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Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the C*-algebra of a CKT family, and prove a universality theorem for C*-algebras generated by these families. In both cases, the C*-algebras generated by quiver algebras play the universal role.


Introduction
In [20], the second author and Jury derived a Wold decomposition for families of partial isometries which satisfy the 'Cuntz-Krieger-Toeplitz' directed graph relations, what we call the ( †) relations. This theorem may also be obtained as a special case of the Wold decomposition recently established by Muhly and Solel [29] for the more general setting of induced representations of tensor algebras. In [20] it was used to investigate the internal structure of free semigroupoid algebras [18,20,21,25,26] and a dilation theory was built around it in [19,30]. In this paper, we apply this Wold decomposition to establish extensions of a number of fundamental theorems to the case of general countable directed graphs with no sinks, whereby the extended results can be regarded as occurring for single vertex graphs.
The next section ( §2) contains background information. In §3 we include an expanded discussion of the Wold decomposition and extend a folklore result for isometries on Hilbert space: We prove that CKT families {S e , P x } of partial isometries and projections are jointly similar precisely when they are jointly unitarily equivalent. We also establish a Szego-type factorization theorem [36,34] for CKT families. In §4 we use the Szego theorem to obtain detailed information [11] on the unit ball of semigroupoid algebras which are partly free [25,26].
Given a countable directed graph G with no sinks, in §5 we show there is a von Neumann inequality for the set of row contractions T = (T 1 , . . . , T n ) of operators on Hilbert space such that the joint actions of the T i are related to G in a natural way. In the case of single vertex graphs, we recover the original form of the von Neumann inequality [41,38] (when n = 1) and Popescu's version [33] (when 2 ≤ n ≤ ∞). If n ≥ 2, the estimates obtained are sharper than the estimates of [33] in general. In fact, we show these estimates respect the partial ordering on directed graphs induced by graph deformations. An immediate consequence of the von Neumann inequality is the existence of functional calculus ( §6) for row contractions based on quiver algebras [21,25,26,28,29,30] and free semigroupoid algebras. There is a long history of both positive and negative results associated with von Neumann inequalities for the multivariable setting, and this work fits in with such efforts. See [1,2,3,6,9,12,17,30,33] for some of the different perspectives. In connection with the current work, see [30] where a more abstract form of the von Neumann inequality is obtained.
In the final section, we apply the Wold decomposition to obtain a pair of universality theorems for directed graph C * -algebras. In particular, a generalization of Coburn's theorem [8,32] for isometries on Hilbert space is established: The C * -algebra generated by any CKT family which satisfies a natural non-unitary condition is isomorphic to the C * -algebra generated by the corresponding quiver algebra. Further, we prove that the C * -algebras generated by quiver algebras are the universal C * -algebras for the CKT relations.

Preliminaries
Let G be a countable directed graph with directed edges E(G) and vertices V(G). Given a finite path w in G, we write w = ywx when the source vertex of w is s(w) = x and the range vertex of w is r(w) = y. (For vertices x ∈ V(G) put r(x) = x = s(x).) Say that G has a source at x ∈ V(G) if there are no edges which finish at x; that is, r(e) = x for e ∈ E(G). Similarly, G has a sink at x ∈ V(G) if s(e) = x for e ∈ E(G). Define the free semigroupoid F + (G) to be the set of all vertices V(G) and all finite paths in G, written in reduced form, with the natural operations of concatenation of paths allowed by the graph structure. A path w = e k · · · e 1 ∈ F + (G) is said to be a loop if s(w) = r(w). If in addition, r(e i ) = r(e j ), for i = j, then w is said to be vertexsimple. A vertex-simple loop w = e k · · · e 1 , e i ∈ E(G), has an entrance if there exists an edge f ∈ E(G) such that r(f ) = r(e i ), for some i, but f = e i . Finally, let P + G be the path algebra generated by G; this is the set of polynomials with complex coefficients and monomials belonging to F + (G).
