cid:16) P S ′ ∈ W j µ S ′ χ S ′ (cid:17) t j . Wefirst compute, for a fixed S ∈ W i and using Lemma 4.18, the terms e | W |− , | W |− ( W ) · ( h W · χ S t i ) · e ( W ) = δ W,W ( S ) e i ( W ) = δ W,W ( S ) e | W |− , ( W ) . Therefore (cid:0) p E · π + ( A ) ∗ · p T − ( E ) · π + ( B ) · p E (cid:1) W = e ( W ) · (cid:16) X i ≥ X S ∈ W i λ S h W · χ S t i (cid:17) ∗ · e | W |− , | W |− ( W ) · (cid:16) X j ≥ X S ′ ∈ W j µ S ′ h W · χ S ′ t j (cid:17) · e ( W )= e ( W ) · (cid:0) λ S ( W ) e | W |− , ( W ) (cid:1) ∗ · e | W |− , | W |− ( W ) · (cid:0) µ S ( W ) e | W |− , ( W ) (cid:1) · e ( W )= λ S ( W ) µ S ( W ) e ( W ) = (cid:0) Ψ( A ⊙ B ) p E (cid:1) W , so the result follows. (cid:3) We now proceed to generalize the above formulas. For this purpose, we first define an idempotentmap P : B [[ t ; T ]] → S [[ t ; T ]] as follows. Lemma 4.22.
With the above notation, there exists an idempotent linear map P : B [[ t ; T ]] → S [[ t ; T ]] such that for each x ∈ B [[ t ; T ]] , we have p T − ( E ) · π + ( x ) · p E = p T − ( E ) · π + ( P ( x )) · p E . Proof.
For i ≥ , let V i be the linear subspace of B i given by span { χ S | S ∈ W i } , and let V ′ i be thelinear subspace of B i spanned by all the projections χ C , where ( ∗ ) i C is a non-empty clopen subset of X of the form (3.1), with s ≥ i ,and such that either s > i or r > .Claim 1: B i = V i + V ′ i .Proof: Recall that B i is spanned by all the characteristic functions χ C , where C is a clopen subsetof X of the form (3.1) with s ≥ i . If s > i or r > , then C is of the form ( ∗ ) i , so that χ C ∈ V ′ i .So we can assume that s = i and r = 0 . Furthermore, if T i ( E ) ∩ C ∩ T − ( E ) is non-empty, then C ∈ W i and so χ C ∈ V i . So we can further assume that T i ( E ) ∩ C ∩ T − ( E ) = ∅ .We can write C = (cid:16) T i ( E ) ∩ C (cid:17) ⊔ (cid:16) G Z ∈ P T i ( Z ) ∩ C (cid:17) . If T i ( E ) ∩ C = ∅ , then χ C is a sum of terms of the form ( ∗ ) i , so that χ C ∈ V ′ i . If T i ( E ) ∩ C = ∅ ,we can further decompose C as C = (cid:16) G Z ∈ P T i ( E ) ∩ C ∩ T − ( Z ) (cid:17) ⊔ (cid:16) G Z ∈ P T i ( Z ) ∩ C (cid:17) by using the assumption T i ( E ) ∩ C ∩ T − ( E ) = ∅ . Note that for each Z ∈ P , either C ∩ T − ( Z ) isempty or it is of the form ( ∗ ) i ; in the latter case we can write any non-empty T i ( E ) ∩ C ∩ T − ( Z ) as T i ( E ) ∩ C ∩ T − ( Z ) = (cid:16) C ∩ T − ( Z ) (cid:17)/(cid:16) G Z ′ ∈ P T i ( Z ′ ) ∩ C ∩ T − ( Z ) (cid:17) . Therefore χ C is a linear combination of terms of the form ( ∗ ) i , and thus χ C ∈ V ′ i . (cid:3) Claim 2: V i ∩ V ′ i = { } . Note that the appearance of the term δ W,W ( S ) already encodes the fact that the term is if | W | 6 = i + 1 . roof: Assume that b ∈ V i ∩ V ′ i and write b = P S ∈ W i λ S χ S , with λ S ∈ K . Since b ∈ V ′ i we have (cid:16) X S ′ ∈ W i χ T i ( E ) ∩ S ′ ∩ T − ( E ) (cid:17) b = X S ∈ W i λ S χ T i ( E ) ∩ S ∩ T − ( E ) . Since { χ T i ( E ) ∩ S ∩ T − ( E ) } S ∈ W i is a family of mutually orthogonal non-zero projections, we get that b = 0 . (cid:3) Therefore B i = V i ⊕ V ′ i for i ≥ . In the base case i = 0 , we need to distinguish between two differentscenarios, depending on whether the intersection E ∩ T − ( E ) is empty or not.Case 1: E ∩ T − ( E ) = ∅ .In this case we have W = ∅ . We take V ′ = B and V = { } .Case 2: E ∩ T − ( E ) = ∅ .Define here V ′ to be the linear subspace of B spanned by all the projections χ C , where ( ∗ ) C is a non-empty clopen subset of X of the form (3.1),and let V = K · χ S ∪ T − ( S ) and V ′′ = K · χ E ∩ T − ( E ) . Analogous computations as in the case for i ≥ show that there is a decomposition B = V ′′ ⊕ V ′ . In particular, V ′ has codimension , so inorder to obtain the decomposition B = V ⊕ V ′ it is enough to show that χ S ∪ T − ( S ) / ∈ V ′ . But if χ S ∪ T − ( S ) ∈ V ′ , we could write it as χ S ∪ T − ( S ) = X r,s ≥ X some Z i λ r,s,Z i χ T − r ( Z − r ) ∩···∩ T − ( Z − ) ∩ Z ∩···∩ T s − ( Z s − ) , and by multiplying the above equality by χ E ∩ T − ( E ) we would get χ E ∩ T − ( E ) = χ E ∩ T − ( E ) · χ S ∪ T − ( S ) = 0 , a contradiction. Hence χ S ∪ T − ( S ) / ∈ V ′ , and we have the desired decomposition. (cid:3) We can now define P as the projection onto the first component in the decomposition B [[ t ; T ]] = (cid:16) Y i ≥ V i t i (cid:17) ⊕ (cid:16) Y i ≥ V ′ i t i (cid:17) = S [[ t ; T ]] ⊕ (cid:16) Y i ≥ V ′ i t i (cid:17) . We check the formula in the statement. Take x = P i ≥ b i t i ∈ B [[ t ; T ]] . We can write it as x = P ( x ) + X i ≥ X C as in ( ∗ ) i λ C χ C t i in the case E ∩ T − ( E ) = ∅ , and as x = P ( x ) + b + X i ≥ X C as in ( ∗ ) i λ C χ C t i if E ∩ T − ( E ) = ∅ . Note that in the latter case χ T − ( E ) · b · χ E = 0 ; hence to prove the formula of thestatement it is enough to check that, for a fixed W = E ∩ T − ( Z ) ∩ · · · ∩ T − k +1 ( Z k − ) ∩ T − k ( E ) ∈ V and C of the form ( ∗ ) i , we have h W · χ T − ( E ) · χ C t i · χ E = 0 . We compute h W · χ T − ( E ) · χ C t i · χ E = χ T k − ( W ) ∩ T − ( E ) ∩ C ∩ T i ( E ) t i . This is zero for C of the form ( ∗ ) i , since either C ⊆ T i ( X \ E ) or C ⊆ T − ( X \ E ) . The resultfollows. (cid:3) We can now generalize the formulas in Lemma 4.21.
Lemma 4.23.
For x, y ∈ B [[ t ; T ]] , the following formulas hold: p E · π + ( x ) ∗ · p T − ( E ) · π + ( y ) · p E = Ψ( P ( x ) ⊙ P ( y )) p E ,p T − ( E ) · π + ( x ) · p E · π + ( y ) ∗ · p T − ( E ) = Ψ( P ( x ) ⊙ P ( y )) p T − ( E ) . roof. By Lemma 4.22 we have p T − ( E ) · π + ( x ) · p E = p T − ( E ) · π + ( P ( x )) · p E for all x ∈ B [[ t ; T ]] .Taking the involution on both sides, we get that p E · π + ( x ) ∗ · p T − ( E ) = p E · π + ( P ( x )) ∗ · p T − ( E ) forall x ∈ B [[ t ; T ]] . Now we obtain, from Lemma 4.21, p E · π + ( x ) ∗ · p T − ( E ) · π + ( y ) · p E = p E · π + ( P ( x )) ∗ · p T − ( E ) · π + ( P ( y )) · p E = Ψ( P ( x ) ⊙ P ( y )) p E , as desired. The proof of the other equality is similar. (cid:3) Recall that S [[ t ; T ]] is a unital ∗ -regular commutative algebra under the Hadamard product ⊙ , theunit being the element e = P i ≥ ( P S ∈ W i χ S ) t i . We obtain: Proposition 4.24.
