ESSENTIAL REPRESENTATIONS OF C^{∗}-CORRESPONDENCES

ILAN HIRSHBERG

Abstract. Let E be a C^{∗}-correspondence over a C^{∗}-algebra A with non-degenerate faithful left action. We show that E admits sufficiently many essential representations (i.e. representations ψ such that \overline{ψ(E)H} = H) to recover the Cuntz-Pimsner algebra O_{E}.

1. Introduction

Let A be a C^{∗}-algebra. Recall that by a C^{∗}-correspondence over A we mean a Hilbert module E over A, along with a fixed left action of A on E via adjointable operators (i.e. a homomorphism A → B(E) – we suppress notation for this homomorphism). T_{E} and O_{E} denote the Pimsner-Toeplitz algebra and the Cuntz-Pimsner algebra, respectively (see [10]). We shall frequently identify A and E with their images in T_{E} and O_{E}.

Throughtout this paper, we shall assume that E is full (i.e. \overline{⟨E, E⟩} = A^{1} ), and that the left action of A on E is faithful and non-degenerate.

By a representation of E on H (c.f. [7, 4, 6]) we mean a pair ψ_{E} : E → B(H) (a linear map), ψ_{A} : A → B(H) (a homomorphism) such that for all e, f ∈ E, a ∈ A we have ψ_{E}(ea) = ψ_{E}(e)ψ_{A}(a), ψ_{E}(ae) = ψ_{A}(a)ψ_{E}(e), ψ_{E}(e)^{∗}ψ_{E}(f) = ψ_{A}(⟨e, f⟩). We will write ψ for both ψ_{A} and ψ_{E}, when it causes no confusion. Note that what we call ‘representation’ is called ‘isometric covariant representation’ in [7]. A representation ψ of E induces a representation of E^{⊗n}, which we denote by ψ as well, by ψ(e_{1} ⊗ ⋯ ⊗ e_{n}) = ψ(e_{1}) ⋯ ψ(e_{n}).

Let E = ⨁_{n=0}^{∞} E^{⊗n} (the Fock module). Each representation π : A → B(H) induces a representation of E on E ⊗_{π} H (where we use π to view H as an A − ℂ correspondence). Following Arveson’s terminology for product systems (c.f. [2]), we call a representation singular if ⋂_{n=0}^{∞} \overline{ψ(E^{⊗n})H} = 0, and essential if \overline{ψ(E)H} = H (essential representations are called ‘fully coisometric’ in [7]).

Recall the following generalization of the Wold decomposition from [7]. Theorem 1.1. Let ψ be a representation of E on H. H decomposes uniquely to a direct sum of invariant subspaces H_{s} ⊕ H_{e}, such that ψ|_{H_{s}} is singular and ψ|_{H_{e}} is essential. Furthermore, the singular part of the representation is unitarily equivalent to a representation induced from the Fock module (as explained above).

Thus it is easy to construct singular representations, however there doesn’t seem to be an obvious way in general to construct essential ones. Any representation of E gives rise to a representation of the Pimsner-Toeplitz algebra T_{E} (see [10], or [6] for a different proof). We will thus refer to representations of T_{E} as ‘essential’ or ‘singular’ if the corresponding representations of E are essential or singular, respectively.

1We follow the convention that when we place two spaces into a bilinear or sesquilinear expression such as ⟨E, F ⟩, or AB, what is meant is the span of all possible pairings.

If a representation of E is essential, then the corresponding representation of T_{E} will factor through the Cuntz-Pimsner algebra O_{E} (see [7]). However, not any representation of O_{E} necessarily arises from an essential representation of E (as a simple example, consider E to be the correspondence over ℂ of dimension ℵ_{0}; here, T_{E} = O_{E} = O_{∞}, and if s_{1}, s_{2}, ... are the generators, and π is the representation, then π comes from an essential representation precisely when ∑_{i=1}^{∞} π(s_{i}s_{i}^{∗}) = 1, with convergence taken in the strong operator topology). It seems, thus, natural to ask whether any C^{∗}-correspondence as above admits an essential representation.

More generally, we can ask the following question. Let π_{ess} be the universal essential representation of T_{E}, i.e. a direct sum of unitary equivalence classes of all essential representations (where we bound the dimension of the target space H to avoid set-theoretic difficulties). By the comments above, π_{ess} factors through O_{E}. We preserve notation and think of π_{ess} as a representation of O_{E}. The main result of this paper is the following. Theorem 1.2. π_{ess} is a faithful representation of O_{E}.

In particular, we see that E always admits essential representations.