Given a family of non-zero partial isometries {S e : e ∈ E(G)} and projections {P x : x ∈ V(G)} which act on a common Hilbert space, consider the following relations: If equality is achieved in (iv) for all x ∈ V(G), then we refer to the relations as ( ‡). Observe that ( ‡) can only occur for G with no sources. We mention that the operator algebras generated by families {S e , P x } which satisfy ( †) have been the focus of intense recent interest. They include so-called Cuntz-Krieger directed graph C * -algebras, free semigroupoid algebras, quiver algebras, etc. (The references [4,10,18,20,21,22,23,24,25,26,28,29,30,37,39] give a starting point for the interested reader.) Let us mention at this point the existence of the universal C * -algebra C * (G) [22,23,24] for the relations ( ‡) for a given directed graph G. From these operator algebra motivations, we refer to ( ‡) as the Cuntz-Krieger (CK) relations, and ( †) as the Cuntz-Krieger-Toeplitz relations (CKT). We next describe the prototypical 'pure model' [19,20] in the Wold decomposition used here. (More details are given in the next section.) Let G be a countable directed graph (with no sinks) and define H G to be the Hilbert space with orthonormal basis given by {ξ w : w ∈ F + (G)}. This generalized 'Fock space' associated with a directed graph was introduced by Muhly [28]. For e ∈ E(G) define partial isometries L e on H G by Observe that L * e L e = P s(e) is the projection onto span{ξ w : r(w) = s(e)}, and that the family {L e , P x } satisfies ( †). For x ∈ V(G), we define the 'tree component' subspace H x = span{ξ w : w = wx}. Notice that these subspaces are reducing for the L e . In fact, the subspaces {H x : x ∈ V(G)} form the unique family of minimal non-zero joint reducing subspaces for the L e [25].
Given a directed graph G, the associated quiver algebra A G is the norm-closed algebra generated by {L e , P x : e ∈ E(G), x ∈ V(G)}, where the P x here are the vertex projections associated with the pure model (1). The associated free semigroupoid algebra L G is the weak operator topology closure of A G . The commutant of L G coincides with L ′ G = R G [25], the wot-closed algebra generated by partial isometries R e , defined on H G (as in (1)) by R e ξ w = ξ we , together with their initial projections which we denote by Q x , x ∈ V(G). The projections Q x are precisely the projections onto the tree component subspaces H x .
We mention that the graph G was proved to be a complete invariant for A G (and L G ) up to unitary equivalence by Kribs and Power [25]. More recently, Solel [37] proved G is an invariant for isometric isomorphism and, independently, the authors [21] proved that G is a banach algebra isomorphism invariant for A G . (In fact, when G has no sources or no sinks it was proved in [21] that G is an algebraic isomorphism invariant for A G .) This is a true departure from the C * -algebra case, where, for instance, it is not hard to see that G is not a * -isomorphism invariant for C * (G). (e.g. For many non-isomorphic graphs, C * (G) is * -isomorphic to the set of compact operators.)

Wold Decomposition
The Wold decomposition from [20] (which assumed G had no sinks) asserts that every set of partial isometries {S e : e ∈ E(G)} which satisfies ( †) on a Hilbert space H is jointly unitarily equivalent to a direct sum where V = (V e ) e∈E(G) satisfies ( ‡), and hence determines a representation of a Cuntz-Krieger directed graph C * -algebra. As indicated above, this result can be derived from a special case of the Wold decomposition from [29]. The restricted ampliations of the L e in this decomposition are said to form the pure part of the dilation. (Notice that the pure model (1) is captured with V e ≡ 0 and α x ≡ 1.) The α x are called the vertex multiplicities in the dilation and are computed from {S e , P x } via the equations More specifically, the joint unitary equivalence (2) arises from the spatial decomposition H = H c ⊕ H p where H c = H ⊥ p and H p = w∈F + (G) ⊕w(S)W are joint reducing subspaces for the S e , with W the wandering subspace for S given by Observe for x ∈ V(G) that P x W = Ran P x − r(e)=x S e S * e is a subspace of W. The subspaces H c and H p have the alternate descriptions where F + (G) d is the set of paths inside F + (G) of length d.