Take s = P Z ∈ P χ Z t = χ X \ E t ∈ B t . Then u := (1 − s ) − = 1 + s + s + · · · ∈ D + satisfies that P ( u ) = e , where e is the unit element of ( S [[ t ; T ]] , ⊙ ) . As a consequence, we have theformulas p E · π + ( u ) ∗ · p T − ( E ) · π + ( x ) · p E = Ψ( P ( x )) p E ,p T − ( E ) · π + ( x ) · p E · π + ( u ) ∗ · p T − ( E ) = Ψ( P ( x )) p T − ( E ) for all x ∈ B [[ t ; T ]] . In particular, inside R B , the ideal generated by p E coincides with the idealgenerated by p T − ( E ) .Proof. We first note that s i = ( χ X \ E t ) i = χ X \ ( E ∪···∪ T i − ( E )) t i = X Z ,Z ,...,Z i − ∈ P χ Z ∩ T ( Z ) ∩···∩ T i − ( Z i − ) t i for i ≥ . It follows from this formula and the computations done in Lemma 4.22 that P ( s i ) = P S ∈ W i χ S t i . To compute P (1) , note first that P (1) = 0 if E ∩ T − ( E ) = ∅ . So assume that E ∩ T − ( E ) = ∅ , and observe that χ S ∪ T − ( S ) + χ ( X \ S ) ∩ ( X \ T − ( S )) = χ S ∪ T − ( S ) + X Z ∈ P Z ∩ T − ( E )= ∅ X Z ′ ∈ P T − ( Z ′ ) ∩ E = ∅ χ Z ∩ T − ( Z ′ ) . By definition, the second part of this expression belongs to the complement V ′ , so that P (1) = χ S ∪ T − ( S ) . Putting everything together, it is clear that P ( u ) = e . The desired formulas follow fromLemma 4.23. In particular, we have p E · π + ( u ) ∗ · p T − ( E ) · π + ( u ) · p E = Ψ( P ( u )) p E = Ψ( e ) p E = p E ,p T − ( E ) · π + ( u ) · p E · π + ( u ) ∗ · p T − ( E ) = Ψ( P ( u )) p T − ( E ) = Ψ( e ) p T − ( E ) = p T − ( E ) . One deduces from this and Proposition 4.16 ( iii ) that the ideal generated by p E coincides with theideal generated by p T − ( E ) inside R B . (cid:3) We now define ∗ -algebras Q and E which resemble the ones defined for a particular case in [6,Lemma 6.10] (see Theorem 6.15 for the exact relation between these constructions). The algebra Q is ∗ -regular, and the ∗ -algebra E is contained in the ∗ -regular closure R B of B in R . Definition 4.25.
With the above notation, we define the ∗ -algebra Q as the ∗ -regular closure of P ( D + ) in the ∗ -regular algebra ( S [[ t ; T ]] , ⊙ , − ) . In other words, Q is the smallest ∗ -regular subalgebraof S [[ t ; T ]] containing P ( D + ) . The ∗ -algebra E is defined as the subalgebra of R generated by D and Ψ( Q ) p E .Let x ∈ D + ⊆ B [[ t ; T ]] , x = P i ≥ b i ( x ) t i . By Lemma 4.22, we can decompose each b i ( x ) = b (1) i ( x ) + b (2) i ( x ) ∈ V i ⊕ V ′ i , being b (1) i ( x ) = P S ∈ W i λ S ( x ) χ S . Then the relative inverse of q := P ( x ) = P i ≥ (cid:16) P S ∈ W i λ S ( x ) χ S (cid:17) t i inside Q is given by (see Observation 4.19) q = X i ≥ (cid:16) X S ∈ W i λ S ( x ) =0 λ S ( x ) − χ S (cid:17) t i , o that qq = P i ≥ (cid:16) P S ∈ W i λ S ( x ) =0 χ S (cid:17) t i .We are now ready to show the desired properties of the ∗ -algebras Q and E . Proposition 4.26.
We have an embedding Ψ( Q ) p E ⊆ p E R B p E . Moreover we have B ⊆ D ⊆ E ⊆ R B , where R B is the ∗ -regular closure of B in R .Proof. Let x ∈ D + , and set u = (1 − s ) − ∈ D + . Due to Proposition 4.24, we have Ψ( P ( x )) p E = p E · π + ( u ) ∗ · p T − ( E ) · π + ( x ) · p E . By Proposition 4.16, we have that π + ( D + ) ⊆ R B , so all the factors of the right-hand side of the aboveequality belong to D ⊆ R B . It follows that Ψ( P ( x )) p E ∈ p E R B p E , and so Ψ( P ( D + )) p E ⊆ p E R B p E .By Proposition 4.20, the map d Ψ( d ) p E defines a ∗ -isomorphism from S [[ t ; T ]] onto p E R p E .It follows that Ψ( Q ) p E is the ∗ -regular closure of Ψ( P ( D + )) p E in p E R p E . Since Ψ( P ( D + )) p E ⊆ p E R B p E and p E R B p E is ∗ -regular, we conclude that Ψ( Q ) p E ⊆ p E R B p E ⊆ R B . From this it isobvious that E ⊆ R B . This shows the result. (cid:3) We will show in Theorem 6.15 that we have the equality E = R B for the algebra B studied in [6,Section 6]. 5. Group algebras arising as Z -crossed product algebras We start this section by first showing how the Fourier transform (sometimes called Pontryagin du-ality) describes the group algebras of several semidirect product groups as Z -crossed product algebras.In this setting, the Atiyah problem for the group algebra is translated to a problem on computingranks inside the corresponding Z -crossed product algebra.For a topological, second countable, locally compact abelian group H one can define its Pontryagindual b H as the set of continuous homomorphisms φ : H → T , also called characters. With thecompact-convergence topology, b H becomes a topological, metrizable, locally compact abelian group.If H is a countable discrete group then b H is compact, and if moreover H is a torsion group then b H istotally disconnected. We refer the reader to [16, Chapter 4] for more information about Pontryaginduality.Suppose now that H is a countable discrete, torsion abelian group. Associated with H , we considerthe subset O ⊆ N given by O = { n ∈ N | there exists an element g ∈ H of order n } . This set inherits the structure of a lattice from that of N . Lemma 5.1.
The set O is a sublattice of ( N , div , gcd , lcm ) . Even more, if n ∈ O , then any divisor d of n belongs to O too, so O is a hereditary sublattice of N .Proof. Given a, b ∈ O , there exist h, g ∈ H such that o ( h ) = a and o ( g ) = b . Take first d to be anydivisor of a , so we write a = da ′ for some a ′ ∈ N . Then the element h a ′ ∈ H has order exactly d , andso d ∈ O . In particular, we have shown that gcd { a, b } belongs to O .We must show now that m := lcm { a, b } belong to O . If both a and b are coprime numbers, it isstraightforward to show that the element hg ∈ H has order exactly ab = m , so we assume that a and b are not coprime. If we let ¶ be the set of prime numbers that appear in the factorizations of a and b , we can write a = Y p ∈¶ p α p , b = Y p ∈¶ p β p , eing α p , β p non-negative integers. Define now a ′ := Y p ∈¶ α p ≥ β p p α p , b ′ := Y p ∈¶ α p <β p p β p so that, by construction, gcd { a ′ , b ′ } = 1 and lcm { a ′ , b ′ } = a ′ b ′ = m = lcm { a, b } . Clearly, the orderof the element e h := h aa ′ is a ′ , and the order of the element e g := g bb ′ is b ′ . We now use the previouscase (since a ′ , b ′ are coprime) to conclude that the element e h e g ∈ H has order exactly a ′ b ′ = m . Thisconcludes the proof of the lemma. (cid:3) For this section, we take K to be any field satisfying the following hypothesis: ( a ) the characteristic of K must be coprime with all n ∈ O ; ( b ) K must contain all the n th roots of unity, for any n ∈ O .We write U ( K ) = [ n ∈ O { all n th roots of unity in K } ⊆ K. It is clear that U ( K ) is a subgroup of the multiplicative group K × . We define the K - Pontryagin dual of H as the set of all homomorphisms b H := { φ : H → K | φ is a morphism of groups } = { φ : H → U ( K ) | φ is a morphism of groups } . This is a group under pointwise product. Note that, since H is endowed with the discrete topology,any element φ ∈ b H will be automatically continuous, whatever would be the topology on U ( K ) .Therefore in the case K = C this definition coincides with the usual Pontryagin dual.We want to mimic the properties of b H for the case K = C to this general setting. Indeed we willshow later that b H does not depend on the particular field K up to isomorphism. To this aim, we needthe notion of compatible family of primitive roots of unity . Definition 5.2.
