OF REDUCED C
∗-ALGEBRAS ASSOCIATED WITH EQUIVALENCE RELATIONS
RADU-BOGDAN MUNTEANU
The reducedC∗-algebraCr∗(R) associated with an ´etale equivalence relationRon a compact metrizable spaceX contains a canonical Cartan algebraCisomorphic toC(X). In this paper we will prove that, whenRis amenable, anyC∗-subalgebra ofCr∗(R) containing C is isomorphic toC∗(S), for some unique subequivalence relationS ofR.
AMS 2010 Subject Classification: Primary 46L05, Secondary 22A22.
Key words: ´etale equivalence relation, reducedC∗-algebra.
1. INTRODUCTION
The reducedC∗-algebras of ´etale equivalence relations are generalizations of the usual matrix ∗-algebras. Given an ´etale equivalence relation R on a compact metrizable space X, one can associate the reduced and the full C∗- algebra, denoted withCr∗(R) andC∗(R), respectively. These twoC∗-algebras, which in general are different, coincide whenRis amenable, as we will assume throughout this paper. The elements of Cr∗(R) can be regarded as elements of C0(R), the space of continuous functions defined on R and vanishing at infinity. Inside Cr∗(R) there exists a canonical Cartan algebra C, consisting of those continuous functions defined on Rand supported on the diagonal of X ×X, that can be identified with C(X), the C∗-algebra of the continuous functions on X. If S is a subequivalence relation of R, the C∗-algebra A(S) consisting of those elements of Cr∗(R) supported on S is a C∗-subalgebra of Cr∗(R) that containsC. In [3] it was proved that anyC∗-subalgebra of Cr∗(R) containingCis of the formA(S), whereSis a uniquely determined subequiva- lence relation of R. In this paper we will prove that for any subequivalence relation S, A(S) is isomorphic to Cr∗(S). Therefore, we will obtain that any subalgebra of Cr∗(R) containing C is isomorphic to Cr∗(S), with S being a uniquely determined subequivalence relation of R. Thus, such subalgebras
REV. ROUMAINE MATH. PURES APPL.,55(2010),5, 381–386
can be studied by using the theory developed for reduced C∗-algebras of ´etale equivalence relations.
The paper is organized as follows. In Section 2 we introduce the notation and we recall some definitions and constructions. In Section 3 we prove the main result.
2. NOTATION AND DEFINITIONS
LetX be a compact metrizable space and Ran equivalence relation on X with countable equivalence classes. We denote bysandr the two canonical projections from R toX, given bys(x, y) =x and r(x, y) =y.
Definition 2.1. IfX is a compact metrizable space, R is an equivalence relation on X, and T a topology on R, we say that (R,T) is ´etale if
(i)T is Hausdorff, second countable andσ-compact;
(ii) the diagonal ∆ ={(x, x)|x∈X}is open in R;
(ii) the mapsr, s:R →X are local homeomorphisms; that is, for every (x, y) inR, we may find an open setU inT such thatr(U) ands(U) are open in X and r:U →r(U) ands:U →s(U) are homeomorphisms;
(iv) if U and V are open sets as above, then the set U V = {(x, z) | (x, y)∈U, (y, z)∈V, for somey} is also open and
(v) if U as above is open, then so is U−1={(x, y)|(y, x)∈U}.
It follows that the diagonal ∆ ={(x, x)|x∈X}is a clopen subset ofR.
Also, ∆ is homeomorphic toX, and so we are justified to identify ∆ with X.
WhenT is understood, we simply say thatRis ´etale. An equivalence relation is also a principal groupoid. The term “´etale” is relatively recent; in the past these have also been known as r-discrete groupoids with counting measure as Haar system (see [3], [4], [5]).
LetCc(R) denote the set of continuous, compactly supported complex- valued functions onR. This is a linear space in an obvious way. The product and the involution are defined by
f∗g(x, y) =X
zRx
f(x, z)g(z, y), f∗(x, y) =f(y, x),
for all f,g inCc(R) and (x, y) in R. In order to show that the product f∗g is again in Cc(R) we use the ´etale property of R.