As a direct consequence of the Wold decomposition, we prove a generalization of the well-known fact that similar isometries are unitarily equivalent. This result also generalizes Popescu's theorem [32] for the case of n-tuples of isometries with orthogonal ranges, which occurs below when G is a single vertex graph with n ≥ 2 edges.  For e ∈ E(G) let

then there is a unitary operator U such that
. Let W and W ′ be the wandering subspaces for S and S ′ respectively. Then In particular, dim W = dim W ′ . But more is true. Recall that the vertex multiplicities in the pure parts of the Wold decompositions for S and S ′ are given by α x = dim P x W and α ′ x = dim P x W ′ . By hypothesis and the previous identity we have . Thus, it is easily seen from the Wold decomposition that there is a unitary U 1 : H p → H ′ p which intertwines the pure parts of S and S ′ ; Since G has no sinks, every vertex projection P x is the initial projection for some S e . Further, recall that H p (respectively H ′ p ) is a reducing subspace for the S e (respectively S ′ e ). It follows that the restricted vertex projections P x | Hp and P ′ x | H ′ p are also intertwined by U 1 . We now show that the restrictions to H c , H ′ c may be spatially intertwined as well. As AS e = S ′ e A and S e A −1 = A −1 S ′ e for e ∈ E(G), equation (4) applied for the S e and the S ′ e implies that AH c = H ′ c . Let

In particular, equation (4) gives
Thus the ( †) relations and the hypothesis implies that for e ∈ E(G), It follows from the adjoint of this identity that

Consider the polar decomposition of
and it follows that Therefore, we may define a unitary U = U 1 ⊕ U 2 on H so that (5) and (7) imply US e = S ′ e U * for e ∈ E(G). As G has no sinks, it follows that the projections P x and P ′ x are also intertwined by the unitary U.
We now present another application of the Wold decomposition that allows us to prove a Szego-type factorization theorem. This theorem generalizes earlier results of Rosenblum and Rovnyak [36, Theorem 3.4] and Popescu [34].
Proof. We need to prove that (ii) implies (i). Let . We will show that the family {T e } e∈E(G) is jointly unitarily equivalent to {S e } e∈E(G) . Towards this end notice that T * e T e = P s(e) and T e T * e ≤ P r(e) . Indeed, , and a similar calculation shows the claimed inequality. Therefore, the collections {P x } x∈V(G) and {T e } e∈E(G) satisfy the requirements of the Wold Decomposition; i.e., the relations ( †), and therefore the earlier discussion applies. First, we determine that the Cuntz-Krieger part of Thus {T e } e∈E(G) is pure. Let W be its wandering subspace and let W x ≡ P x W ⊆ W. Then by the Wold decomposition there exists a unitary operator such that for any e ∈ E(G), such that for any e ∈ E(G), .
In order to establish the joint unitary equivalence between the families . From what we have proved so far, it follows that we need only to to verify β x = α x , for all x ∈ V(G). Fix an x ∈ V(G) and let ξ ∈ W x . Then for all e ∈ E(G) and ζ ∈ H G , we have e S e Y 1/2 for e ∈ E(G) since the same is true for Y . Hence, as G has no sinks, every P x is equal to some initial projection S * e S e = T * e T e and it follows that Therefore, the identity β x = α x holds for all x ∈ V(G), as required.