Let X := { ξ n } n ∈ O be a family of roots of unity in K consisting of primitive ones (sofor each n ∈ O , we choose a primitive n th root of unity ξ n ). The family X is said to be compatible ifthe equation ξ nn · m = ξ m holds for all n, m ∈ O such that n · m ∈ O .So for example for K = C , the family ξ n := e πin is compatible. Lemma 5.3.
A compatible family of primitive roots of unity in K always exists.Proof. If O is finite, then it follows from Lemma 5.1 that O is the set of divisors of some n ∈ N . Inthis case, take a primitive n th root of unity ξ n and set ξ d := ξ n/dn for any divisor d of n .Now assume that O is infinite. Enumerate O = { n i } i ∈ N such that n i < n i +1 for all i ∈ N . Assumefirst that K has characteristic . We first construct a family { η i } i ∈ N such that each η i is a primitive n · · · n th i root of unity, and that η n i +1 i +1 = η i . The elements η i may belong to an algebraic closure of K , but the elements ξ n below belong necessarily to K by our hypothesis.Let η be a primitive n th root of unity, and assume we have constructed η i a primitive n · · · n th i root of unity, such that η n i i = η i − . To construct η i +1 , take first any primitive n · · · n i · n th i +1 root of unity ω i +1 ; then ω n i +1 i +1 is a primitive n · · · n th i root of unity, so of the form η ji for some j coprime with n · · · n i . Let l be the inverseof j modulo n · · · n i . If we can find an integer k such that l := l + n · · · n i · k is coprime with n · · · n i · n i +1 we will be done, since η i +1 := ω li +1 will be a primitive n · · · n th i +1 root of unity, and η n i +1 i +1 = ω l · n i +1 i +1 = η j · li = η i . Let us now proceed to find an integer k such that l = l + n · · · n i · k is coprime with n · · · n i · n i +1 .Assume first that n i +1 is not a prime number; by Lemma 5.1, its divisors must be in O , so they are lready included in n · · · n i . Since l is coprime with n · · · n i , it also is coprime with n i +1 . Thus wemay take k = 0 , and we will be done. If n i +1 is a prime number p , then two cases can happen: (1) if p does not divide l , then l is coprime with both n · · · n i and p = n i +1 , and we may take k = 0 ; (2)if p does divide l , then it certainly cannot divide l + n · · · n i (if that was the case, p would divide n · · · n i , which is impossible), we can thus take k = 1 .Note that, because of our hypotheses on the characteristic of K , the above arguments work mutatismutandis for a field K of arbitrary characteristic (assuming, of course, the required hypotheses).Now, given such a family { η i } i ∈ N of primitive roots of unity, we define the desired compatible one.For each n = n i ∈ O , let n j ∈ O be any element such that n divides the product n · · · n j , andconsider ξ n := η n ··· njn j . We claim that this definition does not depend on the choice of n j , for if n l ∈ O is another elementsuch that n divides n · · · n l , then by assuming n l > n j we compute η n j +1 ··· n l l = η n j +1 ··· n l − l − = · · · = η j . Here we have used the property η n i +1 i +1 = η i for all i ∈ N . The claim follows straightforwardly: η n ··· nln l = η n ··· njn n j +1 ··· n l l = η n ··· njn j . It is easily checked that ξ n defines a primitive n th root of unity. We now verify the compatibilityproperty: given n, m ∈ O such that n · m ∈ O , and n j ∈ O such that n · m divides n · · · n j , wecompute ξ nn · m = η n ··· njn · m nj = η n ··· njm j = ξ m . This concludes the proof of the lemma. (cid:3)
Using Lemma 5.3, we can indeed show that, with our hypothesis on H and K , the group U ( K ) does not depend on the field K , as follows. Lemma 5.4.
With the above hypothesis and notation, we have that U ( K ) does not depend on K .In fact, the map Υ : U ( K ) → U ( C ) which sends ξ an to e πin a for n ∈ O and ≤ a < n is a groupisomorphism.Proof. We first observe that this map is well-defined. Suppose that ξ an = ξ bm , for n, m ∈ O and ≤ a < n , ≤ b < m . Let l = lcm { n, m } , and write l = nl = ml . We have ξ al l = ξ al nl = ξ an = ξ bm = ξ bl ml = ξ bl l , so that al − bl must be a multiple of l . Using this we obtain that Υ( ξ an ) = e πil al = e πil bl = Υ( ξ bm ) . A similar computation shows that Υ is a group homomorphism. Clearly Υ is a bijection and so it isa group isomorphism. (cid:3) Now we can translate the usual metric topology on U ( C ) ⊆ C into a metric topology on U ( K ) . Itturns out that this topology can also be defined directly in terms of the algebraic structure of U ( K ) ,as follows.Take a compatible family of primitive roots of unity X = { ξ n } n ∈ O , so that U ( K ) = [ n ∈ O { , ξ n , ..., ξ n − n } . Define a distance d ξ over U ( K ) in the following way: for two elements ξ an , ξ bm ∈ U ( K ) with ≤ a < n and ≤ b < m (here both n, m ∈ O ), define d ξ ( ξ an , ξ bm ) := (cid:12)(cid:12)(cid:12) an − bm (cid:12)(cid:12)(cid:12) if (cid:12)(cid:12)(cid:12) an − bm (cid:12)(cid:12)(cid:12) ≤ ;1 − (cid:12)(cid:12)(cid:12) an − bm (cid:12)(cid:12)(cid:12) if (cid:12)(cid:12)(cid:12) an − bm (cid:12)(cid:12)(cid:12) > . ote that d ξ is bounded by . It is direct to prove that d ξ is indeed a well-defined distance. We notealso that d ξ is translation invariant, in the sense that d ξ ( g g, g g ) = d ξ ( g , g ) for g , g , g ∈ U ( K ) .This defines a metric topology on U ( K ) . We prove below that in the case K = C , the subspacetopology of U ( C ) ⊆ C coincides with the metric topology induced by d ξ when taking the compatiblefamily ξ n = e πin . It then follows that the isomorphism Υ from Lemma 5.4 is indeed an isomorphismof topological groups. Lemma 5.5.
We have the formula (cid:12)(cid:12)(cid:12) ξ an − ξ bm (cid:12)(cid:12)(cid:12) = 4 sin ( πd ξ ( ξ an , ξ bm )) . Here |·| denotes the complex norm.Proof.
It is a simple computation: (cid:12)(cid:12)(cid:12) ξ an − ξ bm (cid:12)(cid:12)(cid:12) = h cos (cid:16) π an (cid:17) − cos (cid:16) π bm (cid:17)i + h sin (cid:16) π an (cid:17) − sin (cid:16) π bm (cid:17)i = 2 − h cos (cid:16) π an (cid:17) cos (cid:16) π bm (cid:17) + sin (cid:16) π an (cid:17) sin (cid:16) π bm (cid:17)i = 2 h − cos (cid:16) π an − π bm (cid:17)i = 4 sin (cid:16) π (cid:12)(cid:12)(cid:12) an − bm (cid:12)(cid:12)(cid:12)(cid:17) = 4 sin ( πd ξ ( ξ an , ξ bm )) . (cid:3) Proposition 5.6.
