BUNDLES OF C∗-CORRESPONDENCES OVER DIRECTED GRAPHS AND A THEOREM OF IONESCU 6 0 0 JOHN QUIGG 2 n a Abstract. We give a short proof of a recent theorem of Ionescu J which shows that the Cuntz-Pimsner C∗-algebra of a certain cor- 1 respondenceassociatedtoaMauldin-Williamsgraphisisomorphic 2 to the graph algebra. ] A O 1. Introduction . h at In recent decades the study of fractal geometry has led to the in- m troduction of graph-directed iterated function systems [7], also known [ as Mauldin-Williams graphs [2,3]. These are finite directed graphs of 1 contractions among compact metric spaces. Recently, Ionescu [3] as- v sociated a C∗-correspondence to a given Mauldin-Williams graph, and 1 proved that the resulting Cuntz-Pimsner algebra is isomorphic to the 2 5 graph C∗-algebra. Ionescu constructs the isomorphism directly, ex- 1 tending the contractions to the state spaces using Rieffels’s theory of 0 6 Lipschitz metrics on state spaces [9]. Ionescu’s result is perhaps sur- 0 prising, and illustrates important connections among fractal geometry, h/ C∗-correspondences, and graph algebras. Thus, we feel it will be use- t a ful to show how Ionescu’s theorem can be quickly deduced from the m elementary theory of graph algebras. : WerelaxIonescu’shypothesessomewhat: whereasthedirectedgraphs v i in [3] are finite and have neither sources nor sinks, we only require the X graph to be row-finite with no sources. In fact, we only impose these r a conditions to illustrate our method in its simplest form; the general case could be handled with somewhat more effort. Also, instead of contractions among metric spaces, we only require continuous maps among locally compact Hausdorff spaces, together with an equivariant surjection from the infinite path space (see Sec- tion 3 for details). For finite graphs, such continuous maps together with an equivariant surjection constitute a self-similarity structure [6]. Date: February 2, 2008. 2000 Mathematics Subject Classification. Primary 46L08. Key words and phrases. directed graph, C∗-correspondence, graph C∗-algebra, Cuntz-Pimsner algebra. 1 2 JOHN QUIGG As pointed out in [3,6], every Mauldin-Williams graph gives rise to a self-similarity structure. 2. Preliminaries Let E = (E0,E1,r,s) be a (directed) graph, with vertices E0, edges E1, and range and source maps r and s. For u,v ∈ E0 put E1 = uv {e ∈ E1 | r(e) = u,s(e) = v}. Warning: we use the relatively new convention (see [4,8]) regarding the graph algebra C∗(E): the gener- ators go the same way as the edges. Thus, for example, if e ∈ E1 uv then s∗ese = pv and ses∗e ≤ pu, and a finite path e1···en in E satisfies s(ei) = r(ei+1) for i = 1,...,n−1. For simplicity we assume throughout that E is row-finite and has no sources, meaning that each vertex receives a positive but finite number of edges. Let E∗ be the set of finite paths, where vertices are regarded as paths of length 0. Let E∞ denote the set of infinite paths, which under our hypotheses is locally compact Hausdorff when given the product topology. Extend the source and range maps to paths in the obvious way, and for v ∈ E0 put E∗ = {α ∈ E∗ | r(α) = v}, and similarly for v E∞. v For α ∈ E∗ let pα = sαs∗α be the range projection of the generator sα. Put AE = span{pα | α ∈ E∗}, a commutative C∗-subalgebra of C∗(E). It is folklore that there is an isomorphism θ: C (E∞) → 0 AE which takes the characteristic function of the set of infinite paths starting with a finite path α to the generating projection pα. We have θ(C0(Ev∞) = pvAE. Each e ∈ E1 gives rise to a continuous map φE: E∞ → E∞ via uv e v u φE(α) = eα. For f ∈ C (E∞) we have e 0 v E ∗ θ(f ◦φe ) = seθ(f)se. 3. Ionescu’s Theorem Suppose that for each v ∈ E0 we have a C∗-algebra Av, and for each e ∈ Eu1v we have an Au−Av correspondence Xe. Let A = Lv∈E0 Av be the c0-direct sum. Then each Xe can be regarded as a correspondence over A; let X = Le∈E1Xe be the direct sum of these correspondences, with operations (aξ)e = ar(e)ξe, (ξa)e = ξeas(e), hξ,ηiv = X hξe,ηei s(e)=v A THEOREM OF IONESCU 3 for a = (av)v∈E0 ∈ A and ξ = (ξe)e∈E1,η = (ηe)e∈E1 ∈ X. Then X is a correspondence over A. Let OX be the associated Cuntz-Pimsner algebra. Here we are interested in correspondences arising as follows: for each v ∈ E0 let Tv be a locally compact Hausdorff space, and put Av = C0(Tv). For each e ∈ Eu1v let φe: Tv → Tu be a continuous map. and let φ∗e: Au → Av be the associated homomorphism, so that Av becomes a Au−Av correspondence Xe with (right) Hilbert module structure coming from the operations of Av and left module multiplica- tiondefinedusing φ∗e. Then A = Lv∈E0Av canbeidentified withC(T), where T is the disjoint union of {Tv | v ∈ E0}. Let X = Le∈E1Xe as above. If the graph E is finite, each Tv is a compact metric space, and each φe is a contraction, we have a Mauldin-Williams graph, and OX as above was introduced in [1,3]. Ionescu’s proves [3, Theorem 2.3] that OX is isomorphic to the graph algebra C∗(E). Suppose we have a continuous map Φ: E∞ → T satisfying Φ◦φE = e φe ◦ Φ for all e ∈ E1. If E is finite and Φ is surjective, we have a self-similarity structure [6]. Ionescu’s theorem follows from the following result: Theorem 3.1 ([3, Theorem 2.3]). With the above notation, suppose that the continuous map Φ: E∞ → T is surjective. Then the Cuntz- Pimsner algebra OX is isomorphic to the graph algebra C∗(E). Proof. Define π: C (T) → C∗(E) by π(f) = θ(f ◦Φ), and for each e ∈ 0 E1 define ψe: Xe → C∗(E) by ψe(ξ) = seπ(ξ). Routine calculations, using pv = θ(χT ◦ Φ) and commutativity of AE, show that the pair V (ψe,π) is a covariant representation of the correspondence Xe. We can form the direct sum ψ = Le∈E1ψe, giving a Cuntz-Krieger Toeplitz representation (ψ,π) of the correspondence X in C∗(E). Since the range of (ψ,π) contains the generators of C∗(E), an application of the Gauge-Invariant Uniqueness Theorem [5, Theorem 6.4] shows that the associated homomorphism ψ×π: OX → C∗(E) is an isomorphism. (cid:3) References [1] R. Bartholdi, R. Grigorchuk, and V. Nekrashivych, From fractoal groups to fractal sets, Trends in Mathematics: Fractals in Braz 2001, Birkhauser, 2003, pp. 25–18. [2] G. A. Edgar, Measure, topology, and fractal geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. [3] M.Ionescu,OperatorAlgebrasandMauldin Williams Graphs,Preprint(2004). 4 JOHN QUIGG [4] T. Katsura,A class of C∗-algebras generalizing both graph algebras and home- omorphism C∗-algebras, I. Fundamental results,Trans.Amer.Math.Soc.356 (2004), 4287–4322. [5] , On C∗-algebras associated with C∗-correspondences, J. Funct. Anal. 217 (2004), 366–401. [6] J. Kigami, Analysis on Fractals, Cambridge University Press, 2001. [7] R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811–829. [8] I. Raeburn, Graph algebras, CBMS Reg. Conf. Ser. Math., American Mathe- matical Society, Providence, RI, 2005. [9] M. A. Rieffel, Matrics on state spaces, Doc. Math. 4 (1999), 559–600. Department of Mathematics and Statistics, Arizona State Univer- sity, Tempe, Arizona 85287 E-mail address: [email protected]