We have shown that V T e V * = US e U * for e ∈ E(G). In particular, this implies that U * V Y 1/2 S e = S e U * V Y 1/2 , for all e ∈ E(G), and the operator U * V Y 1/2 commutes with {S e } e∈E(G) . Moreover, notice that for e ∈ E(G), Using the fact that R ′ G = L G we obtain the following. Corollary 3.3. Let G be a countable directed graph with no sinks. If Y ∈ B(H) is a positive invertible operator then the following assertions are equivalent:

An Application To The Geometry Of The Unit Ball
In [11], Davidson, Pitts and the first author proved that every operator in the open unit ball of a w*-closed operator algebra generated by n isometries with orthogonal ranges, is the average of isometries. This result applies in particular to the noncommutative Toeplitz algebra L n , thus extending a classical result of Marshall [27] from function theory to the noncommutative setting. Using the Szego type factorization theorem we proved earlier, we extend the result of [11] to a broad class of free semigroupoid algebras.
The techniques of [11] yield a more general result than the one quoted in the introduction. In order to state it, we need the following. It is clear from Corollary 3.3 that any free semigroupoid algebra for a graph with no sinks satisfies the factorization property.
Theorem 4.2. [11] Let A be a norm closed operator algebra satisfying the following two properties: (i) A satisfies the factorization property.
(ii) A contains two isometries with orthogonal ranges.
Remark 4.3. The above result is not explicitly stated as a theorem in [11] but its validity is ascertained on [11, page 118].
On the other hand, in [26] Power and the second author identified a property for a directed graph G that is equivalent to the existence of two isometries in L G with orthogonal ranges.

Von Neumann Inequality
Given a row contraction T = (T 1 , . . . , T n ) of (non-zero) operators acting on a Hilbert space H, consider a family of mutually orthogonal projections P = {P x } x∈J on H which sum to the identity operator and stabilize T in the following sense: Observe that these relations determine a directed graph G with vertex set V(G) ≡ J and n directed edges e 1 , . . . e n , where r(e i ) and s(e i ) are the unique vertices with P r(e i ) T i P s(e i ) = T i . When there is no confusion, we shall write T = (T e ) e∈E(G) if an ordering of the T i has been induced by a fixed projection set P.
Let T = (T e ) e∈E(G) and P = {P x } x∈V(G) satisfy (8) on H for a graph G with no sinks. Then it was shown in [19], and previously in a more abstract form [30] for the general setting of tensor algebras over C *correspondences, that there are partial isometries {S e : e ∈ E(G)} on a space K ⊇ H such that For a fixed directed graph G we shall define the supremum norm ||p|| ∞ of a polynomial p ∈ P + G by ||p|| ∞ ≡ ||p(L G )||, where L G = (L e ) e∈E(G) is the canonical pure model (1) for G. This definition is in line with other settings, where the norm ||p|| ∞ is determined by the pure model in this manner. As the simplest example, recall that the norm of a polynomial p on the unit complex disk satisfies ||p|| ∞ = ||p(U + )||, where U + is the unilateral forward shift [38]. To see this, let S = (S e ) e∈E(G) satisfy ( ‡) and let 0 < r < 1. Then rS = (rS e ) e∈E(G) is a pure row contraction, and hence the minimal partially isometric dilation of rS with respect to {S * e S e : e ∈ E(G)} is given by the form (2) with V e ≡ 0. We claim that the vertex multiplicities for this pure dilation S ′ = (S ′ e ) e∈E(G) satisfy α x ≥ 1 for x ∈ V(G). Indeed, from (9) these multiplicities are determined by Thus if α x = 0, then the ( ‡) relations give which is clearly a contradiction since the P x are assumed to be non-zero in the ( †) relations.
Remark 5.2. In the case of single vertex graphs with n edges, the above theorem recovers the classical von Neumann inequality [41,38] (when n = 1) and Popescu's version [33] (when 2 ≤ n ≤ ∞). We also mention that the conclusion of Theorem 5.1 can be proved through an application of Theorem 3.10 from [30]. However, we note that our direct proof via the Wold decomposition is novel and provides a new perspective on the problem.