For K = C and ξ n = e πin , the subspace topology for U ( C ) ⊆ C coincides with thetopology induced by the distance d ξ .Proof. For a real number | x | ≤ π , there exist positive constants c, C > such that c | x | ≤ | sin( x ) | ≤ C | x | . Applying Lemma 5.5 one gets positive constants A, B > such that A · d ξ ( ξ an , ξ bm ) ≤ (cid:12)(cid:12)(cid:12) ξ an − ξ bm (cid:12)(cid:12)(cid:12) ≤ B · d ξ ( ξ an , ξ bm ) . The result follows. (cid:3)
We can now endow b H with the compact-convergence topology. Recall that, in our setting, thistopology has as a basis the sets B { h ,...,h n } ( φ, ǫ ) := { ψ ∈ b H | d ξ ( ψ ( h i ) , φ ( h i )) < ǫ for all indices i } , where ǫ > , φ ∈ b H and { h , ..., h n } is a finite subset (hence compact) of H . It is easy to show thatthis indeed defines a basis for a topology τ cc in b H . Proposition 5.7.
The topological group ( b H, τ cc ) is a totally disconnected, compact and metrizablegroup.Proof. We have an isomorphism of topological groups b H ∼ = b H C induced by Υ , where b H C is the usualPontryagin dual of H , and it is well-known that b H C has the stated properties. (cid:3) Suppose now that Z acts on H by automorphisms via ρ : Z y H . We write G for the semi-directproduct group H ⋊ ρ Z , so G is generated by t and by any set S consisting of generators of H .We denote by e ρ : Z y KH the action on the group algebra KH extending ρ by linearity, so that KG ∼ = KH ⋊ e ρ Z .The action ρ : Z y H induces another action b ρ : Z y b H by homeomorphisms, defined in the usualway: b ρ n ( φ ) := φ ◦ ρ − n for n ∈ Z and φ ∈ b H. If we write T := b ρ then the action b ρ is generated by T , in the sense that b ρ n ( φ ) = T n ( φ ) for φ ∈ b H and n ∈ Z . This crossed product construction can be generalized by replacing Z with any other countable discrete group Λ , as in[7, Section 2]. However, we will stick into the case Λ = Z for our purposes. bserve that T : b H → b H defines a homeomorphism of the totally disconnected, compact metrizablegroup b H . In the next proposition we establish the relationship between the group algebra KG andthe Z -crossed product C K ( b H ) ⋊ T Z by means of the well-known Fourier transform. Proposition 5.8.
Assume the previous hypothesis and caveats; that is, with O = { n ∈ N | there exists an element g ∈ H of order n } , assume that the characteristic of K is coprime with all n ∈ O , and that, for any n ∈ O , K containsall the n th roots of 1. Let X = { ξ n } n ∈ O be a compatible family of primitive roots of unity in K (seeDefinition 5.2 and Lemma 5.3).Then we can identify the group algebra KG ∼ = KH ⋊ e ρ Z with C K ( b H ) ⋊ T Z via the Fourier transform F : KH → C K ( b H ) , by sending an element h ∈ H of order n to the element n − X j =0 ξ − jn χ U h,j , where U h,j = { φ ∈ b H | φ ( h ) = ξ jn } and χ U h,j denotes the characteristic function of the clopen U h,j ,and then extending it to a map F : KG → C K ( b H ) ⋊ T Z by sending the generator t of Z on KG tothe generator δ t of Z on the Z -crossed product.If moreover K is endowed with an involution − satisfying the compatibility condition ξ n = ξ − n forall n ∈ O , then the Fourier transform preserves the involutions on both KG and the Z -crossed product. See also [7] and [18], where the authors state analogous results. For a proof of Proposition 5.8, see[10]. Recall that the involution on KG is defined by the rule ( λg ) ∗ = λg − for λ ∈ K, g ∈ G andextended by linearity, and the involution on the Z -crossed product is given by ( f · δ t ) ∗ = ( f ∗ ◦ T − ) · δ t − , with f ∗ ( φ ) = f ( φ ) for f ∈ C K ( b H ) and φ ∈ b H , and again extended by linearity. Remark 5.9.
The final condition in Proposition 5.8 that there is an involution on K such that ξ n = ξ − n for all n ∈ O is a non-trivial one when the field K has characteristic p > . When O = { , } ,any involution –in particular the identity involution– works for any field K of characteristic p = 2 (in order to have that p is coprime to ). If O is the set of divisors of p n + 1 for some prime p andsome n ≥ , then one can take the field F p n of p n elements, with the involution ϕ n , where ϕ is theFrobenius automorphism of F p n . If O is infinite, then there is no field of characteristic p > with aninvolution with the property that ξ n = ξ − n for all n ∈ O .Let now K ⊆ C be a subfield of C closed under complex conjugation, which will be the involutionon K . Recall that rk KG denotes the canonical rank function on KG given by the restriction of therank function naturally arising from U ( G ) (see Section 2.3). Our question now is whether we can finda measure b µ on the space b H such that, when applying the construction explained in Theorem 3.1, weend up with a rank function rk A on A = C K ( b H ) ⋊ T Z that coincides with rk KG under the Fouriertransform F . The answer to this question is affirmative in the case rk KG is extremal, and in fact b µ coincides with the normalized Haar measure on b H , as we show in the next proposition. Proposition 5.10.
Let K ⊆ C be a subfield closed under complex conjugation and containing all the n th roots of , for n ∈ O . Then from rk KG we can construct a full T -invariant probability measure b µ on b H , which coincides with the normalized Haar measure on b H .If moreover rk KG is extremal in P ( KG ) then b µ is ergodic, and when applying the construction fromTheorem 3.1 to b µ we end up with a Sylvester matrix rank function rk A on A = C K ( b H ) ⋊ T Z suchthat rk KG = rk A ◦ F .Proof. We first define a finitely additive probability measure µ KG on the algebra K of clopen subsetsof b H by the rule µ KG ( U ) = rk KG ( F − ( χ U )) for every clopen subset U of b H which, by the sameargument as in the proof of [3, Proposition 4.8], can be uniquely extended to a Borel probability easure µ KG on b H . Invariance of µ KG follows from the fact that t is an invertible element, and since rk KG is a faithful rank function it follows that µ KG is full. If moreover rk KG is extremal, then againan argument similar to the one given in the proof of [3, Proposition 4.10] proves that µ KG is ergodic.Now Theorem 3.1 implies that rk KG = rk A ◦ F , as required.Finally, to prove that µ KG coincides with the normalized Haar measure b µ on b H , just note that F − ( χ U ) is a projection in KG for any clopen U ⊆ b H , so its rank coincides with its trace and weobtain µ KG ( U ) = tr KG ( F − ( χ U )) = F − ( χ U )( e ) = Z b H χ U ( φ ) φ ( e ) d b µ ( φ ) = b µ ( U ) . (cid:3) Remark 5.11.
An important observation is that, once we have proven that the Haar measure b µ on b H is T -invariant, this property does not depend on the base field K anymore. So, by assuming nowthat K is any field of arbitrary characteristic p (with p not dividing any natural number n ∈ O )and containing all the n th roots of unity for any n ∈ O , and by assuming ergodicity of b µ , we caninvoke Theorem 3.1 to obtain a ’canonical’ Sylvester matrix rank function on KG , by simply defining rk KG := rk A ◦ F .We can use Proposition 5.10 to prove that the ∗ -regular closure of the group algebra KG inside U ( G ) , which we denoted by R KG , can be identified with R A , the ∗ -regular closure of A inside therank-completion R rk of A with respect to its rank function rk A (recall Theorem 3.1). Theorem 5.12.
Consider the same notation and hypotheses as in Proposition 5.10, and assumethat rk KG is extremal in P ( KG ) . We then obtain a ∗ -isomorphism R KG ∼ = R A . In fact, we havecommutativity of the diagram A ∼ = (cid:15) (cid:15) (cid:31) (cid:127) / / R A ∼ = (cid:15) (cid:15) (cid:31) (cid:127) / / R rk (cid:127) _ (cid:15) (cid:15) KG (cid:31) (cid:127) / / R KG (cid:31) (cid:127) / / U ( G ) . Moreover, the rank-completions of both KG and R KG are ∗ -isomorphic to M K , the von Neumanncontinuous factor over K .Proof. Since U ( G ) is complete with respect to the rk U ( G ) -metric, Proposition 5.10 together withTheorem 3.1 tell us that R rk embeds in U ( G ) , making the previous diagram commutative. In turn,since R rk is itself ∗ -regular, we see that R KG = R ( KG, U ( G )) ∼ = R ( A , R rk ) = R A as ∗ -algebras, as required. The last part follows from Proposition 4.2. (cid:3) The lamplighter group algebra
In this section we apply the constructions given in Sections 3 and 4 to study the lamplightergroup algebra. This algebra is of great relevance because, among other things, it gave the firstcounterexample to the Strong Atiyah Conjecture, see for example [20], [12] and the Introduction.