Letµ be aσ-finite measure on X and let νR(C) =R
|r−1(x)∩C|dµ(x).
The measure µis assumed to be quasi-invariant in the sense ν−1 ∼ν, where ν−1 is the image ofν under the inversion (x, y)→(y, x). For f inCc(R) and
ξ inH=L2(R, νR), we set
IndRµ(f)ξ(x, y) = X
(x,z)∈R
f(x, z)ξ(z, y).
It is easy to check that IndRµ is a bounded representation and the collection of such representations is faithful on Cc(R). The completion of Cc(R), in the norm defined by the equation
kfkr= sup{kIndRµ(f)k, µis a measure onX}, is denoted byCr∗(R) and is called the reducedC∗-algebra of R.
Definition 2.2 ([5]). An ´etale equivalence relation is called amenable if there is a net (fi) in Cc(R) such that
(i) the functions x 7→ P
yRx
|fi(x, y)|2 are uniformly bounded in the sup- norm;
(ii) the functions (x, y)7→) P
zRx
fi(x, z)fi(y, z) converge to 1 uniformly on any compact subset of R.
Because our equivalence relations are assumed to be second countable, sequences suffice in the definition of amenability.
To an ´etale equivalence relation R one can also associate the so-called theC∗-algebra ofRthat is denoted by C∗(R). In general,C∗(R) and Cr∗(R) are different. WhenR is amenable, Cr∗(R) =C∗(R).
We recall that the elements of Cr∗(R) can be regarded as functions in C0(R), the space of continuous functions on R which vanish at infinity [5, Proposition 4.2]. The set of continuous function defined on R and supported on the diagonal ∆ of X×X form an abelian subalgebra of Cr∗(R), that we denote by C. The restriction map P : C∗(R) → C is a faithful conditional expectation [5, Proposition 4.8]. The properties of the subalgebra C may be summarized by the notion of Cartan algebra.
Definition 2.3 ([6]). A ∗-subalgebra B of a C∗-algebra A is called a Cartan algebra if
(i)B contains an approximate unit of A;
(ii)B is maximal abelian;
(iii) N(B) ={a∈A:aBa∗ ⊆B, a∗Ba⊆B}, the normalizer ofB inA, generates A;
(iv) there exists a faithful conditional expectation ofA onto B.
In [5], the concept of Cartan algebra for C∗-algebras was defined in a different way. The definition given here it is the natural analogue of the concept of Cartan algebra for von Neumann algebras, introduced by Feldman and Moore [1]. We have thatC is a Cartan subalgebra ofCr∗(R) [6, Theorem 5.2].
Since X is identified with ∆, a function f ∈ C(X) can be regarded as a function defined on R, supported on ∆ also denoted byf, and given by
(1) f(x, y) =
f(x, x) ifx=y, 0 ifx6=y.
In this way, C(X) can be identified with C⊆Cr∗(R).
Note that ifµis a quasi-invariant measure on Xand supp(µ) =X, then kIndRµ(f)k= kfkr, for allf ∈Cc(R). Therefore, Cr∗(R) can be seen as the completion of IndR(Cc(R)) inB(H).
ReducedC∗-algebras associated to ´etale equivalence relations are a kind of generalized matrix algebras. Notice that, if X = {1,2, . . . , n} and R = X×X, thenCr∗(R) is justMn(C) andC'C(X) is the subalgebra of diagonal matrices of Mn(C).
3. SUBALGEBRAS OF Cr∗(R) WHICH CONTAIN C 'C(X) In this section R is an amenable ´etale equivalence relation on X. We assume thatU in Definition 1 (iii), may be chosen to be open and compact.
Another equivalence relation S on X, which is an open subset of R, is called subequivalence relation of R. For a subequivalence relation S of R we define
A(S) ={f ∈Cr∗(R), f = 0 on R \ S}.