This paper is structured as follows. In section 2, we adapt to our setting parts of Arveson’s theory of decreasing weights ([1, 2]) and its discrete version due to Fowler ([3]). The adaptation is straightforward, but we nonetheless include full proofs for the reader’s convenience. This will give us a sufficient criterion for a state on T_{E} to give rise, via the GNS representation, to an essential representation of E. In section 3, we show how to construct sufficiently many such states to deduce Theorem 1.2. We end by making some remarks on connections to endomorphisms. We mention that shortly after this paper has been submitted for publication, Michael Skeide ([12]) found a different approach to the results presented herein, which can also be used to settle the W^{∗}-case (see section 4).

I would like to thank Takeshi Katsura for a helpful discussion, Michael Skeide for reading an earlier draft of this paper, and the referee for a careful reading.

2. Functionals on the algebra of operators of bounded support

We fix some terminology about topologies on B(E). Let T_{λ} be a net. We say that T_{λ} → 0 in the strong topology if ∥T_{λ}e∥_{E} → 0 for all e ∈ E, and in the strict topology if ∥T_{λ}A∥_{B(E)}, ∥AT_{λ}∥_{B(E)} → 0 for all A ∈ K(E) (i.e. the strict topology of B(E) coming from its being the multiplier algebra of K(E)). Note that the strong topology is coarser than the strict topology. We say that a linear functional on B(E) is strictly continuous if it is continuous with respect to the strict topology. Suppose φ is a state on K(E), then we can (uniquely) extend φ to a strictly continuous state on B(E) (using a GNS representation). Sesquilinear forms here will always be linear on the right, and conjugate-linear on the left.

Let P_{n} ∈ B(E) be the projection onto the submodule ⨁_{k=0}^{n} E^{⊗k}. Denote J_{n}(E) = P_{n}B(E)P_{n}, and let J_{c}(E) = ⋃_{n=0}^{∞} J_{n}(E) (no closure), the ∗-algebra of operators of bounded support. Let Q_{n} = 1 − P_{n−1}, R_{n} = P_{n} − P_{n−1} (i.e. R_{n} is the projection onto E^{⊗n}, and Q_{n} is the projection onto ⨁_{k=n}^{∞} E^{⊗k}). Let β : B(E) → B(Q_{1}E) be the representation given by β(T) = T ⊗ 1_{E}, where we identify Q_{1}E with E ⊗_{A} E (notice that to do so, we use the nondegeneracy of the left action). We think of β as an endomorphism of B(E), where we identify B(Q_{1}E) with Q_{1}B(E)Q_{1}. We now let δ : B(E) → B(E) be the self-adjoint linear map given by δ(T) = T − β(T). Notice that β(J_{c}(E)) ⊆ J_{c}(E) and therefore also δ(J_{c}(E)) ⊆ J_{c}(E).

Define an unbounded map λ : J_{c}(E) → B(E) by λ(T) = ∑_{k=0}^{∞} β^{k}(T), where the convergence is taken in the strong topology. Now, β(λ(T)) = λ(T)−T, so δ(λ(T)) = T. Similarly, λ(δ(T)) = T, so in particular, we have that λ(J_{c}(E)) ⊇ J_{c}(E). λ is clearly positive, being a sum of homomorphisms.

Notice that we have δ(xy) = xδ(y)+δ(x)y−δ(x)δ(y). If x ∈ J_{n}(E) and y ∈ J_{m}(E), a simple computation shows that

λ(x)λ(y) = λ (xy + ∑_{k=1}^{n} xβ^{k}(y) + ∑_{k=1}^{m} β^{k}(x)y) .

This implies the following lemma. Lemma 2.1. If x ∈ J_{n}(E) and y ∈ J_{m}(E) then λ(x)λ(y) ∈ λ(J_{n+m}(E)). In particular, λ(J_{c}(E)) is a unital ∗-algebra.

Denote T̂_{E} = \overline{λ(J_{c}(E))}. Notice that β preserves T̂_{E}. We shall denote the restriction of β to this C^{∗}-algebra by β as well.

Let ξ ∈ E^{⊗n}, η ∈ E^{⊗m}, then β(T_{ξ}T_{η}^{∗})ζ = T_{ξ}T_{η}^{∗} ⊗ 1_{E}ζ = 0 if ζ ∈ E^{⊗k}, k ≤ m, and otherwise, we have β(T_{ξ}T_{η}^{∗})ζ = T_{ξ}T_{η}^{∗}ζ. Consequently, we have that δ(T_{ξ}T_{η}^{∗}) = ξ ⊗ η^{∗} ∈ K(E). Therefore, we have that δ(T_{E}) ⊆ K(E), and λ(J_{c}(E) ∩ K(E)) is dense in T_{E}. Remark 2.2. The fact that δ(T_{E}) ⊆ K(E) appears in [10], where it is used to obtain an element of KK(T_{E}, A) (shown to be invertible) from the pair (id, β) (in our notation). Definition 2.3. A linear functional ω on J_{c}(E) is said to be decreasing if ω ∘ δ is a positive functional on J_{c}(E).