Definition 6.1.
The lamplighter group Γ is the wreath product of the finite group Z of two elementsby Z . In other words, Γ = Z ≀ Z = (cid:16) M i ∈ Z Z (cid:17) ⋊ σ Z where the semidirect product is taken with respect to the Bernoulli shift σ : Z y L i ∈ Z Z defined by σ n ( x ) i = x i + n for x = ( x i ) ∈ M i ∈ Z Z . In terms of generators and relations, if we denote by t the generator corresponding to Z , and by a i the generator corresponding to the i th copy of Z , we have the presentation Γ = h t, { a i } i ∈ Z | a i , a i a j a i a j , ta i t − a i − for i, j ∈ Z i . ow the Fourier transform (if K is any field with involution of characteristic different from ) givesa ∗ -isomorphism K Γ ∼ = C K ( X ) ⋊ T Z , where X = { , } Z is the Cantor set and T is the shift map,namely T ( x ) i = x i +1 for x ∈ X . The isomorphism is given by the identifications χ X , t t, a i χ U i − χ X \ U i where U i = { x ∈ X | x i = 0 } . Note that, in particular, the elements e i = a i are idempotents in K Γ ,and so are f i = 1 − e i . They correspond to the characteristic functions of the clopen sets consistingof all the elements in X having a (resp. a ) at the i th component, respectively.6.1. The approximating algebras A n for the lamplighter group algebra. We now give aconcrete family of pairs { ( E n , P n ) } n ≥ as in Section 3, together with the corresponding algebras A n = A ( E n , P n ) for the algebra A = C K ( X ) ⋊ T Z corresponding to the lamplighter group.We will follow the same notation as in [18, Section 3]: given ǫ − k , ..., ǫ l ∈ { , } , the cylinder set { x = ( x i ) ∈ X | x − k = ǫ − k , ..., x l = ǫ l } will be denoted by [ ǫ − k · · · ǫ · · · ǫ l ] . It is then clear that abasis for the topology of X is given by the collection of clopen sets consisting of all the cylinder sets.We have a natural measure µ on X given by the usual product measure, where we take the (cid:0) , (cid:1) -measure on each component { , } . It is well-known (see [26, Example 3.1]) that µ is an ergodic, fulland shift-invariant probability measure on X . In fact, rk K Γ ( F − ( χ [ ǫ − k ··· ǫ ··· ǫ l ] )) = 12 l + k +1 = µ ([ ǫ − k · · · ǫ · · · ǫ l ]) . It follows from Theorem 3.1 that rk K Γ ◦ F − coincides with rk A , where A = C K ( X ) ⋊ T Z . Inparticular, the set of ℓ -Betti numbers arising from Γ with coefficients in K can be also computed bymeans of rk A : C (Γ , K ) = [ k ≥ { k − rk A ( A ) | A ∈ M k ( A ) } , which is closely related to C ( A ) = rk A (cid:16) S k ≥ M k ( A ) (cid:17) . In fact, they are both subsemigroups of ( R + , +) which generate the same subgroup G (Γ , K ) = G ( A ) , see Subsection 2.3.Let us start our analysis of K Γ ∼ = C K ( X ) ⋊ T Z using the dynamical approximation from Section3. For n ≥ , we take E n = [1 · · · · · · (with n + 1 one’s) for the sequence of clopen sets, whoseintersection gives the point y = ( ..., , , , ... ) ∈ X which is a fixed point for the shift map T . Wetake the partitions P n of the complements X \ E n to be the obvious ones, namely P n = { [00 · · · · · · , [00 · · · · · · , ..., [01 · · · · · · } . Write A n := A ( E n , P n ) for the unital ∗ -subalgebra of A = C K ( X ) ⋊ T Z generated by the partialisometries χ Z t , Z ∈ P n . It is easily seen that A n coincides with the unital ∗ -subalgebra of A generatedby the partial isometries s i = e i t for − n ≤ i ≤ n , where recall that each e i is the projection in K Γ given by a i (equivalently, the characteristic function of the clopen set [0 i ] ), and we put f i = 1 − e i .We have, for each n ≥ , inclusions A n ⊆ A n +1 . The quasi-partition P n consists of the translates ofthe sets W ∈ V n of the following types:a) W = [11 · · · · · · of length (there are n + 2 one’s);b) W = [11 · · · · · · · · · · · · of length n + 2 (there are n + 2 one’s, and a zero);c) W ( ∗ , ∗ , ..., ∗ , ∗ ) = [11 · · · · · · ∗ ∗ · · · ∗ ∗ · · · · · · of length (2 n + 3) + l ,where in the last type l ≥ is the number of ∗ , and each ∗ can be either a zero or a one, but with atmost n consecutive one’s. It can be checked by hand that indeed P n forms a quasi-partition of X ,namely that X W ∈ V n | W | µ ( W ) = 1 . To this aim, we first need a definition. We write m = 2 n + 1 . It can also be done by taking an even number of one’s at each level n ; we are taking an odd number for notationalconvenience. efinition 6.2. For k ∈ Z + we define the k th m -acci number, denoted by Fib m ( k ) , recursively bysetting Fib m (0) = 0 , Fib m (1) = Fib m (2) = 1 , Fib m (3) = 2 , . . . , Fib m ( m −
1) = 2 m − , and for r ∈ Z + , Fib m ( r + m ) = Fib m ( r + m −
1) + · · · + Fib m ( r ) . This sequence is also known in the literature as the m -step Fibonacci sequence, see for example [15]and [33]. Lemma 6.3.
For k ≥ , Fib m ( k ) is exactly the number of possible sequences ( ǫ , ..., ǫ l ) of length l = k − that one can construct with zeroes and ones, but having at most m − consecutive one’s.Proof. A simple combinatorial argument yields the result. (cid:3)
By using the summation rules X k ≥ Fib m ( k )2 k = 2 m − , X k ≥ k Fib m ( k )2 k = 2 m − ( m + 1)2 m − , whose proofs can be found in [10, Lemma 3.2.3], we compute X W ∈ V n | W | µ ( W ) = 12 m +1 + 12 m (cid:16) X k ≥ m Fib m ( k )2 k + X k ≥ k Fib m ( k )2 k (cid:17) = m + 12 m +1 + (cid:16) − m + 12 m +1 (cid:17) = 1 , as we already know.Recall from Section 3 that we have faithful ∗ -representations π n : A n ֒ → R n , x ( h W · x ) W . Inour situation, we have the concrete expression R n = K × Q k ≥ M m + k ( K ) Fib m ( k ) .If now K is a subfield of C closed under complex conjugation, then by Theorem 5.12 we can identify R K Γ ∼ = R A , and in fact the ∗ -regular closure of each A n inside U (Γ) coincides with R n = R ( A n , R n ) ,and the same for R ∞ = R ( A ∞ , R ∞ ) . In particular, Proposition 4.2 applies to give the following result,already proved by Elek in [13] for K = C . Corollary 6.4.
Let K be a subfield of C closed under complex conjugation, and let R rk to be therank-completion of R K Γ inside U (Γ) with respect to rk U (Γ) . Then R rk ∼ = M K as ∗ -algebras over K ,where M K denotes the von Neumann continuous factor over K . The algebra of special terms for the lamplighter group algebra.