Partially Ordered Sets of Directed
Graphs. An alternative perspective on Theorem 5.1 is the following: Given a countable directed graph G with no sinks, the theorem shows there is a corresponding von Neumann inequality with estimate given by (for this discussion) ||p|| G,∞ ≡ ||p(L G )|| for p ∈ P + G , recalling that L G = (L e ) e∈E(G) . On the other hand, every directed graph determines a partially ordered set of directed graphs through its deformations. A directed graph G 2 is a deformation of G 1 if G 2 is obtained from G 1 by identifying certain vertices in G 1 . The partial order is defined by: G 1 ≤ G 2 if and only if G 2 is a deformation of G 1 . (This perspective was also discussed in [19] in the context of dilation theory, where it was shown that every row contraction T generates such a partially ordered set of directed graphs through its family of minimal partially isometric dilations.) We observe below that the von Neumann inequality estimates respect the ordering in such a partially ordered set of directed graphs.
Let C n be the directed graph with a single vertex and n directed loop edges. The Sz.-Nagy and Popescu estimates are given by ||p(L Cn )||, where p is allowed to be any polynomial in n noncommuting variables when n ≥ 2. Let G be a countable directed graph with n edges. Then L G = (L e ) e∈E(G) is a pure row contraction (in the n-tuple sense [35]) and hence its (unique) minimal isometric dilation [7,15,35] is given by a multiple of L Cn = (L 1 , . . . , L n ), where the L i are the creation isometries on the full n-variable Fock space. Thus, given L G = (L e ) e∈E(G) and such a polynomial p, we shall write p(L G ) for the evaluation of p at the L e which arises from the ordering of the L e induced by the minimal isometric dilation of L G . Corollary 5.3. If G 1 ≤ G 2 are directed graphs with no sinks and n directed edges, then for every polynomial p in n noncommuting variables Proof. The relation G 1 ≤ G 2 can be seen to induce a G 2 -ordering on L G 1 , where the corresponding projections which stabilize L G 1 are given by the sums of the initial projections for L e , e ∈ E(G 1 ), which are naturally induced by the deformation. Hence the first inequality is a consequence of Theorem 5.1. The last inequality follows from Popescu's von Neumann inequality since L G 1 and L G 2 are both row contractions and n-tuples.
It is not hard to see that ||p|| G,∞ gives a sharper estimate than ||p(L Cn )|| in general when n ≥ 2. Consider the following simple illustration of this fact.
is a row contraction which is stabilized by P = {P 1 , P 2 } where P i ≡ P H are the projections of H (2) onto its coordinate spaces. Let G be the graph with two vertices x, y and edges e = yex, f = xf y. Then L G = (L e , L f ) in this case. Observe that C 2 is a deformation (the only one) of G. Let L C 2 = (L 1 , L 2 ), where L 1 , L 2 are the creation isometries on the full 2-variable Fock space. Then L e + L f is an isometry and L 1 + L 2 = √ 2L where L is an isometry, and hence Further, the L e + L f estimate is best possible in this example since ||V 1 + V 2 || = 1 if ||T i || = 1 for i = 1 or 2.

Functional Calculus
The von Neumann inequality leads to a natural functional calculus determined by quiver algebras and, in the case of pure row contractions, by free semigroupoid algebras. Observe that when T = (T e ) e∈E(G) is a row contraction with G-ordering induced by projections P = {P x } which satisfy (8) for T , we may consider all polynomials {p(T ) : p ∈ P + G } in the generators T e , P x . The following result is an immediate consequence of Theorem 5.1 and may also be obtained as a special case of Theorem 3.10 from [30]. x for e ∈ E(G) and x ∈ V(G). Proof. Condition (11) means precisely that the minimal partially isometric dilation V = (V e ) e∈E(G) of T with respect to {P ′ x } is pure [19]. By the Wold decomposition for V , we may define operators f (V ) for all f ∈ L G , where f (V ) = wot − lim k p k (V ) for some sequence p k ∈ P + G . From the properties of the minimal dilation we have p k (T ) = P H p k (V )| H . Thus, p k (T ) converges to an operator f (T ) ≡ wot −lim k p k (T ) ∈ B(H). Finally, by the von Neumann inequality and a standard approximation argument we have ||f (T )|| ≤ ||f (V )|| ≤ ||f (L)||, and the result follows. Remark 6.3. It should be possible to prove an L G functional calculus for more general row contractions. For instance, as in [38,5,32] this could be accomplished by first identifying 'characteristic functions' for an appropriate notion of completely non-unitary row contractions here.