We recall a simple technical lemma characterizing when positive linear functionals on a normed algebra are bounded (see [2], lemma 4.9.4). We include a proof for the reader’s convenience, but just state it for the unital case, which is the one we need. Lemma 2.4. Let A be a unital normed complex algebra with isometric involution, such that ∥1∥ = 1. Let φ be a linear functional on A, which is positive, in the sense that φ(x^{∗}x) ≥ 0 for all x ∈ A. If lim sup_{n→∞} |φ(x^{n})|^{1/n} ≤ ∥x∥ for all x, then φ is bounded, with norm φ(1).

Proof. By the Cauchy-Schwarz inequality, |φ(x)| = |φ(1 · x)| ≤ (φ(1)φ(x^{∗}x))^{1/2}. Applying this again to the right hand side n times yields

|φ(x)| ≤ φ(1)^{1/2+1/4+...+1/2^{n}}φ ((x^{∗}x)^{2^{n−1}})^{1/2^{n}}

and by taking lim sup, we get that |φ(x)| ≤ φ(1)∥x^{∗}x∥^{1/2} ≤ φ(1)∥x∥, as required. □

The following proposition is a straightforward generalization of [3], Proposition 1.5. Proposition 2.5. ω ∘ δ is positive on λ(J_{c}(E)) if and only if ω is positive and decreasing (i.e. if both ω and ω ∘ δ are positive on J_{c}(E)) in which case ∥ω ∘ δ∥ = ω(P_{0}).

Proof. If ω ∘ δ is positive on λ(J_{c}(E)), then it is also positive on J_{c}(E) ⊆ λ(J_{c}(E)), and ω = ω ∘ δ ∘ λ is positive, as it is a composition of positive functionals.

Now, suppose ω is positive and decreasing. Since ω is positive, we can use it to view J_{c}(E) as an semi-inner product space, and complete it to a Hilbert space H.

Denote by x̂ the image of x ∈ J_{c}(E) in H (i.e. the coset of x module the null-space of ω in the completion). ω is decreasing, so ω(x^{∗}x) − ω(β(x^{∗}x)) ≥ 0 for all x ∈ J_{c}(E). Therefore, the map x̂ ↦ β̂(x) is well defined, and extends (uniquely) to a linear contraction T : H → H.

Set

T_{k} = \begin{cases}T^{k} & | & k ≥ 0 \\ T^{∗−k} & | & k < 0 \end{cases}

Seen as operator valued function ℤ → B(H), the map k ↦ T_{k} is positive definite ([13]). One checks that for x ∈ J_{c}(E), we have

δ(λ(x)^{∗}λ(x)) = x^{∗}λ(x) + λ(x^{∗})x − x^{∗}x = ∑_{n=0}^{∞} x^{∗}β^{n}(x) + ∑_{n=1}^{∞} β^{n}(x^{∗})x

(a finite sum). Notice that

⟨x̂, T_{k}x̂⟩_{H} = \begin{cases}ω(x^{∗}β^{k}(x)) & | & k ≥ 0 \\ ω(β^{−k}(x^{∗})x) & | & k < 0 \end{cases}

So, we have ω ∘ δ(λ(x)^{∗}λ(x)) = ∑_{k=−∞}^{∞} ⟨x̂, T_{k}x̂⟩ ≥ 0 (being the value of the Fourier transform at z = 1 of the finitely supported positive definite function on ℤ given by k ↦ ⟨x̂, T_{k}x̂⟩).