We now interpret theresults in Subsection 4.2 for the lamplighter group algebra. In particular, we show that the corre-sponding algebra of special terms S n [[ t ; T ]] is an integral domain. Our notation here is a little bitdifferent from that section: we write A n, [[ t ; T ]] instead of B [[ t, T ]] to denote the set of infinite sums X i ≥ b i ( χ X \ E n t ) i = X i ≥ b i t i , where b i ∈ A n,i := χ X \ ( E n ∪···∪ T i − ( E n )) A n, with A n, = C K ( X ) ∩ A n . We then have a representation π n : A n, [[ t ; T ]] → R n , π n ( a ) = ( h W · a ) W .Following Definition 4.14, we denote by ( D n ) + the division closure of ( A n ) + = L i ≥ A n,i t i in A n, [[ t ; T ]] . We write S n [[ t ; T ]] to denote the subspace of A n, [[ t ; T ]] consisting of those elements P i ≥ b i t i such that each b i belongs to span { χ S | S ∈ W i } (see Definition 4.17). These are easy todescribe here: noting that E n ∩ T − ( E n ) = ∅ , we have that the special term of degree is given by S = T − ( S ) = [1 · · · · · · | {z } n ] , which corresponds to χ S = f − n +1 · · · f · · · f n − f n ; the special one of degree i = 2 n + 1 is S = [11 · · · · · · | {z } n · · · · · · | {z } n ] , orresponding to χ S t n +1 = f − n f − n +1 · · · f − n · · · f − n − e − n f − n +1 · · · f · · · f n − f n t n +1 ; finally, withdegree i ≥ n + 2 , we have the elements S = [11 · · · · · · | {z } n ∗ ∗ · · · ∗ ∗ | {z } i − (2 n +2) · · · · · · | {z } n ] corresponding to χ S t i = f − n − i +1 f − n − i +2 · · · f − i +1 · · · f − i + n e − i + n +1 ( ∗ ) − i + n +2 · · · ( ∗ ) − n − e − n f − n +1 · · · f · · · f n − f n t i , with ( ∗ ) j ∈ { e j , f j } having no more than n consecutive f j ’s. The next lemma shows that thelamplighter group algebra has some special properties that are reflected in S n [[ t ; T ]] . Lemma 6.5.
The space S n [[ t ; T ]] becomes a subalgebra of A n, [[ t ; T ]] , and even an integral domain.Proof. We show that if S ∈ W i , S ′ ∈ W j then S ∩ T i ( S ′ ) ∈ W i + j (here i, j ≥ n + 2 , the other casescan be also checked in a similar way). We have χ S = f − n − i +1 · · · f − i +1 · · · f − i + n e − i + n +1 a − i + n +2 · · · a − n − e − n f − n +1 · · · f · · · f n ,χ S ′ = f − n − j +1 · · · f − j +1 · · · f − j + n e − j + n +1 b − j + n +2 · · · b − n − e − n f − n +1 · · · f · · · f n , with a k , b k ∈ { e k , f k } with no more than n consecutive f k ’s, so that χ S t i · χ S ′ t j = χ S ∩ T i ( S ′ ) t i + j = f − n − j − i +1 · · · f − j − i +1 · · · f − j − i + n e − j − i + n +1 b − j − i + n +2 · · · b − n − i − e − n − i · f − n − i +1 · · · f − i f − i +1 · · · f − i + n e − i + n +1 a − i + n +2 · · · a − n − e − n f − n +1 · · · f · · · f n t i + j . Now it is clear that S ∩ T i ( S ′ ) ∈ W i + j . This shows that S n [[ t ; T ]] is a subalgebra of A n, [[ t ; T ]] . Toshow that S n [[ t ; T ]] is a domain, consider two non-zero elements a, b ∈ S n [[ t ; T ]] , and let χ S t i and χ S ′ t j be terms in the support of a and b respectively, of smallest degree. By the computation above χ S t i · χ S ′ t j = χ S ∩ T i ( S ′ ) t i + j is a non-zero term of smallest degree in ab . This shows that ab = 0 . Notethat the special term χ S ∪ T − ( S ) = χ S = χ T − ( S ) is the unit of the algebra S n [[ t ; T ]] . (cid:3) Define S n [ t ; T ] ⊆ S n [[ t ; T ]] as the set of elements of S n [[ t ; T ]] with finite support, i.e. of the form P ri =0 b i t i with b i belonging to the linear span of the special elements of degree i , and r a positiveinteger. Proposition 6.6.
For n ≥ , S n [ t ; T ] is a free K -algebra with infinitely many generators, and S n [[ t ; T ]] is a free power series K -algebra with infinitely many generators.Proof. We say that a special term χ S t i of the form f − n − i +1 · · · f − i +1 · · · f − i + n e − i + n +1 a − i + n +2 · · · a − n − e − n f − n +1 · · · f · · · f n t i is pure if there are no more than n − consecutive f j ’s in the a − i + n +2 · · · a − n − sequence. Denoteby P u the set of pure elements. Then every special term χ S t i can be written uniquely as a productof pure terms, so we obtain an isomorphism K hh x b | b ∈ P u ii ∼ = S n [[ t ; T ]] , x b b, χ [1 ··· ··· which restricts to an isomorphism K h x b | b ∈ P u i ∼ = S n [ t ; T ] . (cid:3) In the next subsection we provide the description of S n [[ t ; T ]] for the case n = 0 .We now observe that, for n ≥ , the ∗ -regular algebra Q n and so the ∗ -algebra E n , correspondingto the algebra A n (described in Definition 4.25), contain a well-known large ∗ -subalgebra.With the notation used in the proof of Proposition 6.6, denote by K rat h x b | b ∈ P u i the algebraof non-commutative rational series, which is by definition the division closure of K h x b | b ∈ P u i in K hh x b | b ∈ P u ii , see [9]. Note that K rat h x b | b ∈ P u i = [ F K rat h x b | b ∈ F i , where F ranges over all the finite subsets of P u . We see K rat h x b | b ∈ P u i as a subalgebra of S n [[ t ; T ]] via the identification K hh x b | b ∈ P u ii ∼ = S n [[ t ; T ]] provided by Proposition 6.6. By [9, Theorems 1.5.5and 1.7.1], the algebra K rat h x b | b ∈ P u i is even a ∗ -subalgebra of ( S n [[ t ; T ]] , ⊙ , − ) . We will denote he algebra of rational series endowed with the Hadamard product ⊙ by K rat h x b | b ∈ P u i ◦ . Notethat, since P u is infinite, the algebra K rat h x b | b ∈ P u i ◦ is not unital, but it has a local unit, namelythe family { (1 − P b ∈ F x b ) − } F , where F ranges over all the finite subsets of P u . Proposition 6.7.
Let n ≥ and let Q n be the ∗ -regular closure of P (( D n ) + ) in ( S n [[ t ; T ]] , ⊙ , − ) .Then Q n contains the ∗ -regular closure of K rat h x b | b ∈ P u i ◦ in ( S n [[ t ; T ]] , ⊙ , − ) .Proof. Using the identifications given in the proof of Proposition 6.6, we only need to check that thedivision closure D of S n [ t ; T ] in S n [[ t ; T ]] is contained in P (( D n ) + ) .Let us denote by g the unit of S n [[ t ; T ]] , that is g = χ S ∩ T − ( S ) = χ S = χ T − ( S ) . Then thedivision closure of (1 − g ) K + S n [ t ; T ] in (1 − g ) K + S n [[ t ; T ]] is precisely (1 − g ) K + D . Moreover wehave inclusions of unital algebras (1 − g ) K + S n [ t ; T ] ⊆ ( D n ) + ∩ ((1 − g ) K + S n [[ t ; T ]]) ⊆ (1 − g ) K + S n [[ t ; T ]] , and ( D n ) + ∩ ((1 − g ) K + S n [[ t ; T ]]) is inversion closed in (1 − g ) K + S n [[ t ; T ]] , so (1 − g ) K + D ⊆ ( D n ) + ∩ ((1 − g ) K + S n [[ t ; T ]]) ⊆ ( D n ) + . We thus get D = P ( D ) ⊆ P (( D n ) + ) , as desired. (cid:3) To close this subsection, we compute the ∗ -regular closure of K rat h X i ◦ in K hh X ii ◦ . To this end,we first analyze the ∗ -regular closure in the setting of commutative rings.For any unital ring T , we denote by B ( T ) the Boolean algebra of central idempotents of T . Recallthat e ∧ f = ef and e ∨ f = e + f − ef for e, f ∈ B ( T ) .Let R be a commutative unital ∗ -regular ring and let S be a unital ∗ -subring of R . Observe that theidempotents of R are necessarily self-adjoint. We want to obtain a simplified form of the constructionof the ∗ -regular closure of S in R .Write(6.1) E ( S ) = { e ∈ B ( R ) | there exists a ∈ S such that aR = eR } . We can think of the elements of E ( S ) as being the supports of the elements of S . Note that B ( S ) ⊆ E ( S ) ⊆ B ( R ) and that for a ∈ S and e ∈ E ( S ) such that aR = eR , we have (1 − e ) R = Ann R ( a ) ,the annihilator of a in R . Lemma 6.8.