A(S) is aC∗-subalgebra ofC∗(R) that containsC 'C(X). Also, any function in Cc(S), the space of continuous compactly supported functions on S, can be seen as a function in Cc(R), by extending it with zero on R \ S. In [3], it is proved that any C∗-subalgebra of Cr∗(R) containing C is of the form A(S) for a unique subequivalence S and the correspondenceS 7→A(S) is an inclusion preserving bijection between the collection of subequivalence of R and C∗-subalgebras ofCr∗(R) containingC. In this section we prove that any C∗-subalgebra of Cr∗(R) of the form A(S), with S a subequivalence relation of R, is isomorphic toCr∗(S), and therefore, we will have the following result:
Theorem 3.1. Any C∗-subalgebra of Cr∗(R) that contains C 'C(X) is isomorphic to a certain Cr∗(S) for a unique subequivalence S of R.
Proof. The theorem results from [3, Theorem 4.1] and the following proposition.
Proposition3.2. Let S be a subequivalence ofR. ThenA(S)'Cr∗(S).
Proof. Consider µ a quasi-invariant probability measure on X with supp(µ) = X and let IndRµ and IndSµ be the representations induced by
µ corresponding to the equivalence relations R and S, respectively. Clearly, the measure νS is the restriction of the measure νR to S. Any function f defined on S can be seen as a function defined on R which is zero on R \ S and is equal to f on R. Thus, the Hilbert space L2(S, νS) can be regarded as a subspace of H = L2(R, νR), denoted by H0 and Cc(S) can be seen as a subspace of Cc(R) which is norm dense in A(S) (see [3]). Therefore, since supp(µ) =X, IndRµ(Cr∗(R)) can be identified with the norm closure of IndRµ(Cc(R)) in B(H) and A(S) ⊆Cr∗(R) can be identified with the norm- closure of IndRµ(Cc(S)) inB(H). Let π denote the restriction to H0 of the representation IndRµ of Cc(S). The identification of H0 with L2(S, νS), al- lows us to identify π with IndSµ and further, to identify the norm closure of π(Cc(S)) inB(H0) withCr∗(S). LetN be the von Neumann algebra acting on H, obtained as the weak closure ofπ(Cc(S)). We denote bye the orthogonal projection from H onto H0, which is given by e(f) =f|S, forf ∈H. We will show thate∈N0, whereN0 is the commutant ofN inB(H), and the central projection z(e) ofeis 1. This will imply thatNe'N. The isomorphism from N toNe is given byx7→exe. This correspondence is norm preserving, being a ∗-isomorphism. It follows that kIndRµ(f)k=kπ(f)k for all f ∈Cc(S) and so, A(S)'Cr∗(S).
The only thing remaing to be proved is thate∈N0andz(e) = 1. SinceN leaves H0 invariant, it follows thate∈N0. The weak closure of IndRµ(Cc(R)) in B(H) is M = M(X, µ,R), the von Neumann algebra associated with the equivalence relation Ron (X, µ) by Feldman and Moore [1]. Any function in L∞(X, µ), can identified with a function defined on Ras in (1). In particular, If A is a measurable subset of X, the characteristic function 1A is identified with 1A×A∩∆. With this identification, it makes sense to consider IndRµ(f), for f ∈ L∞(X, µ). Let A = {IndRµ(f), f ∈ L∞(X, µ)}. Then, A coincides with the weak closure of IndRµ(C) in B(H) and, by [1, Proposition 2.9], A is a maximal abelian subalgebra of M, isomorphic to L∞(X, µ). Since A ⊂ N ⊂ M, it results that A ' L∞(X, µ) is also maximal abelian in N, and consequently,Z(N) =N∩N0, the center ofN, is contained inA 'L∞(X, µ).
Therefore, we can identify z(e), the central support of e, with IndRµ(1A) for a certain measurable set A ⊆ X. We have 1A×A∩∆ = IndRµ(1A)1∆ = z(e)e(1∆) =e(1∆) = 1∆. It follows that A=X up to a set of measure zero, and so, z(e) = 1.
Acknowledgement.This work is part of author’s Ph.D. Thesis, written at the Uni- versity of Ottawa. The author is grateful to his Ph.D. supervisor, Professor Thierry Giordano, for guidance and support.
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Received 20 February 2010 University of Bucharest Faculty of Chemistry
Dept. of Physics and Applied Mathematics 4-12 Bd. Regina Elisabeta, Sector 1
Bucharest, Romania radubog@yahoo.com