It remains to prove that ω∘δ is bounded. For that, we use Lemma 2.4. Notice that ω ∘ δ(1) = ω(P_{0}). So, it remains to show that for any k, and any x ∈ J_{k}(E), we have lim sup_{n→∞} |ω ∘ δ(λ(x)^{n})|^{1/n} ≤ ∥λ(x)∥. By Lemma 2.1, λ(x)^{n} ∈ λ(J_{kn}(E)). Note also that ∥δ∥ = 2. So, |ω ∘ δ(λ(x)^{n})| ≤ ∥ω|_{J_{kn}(E)}∥∥δ((λ(x)^{n})∥ ≤ 2ω(P_{kn})∥λ(x)∥^{n}. Now, P_{kn} = ∑_{j=0}^{kn} β^{j}(P_{0}). Since ω is decreasing, we have then that ω(P_{kn}) ≤ (kn + 1)ω(P_{0}). So, |ω ∘ δ(λ(x)^{n})|^{1/n} ≤ (2(kn + 1)ω(P_{0}))^{1/n} ∥λ(x)∥, which gives us the required inequality. □

Slightly abusing notation, we denote the (necessarily unique) extention of ω ∘ δ to T̂_{E} by ω ∘ δ as well. Notice that if φ is a positive linear functional on T̂_{E}, then ω = φ ∘ λ is positive and decreasing, so any positive linear functional on T̂_{E} is obtained in this way.

A linear functional ω on J_{c}(E) is said to be locally strict if for all n, the restiction of ω to J_{n}(E) is strictly continuous.

Suppose ω_{1}, ω_{2} are two locally strict decreasing functionals on J_{c}(E), and the restrictions of ω_{i} ∘ δ i = 1, 2 agree on T_{E} = \overline{λ(J_{c}(E) ∩ K(E))}, then ω_{1} and ω_{2} agree on J_{c}(E) ∩ K(E), and since they are locally strict, ω_{1}, ω_{2} agree everywhere.

Suppose now that φ is a positive linear functional on T_{E}. Define ω on J_{c}(E)∩K(E) by ω = φ ∘ λ. ω is positive (and bounded) on J_{n}(E) ∩ K(E), and therefore extends, uniquely, to a strictly continuous functional on J_{n}(E). We view ω as a locally strict functional on J_{c}(E). Proposition 2.6. ω, obtained as described in the previous paragraph, is positive and decreasing.

The proof is an easy generalization of that of [3], Theorem 1.9.

Proof. It suffices to show that ω ∘ δ is positive on λ(J_{c}(E)). Notice that we already know that ω ∘ δ is positive on λ(J_{c}(E) ∩ K(E)). For any x ∈ J_{c}(E), define a function f_{x} : ℤ → ℂ by

f_{x}(k) = \begin{cases}ω(x^{∗}β^{k}(x)) & | & k ≥ 0 \\ ω(β^{−k}(x^{∗})x) & | & k < 0 \end{cases}

As in the proof of proposition 2.5, we see that ω ∘ δ(λ(x)^{∗}λ(x)) = ∑_{k=−∞}^{∞} f_{x}(k) (only finitely summands are non-zero), so it will suffice if we could show that f_{x} is in fact positive definite. Since ω is locally strict, f_{x} is a pointwise limit of functions of the form f_{y}, where y ∈ J_{c}(E) ∩ K(E). It therefore suffices to show that f_{x} is positive definite when x ∈ J_{c}(E) ∩ K(E), and therefore, it suffices to show that the Fourier transform f̂_{x}(z) ≥ 0 for all z ∈ 𝕋.

Suppose x ∈ J_{N}(E) ∩ K(E). For z ∈ 𝕋, let u_{z} = ∑_{n=0}^{N} z^{n}R_{n}. Notice that u_{z}x ∈ J_{N}(E) ∩ K(E), and a simple computation shows that f_{u_{z}x}(k) = z̄^{k}f_{x}(k). So, f̂_{x}(z) = ∑_{k=−∞}^{∞} f_{x}(k)z̄^{k} = ∑_{k=−∞}^{∞} f_{u_{z}x}(k) = ω ∘ δ(λ(u_{z}x)^{∗}λ(u_{z}x)) ≥ 0, as required. □

Thus, for any positive linear functional φ on T_{E} we have a positive linear functional φ̂ on T̂_{E}, given by taking ω = φ ∘ λ, extending it to J_{c}(E), and composing with δ. Note that any approximate unit of A is also an approximate unit of T_{E} (when we view A as a subalgebra of T_{E}), since we assumed that the left action of A on E is non-degenerate. Therefore, any state φ of T_{E} restricts to a state of A. In the above notation, we have that ω(P_{0}) = ∥φ|_{A}∥, so, if φ is a state, then so is φ̂.

Remark 2.7. The map S(T_{E}) → S(T̂_{E}) given by φ ↦ φ̂ is clearly affine, although the reader is cautioned that generally it is not continuous. Definition 2.8. We call a state of T_{E} essential if its associated GNS representation, restriced to the canonical copies of A,E in T_{E}, is essential.

Let φ be a state of T_{E}, and let ω be its associated locally strict functional. Claim 2.9. If φ = φ̂ ∘ β then φ is essential.