Let R be a commutative unital ∗ -regular ring and let S be a unital ∗ -subring of R . Theset E ( S ) is closed under meets in B ( R ) . Consequently, E ( S ) is a Boolean subalgebra of B ( R ) if andonly if E ( S ) is closed under complements.Proof. Let a, b ∈ S , and suppose that aR = eR and bR = f R for e, f ∈ B ( R ) . We then have: ( e ∧ f ) R = ef R = ( eR )( f R ) = ( aR )( bR ) = ( ab ) R, which shows that e ∧ f ∈ E ( S ) . The second part follows from the De Morgan laws. (cid:3) We can obtain now the description of the ∗ -regular closure. Proposition 6.9.
Let R be a unital commutative ∗ -regular ring and let S be a unital ∗ -subring of R . Let E ( S ) ⊆ B ( R ) be the set defined in (6.1) , and let B S be the smallest Boolean subalgebra of B ( R ) containing E ( S ) . Let S B S be the subring of R generated by S and B S . Then the ∗ -regularclosure R of S in R coincides with the classical ring of quotients Q cl ( S B S ) of S B S . Moreover wehave B ( R ) = B S .Proof. We first show that Q cl ( S B S ) naturally embeds in R . Observe that every element x ∈ S B S can be written in the form x = P ni =1 e i s i , where s i ∈ S for all i and ( e i ) is a sequence of pairwiseorthogonal elements of B S . Now let f i ∈ E ( S ) such that s i R = f i R . We have ( e i s i ) R = ( e i f i ) R , and h i := e i f i = e i ∧ f i ∈ B S , because B S is a Boolean subalgebra of B ( R ) . Hence, by replacing each e i by h i , we can further assume that ( e i s i ) R = e i R . It follows that the support projection of x in R isexactly P ni =1 e i = ∨ ni =1 e i ∈ B S . So we have shown that the support projection in R of any elementof S B S belongs to B S . et z be a non-zero-divisor of S B S , and write zR = eR for some idempotent e ∈ R . By what weproved above we have e ∈ B S . Since (1 − e ) z = 0 , we conclude that e = 1 . Hence z is invertible in R and therefore there is a unique embedding of Q cl ( S B S ) into R extending the canonical embedding S ֒ → R .Let R denote the ∗ -regular closure of S in R . We first show that Q cl ( S B S ) ⊆ R . It is clear that E ( S ) ⊆ R , so that B S ⊆ B ( R ) and S B S ⊆ R . Now since all non-zero-divisors of S B S are invertiblein R , we get that Q cl ( S B S ) ⊆ R .For the other inclusion R ⊆ Q cl ( S B S ) , it suffices to show that Q cl ( S B S ) is a ∗ -regular subring of R . First notice that since S is a ∗ -subring of R , S B S is a ∗ -subring of R too. It follows that Q cl ( S B S ) is a ∗ -subring of R . Now let x = ab − ∈ Q cl ( S B S ) , where a ∈ S B S , and b a non-zero-divisor in S B S . We proved above that the support projection, say e , of a belongs to B S . Hence a + (1 − e ) is a non-zero-divisor in S B S , so that it is invertible in Q cl ( S B S ) . Now it is easily checked that thequasi-inverse of x is eb ( a + (1 − e )) − ∈ Q cl ( S B S ) , so that Q cl ( S B S ) is a ∗ -regular ring. (cid:3) Let X be an infinite countable set and write X = S n ≥ X n , where X n ⊆ X n +1 and | X n | = n for all n . Let R n rat be the ∗ -regular closure of K rat h X n i ◦ in K hh X n ii ◦ , and let R rat be the ∗ -regular closureof K rat h X i ◦ in K hh X ii ◦ . Then we have R rat = (cid:16) [ n ≥ R n rat (cid:17) + 1 · K. Therefore, it is enough to compute the ∗ -algebras R n rat .For a finite set X , denote by X ∗ the free monoid generated by X . For a = P w ∈ X ∗ a w w ∈ K hh X ii ,denote by supp ( a ) the support of a , that is, the set of all w ∈ X ∗ such that a w = 0 . The annihilator ann ( a ) is defined as the complement of supp ( a ) in X ∗ . The Boolean algebra B ( K hh X ii ◦ ) is isomorphicto the power set P ( X ∗ ) of X ∗ . A subset of X ∗ is usually called a language (see [9, p. 4]).The following proposition is the key to obtain some new irrational ℓ -Betti numbers arising fromthe lamplighter group, see [4, Subsection 4.3]. Proposition 6.10.
Let K be a field with involution, and let X be a finite non-empty set. Let K bethe set of all the supports of elements from K rat h X i , and let B be the Boolean subalgebra of P ( X ∗ ) generated by K . Then the ∗ -regular closure of K rat h X i ◦ in K hh X ii ◦ is the ∗ -algebra of all formalpower series whose support belongs to B .Proof. Note that the set K corresponds exactly with the set E ( S ) described in Proposition 6.9. Hencethe result follows from that proposition. (cid:3) Remark 6.11.
By [9, Section 3.4], if K is a field of characteristic zero, and | X | > , then the set K of Proposition 6.10 is not closed under complementation, and so it is not a Boolean subalgebra ofsubsets of X ∗ . It follows that B is strictly larger than K , and in particular properly contains theBoolean algebra of all the rational languages over X , see [9, Chapter 3].6.3. Analysis of the algebra A . Here we analyze the example in [6, Section 6] in the light of thetheory developed in the present paper. The algebra constructed there coincides with the algebra A ,the first of the approximating algebras considered in the preceding two sections. In [6], a concretedescription of the ∗ -regular closure of the ∗ -algebra A is obtained, and it is shown that G ( A ) = Q .Recall that G ( A ) denotes the subgroup of R generated by the set of ℓ -Betti numbers C ( A ) arisingfrom A . Note that A coincides with the semigroup algebra K F of the monogenic free inversemonoid F (see [6]).We will denote by D + and D − the subalgebras of A , [[ t ; T ]] and A , [[ t − ; T − ]] introduced inDefinition 4.14.Now recall from [6] that the algebra Σ − A embeds naturally in R = Q i ≥ M i ( K ) . Here Σ is theset of all the polynomials of the form f ( s ) for f ( x ) ∈ K [ x ] with f (0) = 1 , and s = χ X \ E t = χ [0] t .Denoting by B the image of Σ − A in R , the ∗ -subalgebra of R generated by B + B ∗ will bedenoted by T . This algebra was considered in [6] with the notation D (see [6, Proposition 6.8]).We will show below that indeed T = D , where, as in Section 4, D denotes the ∗ -subalgebra of R enerated by π + ( D + ) + π − ( D − ) = π + ( D + ) + π + ( D + ) ∗ . Thus in the end the algebras denoted by D in each of the two papers do agree. Proposition 6.12.