A Generalization of Coburn's Theorem
In [8] Coburn showed that the C * -algebra generated by a non-unitary isometry V is isomorphic to the Toeplitz C * -algebra, via a map that sends V to the forward shift. This result was generalized by Popescu to Cuntz-Toeplitz C * -algebras [32]. In this section we consider the generalization of Coburn's Theorem to C * -algebras generated by partial isometries associated with more general directed graphs.
The proof of Coburn's theorem depends on the universality of the algebra of continuous functions on the circle as the C * -algebra of a graph G consisting of one vertex and one loop-edge. The following result of Szymanski [39] generalizes this fact to arbitrary graph C *algebras.
Theorem 7.1. [39] Let G be a countable directed graph with no sources and ρ : C * (G) −→ A be a * -homomorphism into a C * -algebra A. Then ρ is injective if and only if the following two conditions are satisfied: Proof. Since G has no sources, the projections L x , x ∈ V(G), are infinite dimensional and thus L x + K = 0. Furthermore, and so the collections {L x + K} x∈V(G) and {L e + K} e∈E(G) satisfy the relations ( ‡). Hence there exists a * -homomorphism ρ : C * (G) −→ C * (A G )/ K with ρ(P x ) = L x + K and ρ(S e ) = L e + K for all x ∈ V(G) and e ∈ E(G). We now use Szymanski's result to show that ρ is injective.
Towards this end, notice that we have already verified condition (i) in Theorem 7.1. To verify the other condition let w ∈ F + (G) be any loop. Consider the direct sum decomposition H G = L x (H G )⊕L x (H G ) ⊥ and notice that with respect to this decomposition, L w admits the 2×2 matrix form L 0 0 0 , with L unitarily equivalent to some ampliation S (α) of the forward shift S. It is well known that if λ ∈ T, then the range of S −λI is not closed. Therefore the same is true for S (α) −λI and so λ belongs to the essential spectrum of S (α) ; i.e., λ ∈ σ(ρ(S w )), as desired.
With a standard 'diagram chasing' argument, Corollary 7.2 can be used to show the essential norm coincides with the operator norm for any element of A G . However, we can prove this directly for a larger algebra, the free semigroupoid algebra L G , hence we include a proof of this fact. Proposition 7.3. The following assertions are equivalent for a countable directed graph G: (i) G has no sources.
Proof. If x is a source vertex in G, then P x is a rank one projection inside A G (and L G ). Conversely, suppose G has no sources. Let d ≥ 1 be a positive integer. For all x ∈ V(G) we may choose a path w x ∈ F + (G) of length d such that r(w x ) = x. As the w x are distinct paths of the same length, notice that the partial isometries R wx , x ∈ V(G), have pairwise orthogonal ranges. Thus, we may define an isometry R (d) = x∈V(G) R wx in the commutant L ′ G = R G (where the sum converges wot in the infinite vertex case); Hence, if A belongs to L G we have On the other hand, {R (d) ξ} d≥1 converges weakly to zero for any choice of vector ξ ∈ H G , and thus ||A|| = ||A|| e as claimed.
Corollary 7.4. If G be a directed graph with no sources, then the compact perturbations L G + K are norm closed.