With the above notation, we have T = D .Proof. Note that for any f ( s ) ∈ Σ , we clearly have that f ( s ) − ∈ D + . Since obviously A ⊆ D , weget that B = Σ − A ⊆ D , and since D is a ∗ -algebra we get T ⊆ D .We may consider the algebra A , (( t ; T )) consisting of skew Laurent power series P i ≥− n b i t i , where n ∈ Z + and b i ∈ B i for all i . Observe that B ⊆ A , (( t ; T )) . Set B + := B ∩ A , [[ t ; T ]] . We aim toshow that π + ( D + ) ⊆ B + . This would imply that D ⊆ T , and thus we would get the desired equality D = T .Consider the natural onto homomorphism ρ : B → K ( x ) sending s to x and s ∗ to x − . The kernelof ρ is the ideal of B generated by − ss ∗ and − s ∗ s . We denote by ρ + the restriction of the map ρ to B + .To show that π + ( D + ) ⊆ B + , it is enough to prove that B + is closed under inverses in A , [[ t ; T ]] ,because D + is the division closure of A , [ t ; T ] in A , [[ t ; T ]] .Observe that ρ restricts to a surjective homomorphism ρ : A , → K , which can be described asfollows. Given any expression z = P U ∈ P λ U χ U , where P is a partition of X into clopen basic subsetsas in [3, Lemma 3.8], there is a unique U ∈ P such that χ U does not belong to the kernel of ρ , namely U = [0 i · · · for some i . Then we have ρ ( z ) = λ U = z (( ..., , , , ... )) . Note that ρ ( T j ( z )) = ρ ( z ) for all j ∈ Z . This enables us to extend ρ to a well-defined homomorphism A , (( t ; T )) → K (( x )) ,also denoted by ρ , which is given by ρ ( ∞ X i = − n a i t i ) = ∞ X i = − n ρ ( a i ) x i . We have the following commutative diagram B + / / ρ + (cid:15) (cid:15) A , [[ t ; T ]] ρ + (cid:15) (cid:15) K [ x ] ( x ) / / K [[ x ]] where ρ + from the right-hand side is the restriction of ρ : A , (( t ; T )) → K (( x )) to A , [[ t ; T ]] . Theimage of the map K [ x ] ( x ) → K [[ x ]] is of course the algebra of rational series. Now let z = P i ≥ b i t i ∈ B + be invertible in A , [[ t ; T ]] . By Lemma 4.13, we can assume without loss of generality that b = 1 .Then ρ + ( z ) must be invertible in K [ x ] ( x ) , so that there are f ( x ) , g ( x ) ∈ K [ x ] such that f (0) = g (0) = 1 and ρ + ( z ) = f ( x ) g ( x ) − . Now f ( s ) g ( s ) − ∈ B + and we have z = f ( s ) g ( s ) − + y, where y belongs to the ideal I > := I ∩ (cid:0) L i ≥ A ,i t i (cid:1) , where I is the ideal of B generated by − ss ∗ and − s ∗ s . Therefore we have that zg ( s ) f ( s ) − = 1 + y ′ , where y ′ ∈ I > , and we need to show that y ′ is invertible. Indeed, we will show that I > is a nil-ideal, that is, that every element of I > isnilpotent.By [6, Lemma 4.7], each element of I can be expressed as a (finite) linear combination of terms ofthe following forms:(A) f − s i (1 − ss ∗ )( s ∗ ) j , for i, j ≥ and f ∈ Σ ,(B) ( s ∗ ) i (1 − s ∗ s ) s j f − , for i, j ≥ and f ∈ Σ ,(C) ( s ∗ ) i (1 − s ∗ s ) s j f − (1 − ss ∗ )( s ∗ ) k , for i, j, k ≥ and f ∈ Σ ,(D) elements from soc( A ) .Let y be an element in I > . Let { h n } n be the canonical central projections of A . Since the constantterm of y is , we see that each matrix h n · y must be a strictly lower triangular matrix. We want toshow that there is a fixed integer R such that ( h n · y ) R = 0 for all n . Now taking into account theforms (A), (B),(C), (D) above, we see that there are positive integers N and r such that for all n ≥ N he matrix h n · y has the property that it is strictly lower triangular and that ( h n · y ) ij = 0 for all ( i, j ) such that i ≤ n − r and j ≥ r + 1 . The result then follows from the next straightforward lemma. (cid:3) Lemma 6.13.
Let < r < n . Suppose that A = ( a ij ) ≤ i,j ≤ n is a strictly lower triangular matrix suchthat a ij = 0 whenever ( i, j ) satisfies that i ≤ n − r and j ≥ r + 1 . Then A r +1 = 0 .Proof. For ≤ t ≤ r + 1 , we show by induction that A t = ( c ij ) , with c ij = 0 whenever i ≤ j + t − and whenever ( i, j ) satisfies that i ≤ n − r + t − and j ≥ r + 1 , and also whenever ( i, j ) satisfies that i ≤ n − r and j ≥ r − t + 2 . Moreover if t ≥ then c n − r +1 ,r +3 − t ′ = c n − r +2 ,r +4 − t ′ = · · · = c n − r + t ′ − ,r = 0 for ≤ t ′ ≤ t .Assume the result holds for t ≤ r . We will show it holds for t + 1 . Write D := A t +1 . It is asimple matter to show that d ij = 0 whenever i ≤ j + t . Now assume that the pair ( i, j ) satisfies that i ≤ n − r + t and that j ≥ r + 1 . Suppose that there is a non-zero term c ik a kj contributing to theterm d ij . Then i > k + t − and k > j . Therefore we have k + t − < i ≤ n − r + t, which implies that k − < n − r so that k ≤ n − r . But now since k ≤ n − r and j ≥ r + 1 we havethat a kj = 0 by hypothesis. So all the products c ik a kj are and we get that d ij = 0 .Now assume that ( i, j ) satisfies that i ≤ n − r and j ≥ r − t + 1 . Proceeding as above the term c ik a kj is either or i > k + t − and k > j . We get k > j ≥ r − t + 1 and so k ≥ r − t + 2 . Since i ≤ n − r and k ≥ r − t + 2 , we get that c ik = 0 by the induction hypothesis, and so c ik a kj = 0 .Therefore d ij = 0 .It remains to show the last statement. Suppose first that t = 3 . We have to show that d n − r +1 ,r = 0 .Assume there is a term c n − r +1 ,k a k,r which is non-zero, where here A = ( c ij ) . Then we must have n − r + 1 > k + 1 and k > r and thus k ≥ r + 1 so that c n − r +1 ,k = 0 by what we have proved before,and we get a contradiction. Hence d n − r +1 ,r = 0 . Using induction, we assume the result true for all ≤ t ′ ≤ t ≤ r and we show it for t + 1 . Setting A t +1 = ( d ij ) , we need to show that d n − r + s,r + s − t +1 = 0 for ≤ s ≤ t − (this is the case t ′ = t + 1 of the statement, the cases where ≤ t ′ ≤ t follow the samepattern). Write A t = ( c ij ) and let s be an integer such that ≤ s ≤ t − . Let c n − r + s,k a k,r + s − t +1 benon-zero. By the induction hypothesis and what we proved before, we known that c n − r + s,j = 0 for j ≥ r + s − t + 2 . Therefore we get that k < r + s − t + 2 . We also have that k > r + s − t + 1 , since a k,r + s − t +1 = 0 , so k ≥ r + s − t + 2 , and we get a contradiction.Therefore we have proven that A r +1 is a matrix consisting of a r × r lower diagonal matrix at theleft lower corner, and the rest of the entries are . It follows that A r +1 = 0 . (cid:3) We now proceed to describe the special elements S [[ t ; T ]] at this level. There is exactly one specialterm for each degree i ≥ . For i = 0 , it is given by S = T − ( S ) = X and the correspondingelement inside the algebra is χ X = 1 . For i ≥ we get the element S = [0 i · · · , correspondingto χ S t i = χ [0 i ··· t i . Note that s i = χ [0 i ··· t i for i > , where s = χ X \ E t , and so S [[ t ; T ]] can beisomorphically identified with the algebra of formal power series K [[ x ]] , the isomorphism sending s x .The isomorphism given in Proposition 4.20 coincides exactly with the isomorphism ψ given in [6,paragraph preceding Proposition 6.8], and formulas (6.3) and (6.4) from [6] are deduced from Lemma4.21.As in [6], we denote by R ◦ the subalgebra of rational series in K [[ x ]] , endowed with the Hadamardproduct ⊙ . It is not closed under inversion in ( K [[ x ]] , ⊙ ) . Let Q be the classical ring of quotients of R ◦ in ( K [[ x ]] , ⊙ ) , which coincides with its ∗ -regular closure R ( R ◦ , K [[ x ]]) (see [6, Lemma 6.10], whereit is denoted by Q ). The algebra Q is not to be confused with our algebra Q which, by Definition4.25, is the ∗ -regular closure of P ( D + ) inside ( K [[ x ]] , ⊙ ) . Lemma 6.14.
We have Q ⊆ Q . roof. Once we show the inclusion R ◦ ⊆ P ( D + ) we will be done since Q , being the ∗ -regular closureof P ( D + ) in ( K [[ s ]] , ⊙ ) , will contain Q .So let p ( s ) ∈ R ◦ . By definition, p ( s ) ∈ K [[ s ]] , so that P ( p ( s )) = p ( s ) . (Recall that P is theidempotent map given in Lemma 4.22.) Hence p ( s ) ∈ P ( D + ) , as required. (cid:3) We are now in a position to prove that the ∗ -subalgebra E of R = R ( A , R ) generated by D and Ψ( Q )(1 − ss ∗ ) coincides with the ∗ -subalgebra of R generated by T and Ψ( Q )(1 − ss ∗ ) , which inturn coincides with R by [6, Theorem 6.13]. Let E denote the latter ∗ -subalgebra. Theorem 6.15.
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Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra(Barcelona), Spain.
E-mail address : [email protected] (J. Claramunt) Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900Florianópolis SC, Brazil.
E-mail address : [email protected]@posgrad.ufsc.br