The next result shows that C * (A G ) is the universal C * -algebra for the relations ( †) and it is known to the specialists (compare with [31,16]). Nevertheless, we feel that the short proof presented here is new and we therefore include it.
for all e ∈ E(G). Proof. From the discussion in Section 3, it is clear that the Wold decomposition (which was proved for G with no sinks) implies the existence of C * -homomorphisms where ρ is the * -isomorphism identified by Corollary 7.2. If ψ ≡ ρ • π, then the desired map here is ). Tracing the definitions of these maps shows that τ (L u ) = S ′ u for all u ∈ F + (G).
The previous theorem leads to a natural C * -algebra extension of the von Neumann inequality Theorem 5.1 (see also [32] for the case of isometries with orthogonal ranges). Though note that the proof of this result relies on Szymanski's Theorem, whereas our proof of Theorem 5.1 is direct. Let P 0 G be the set of polynomials generated by all indeterminant monomials of the form vw with v, w ∈ F + (G) and where x = x for x ∈ V(G). Given T = (T e ) e∈E(G) and p ∈ P 0 G , we write p(T, T * ) for the polynomial p evaluated at the generators T e (for e ∈ P 0 G ) and T * e (for e ∈ P 0 G ) and P x (for x = x ∈ P 0 G ∩ V(G)). Corollary 7.6. Let G be a countable directed graph with no sources and no sinks. Let T = (T e ) e∈E(G) be a row contraction on H stabilized as in (8) Proof. Let S = (S e ) e∈E(G) be the minimal partially isometric dilation of T with respect to the given projection family. From the basic properties of the minimal dilation, it follows that p(T, T * ) = P H p(S, S * )| H for p ∈ P 0 G .
The next theorem is the promised generalization of Coburn's Theorem. Given a collection {S ′ e } e∈E(G) of isometries, we isolate a condition which is the appropriate extension of an isometry being non-unitary here (condition (13) below). We show that any collection of partial isometries satisfying this condition generates a C * -algebra isomorphic to C * (A G ).
Theorem 7.7. Let G be a countable directed graph with no sources and no sinks and let C * (A G ) be the C * -algebra generated by the quiver algebra A G . Let {P ′ x } x∈V(G) and {S ′ e } e∈E(G) be families of projections and partial isometries respectively, acting on a Hilbert space H and satisfying ( †). If, (13) r(e)=x S ′ e (S ′ e ) * = P ′ x for x ∈ V(G), then there exists an injective * -homorphism φ : C * (A G ) → B(H) such that φ(L e ) = S ′ e for all e ∈ E(G). In particular, the C * -algebra C * ({S ′ e } e∈E(G) ), generated by the collection {S ′ e } e∈E(G) , is isomorphic to C * (A G ). where, S ′ e (S ′ e ) *   .
Let W x = P ′ x W = 0 and let ζ x ∈ W x be a unit vector. Notice that the subspaces M x ≡ span{S w ζ x : w ∈ F + (G)} are reducing for {S ′ e } e∈E(G) and the restrictions S ′ e | Mx , e ∈ E(G), are jointly unitarily equivalent to L e | Qx , e ∈ E(G), respectively. Therefore, the subspace M = ⊕ x∈V(G) M x is reducing for {S ′ e } e∈E(G) and the restrictions S ′ e | Mx , e ∈ E(G), are jointly unitarily equivalent to L e , e ∈ E(G). We have therefore established the existence of an epimorphism ψ : C * ({S ′ e } e∈E(G) ) −→ C * (A G ) mapping generators to generators. It remains to show that ψ is injective.
Remark 7.8. It was pointed to us by the referee that Theorem 7.7 has been previously obtained by Fowler and Raeburn [13, Theorem 4.1]. Their approach is quite different from ours since they make an extensive use of Hilbert C * -modules as well as their own methods [14] for analyzing semigroup crossed products. Our proof is shorter and elementary; we only require Szymanski's result, whose proof in [39] is essentially self-contained.
an NSERC research grant and start up funds from the University of Guelph. We would like to thank the referee for a number of helpful comments.