Proof. Let (π, H, Ω) be the GNS representation of T_{E} with respect to φ. Let Q′ be the projection onto ⋂_{n=0}^{∞} \overline{π(E^{⊗n})H}, then Q′H is invariant, the restriction of π to Q′H is essential, and Q′H is the largest subspace of H which satisfies this property. Thus, φ is essential if and only if Q′Ω = Ω, i.e. if ∥Q′Ω∥ = ⟨Ω, Q′Ω⟩ = 1. Let Q′_{n} be the projection onto \overline{π(E^{⊗n})H} (so Q′_{n} → Q′ in the strong operator topology, and so, ⟨Ω, Q′_{n}Ω⟩ → ⟨Ω, Q′Ω⟩). If a_{k}^{(n)} is an approximate identity for K(E^{⊗n})

then π (λ(a_{k}^{(n)})) → Q′_{n} in the strong operator topology. Therefore, ⟨Ω, Q′_{n}Ω⟩ =

lim_{k} ⟨Ω, λ(a_{k}^{(n)})Ω⟩ = lim_{k} φ(λ(a_{k}^{(n)})) = lim_{k} ω(a_{k}^{(n)}) = ω(R_{n}) = φ̂(Q_{n}) = φ̂ ∘ β^{n}(1). Thus, if φ = φ̂ ∘ β, then, if (a_{k}) is an approximate unit for A, then φ̂ ∘ β^{n}(1) = lim_{k} φ̂ ∘ β^{n}(a_{k}) = lim_{k} φ(a_{k}) = 1 and so φ is essential. □

Claim 2.10. φ = φ̂ ∘ β if and only if ω = ω ∘ β.

Proof. Note that λ ∘ β = β ∘ λ, and δ ∘ β = β ∘ δ. So if φ = φ̂ ∘ β then ω = φ̂ ∘ λ, and ω ∘ β = φ̂ ∘ λ ∘ β = φ̂ ∘ β ∘ λ = φ ∘ λ = ω. Conversely, if ω = ω ∘ β then φ = ω ∘ δ = ω ∘ β ∘ δ = φ̂ ∘ β. □

3. Construction of essential states

By Claims 2.9 and 2.10, we can construct essential states on T_{E} as follows. Suppose we have a sequence of strictly continuous states ν_{n} on B(E^{⊗n}), such that ν_{n+1} ∘ β = ν_{n} for all n. Viewing each such ν_{n} as a positive linear functional on J_{c}(E) (with support R_{n}), we can form ω = ∑_{n} ν_{n}. ω, then, is clearly a locally strict decreasing functional, and ω ∘ β = ω, so the associated state φ on T_{E} is essential.

Let ν_{0} be an arbitrary state on A (extended as strictly continuous state to M(A) ≅ B(E^{⊗0}) if A is non-unital). We can push forward ν_{0} to β(B(E^{⊗0})), thought of as a C^{∗}-subalgebra of B(E), and extend it to a state ν_{1} on B(E). Now push ν_{1} to the subalgebra β(B(E)) of B(E^{⊗2}), etc. The problem is that while we can always extend the state ν_{n}, thought of as a state of β(B(E^{⊗n})) to all of B(E^{⊗n+1}), it is not necessarily the case that the extension can be chosen to be strictly continuous, as the following example shows. Example 3.1. Let A = C([0, 1]), E = C([0, 1))⊕C((0, 1]) (with the standard inner product), and take the left action to be

f · (g_{1}, g_{2})(x) = (f(x/2)g_{1}(x), f((x + 1)/2)g_{2}(x))

Any state ν_{0} on A is given by a measure on [0, 1], and it is easy to check that we can choose a strictly continuous extension to B(E) if and only if 1/2 is not an atom of this measure.

We will show, though, that there is a dense set of states of A which do admit a strictly continuous sequence of extensions as above. To that end, we start with some technical lemmas. Lemma 3.2. Let

X_{0} ⟵^{f_{0,1}} X_{1} ⟵^{f_{1,2}} X_{2} ⟵^{f_{2,3}} ⋯

be an inverse system of surjective continuous maps, where X_{n}, n = 0, 1, ... are compact Hausdorff spaces. Denote by f_{n,m} : X_{m} → X_{n}, n < m the map obtained by composition (and f_{n,n} will be the identity). Let X_{∞} be the inverse limit, and let f_{n,∞} : X_{∞} → X_{n} be the canonical maps.

Suppose we have dense open sets U_{n} ⊆ X_{n}, n = 0, 1, 2, ... such that f_{n,n+1}(U_{n+1}) ⊆ U_{n} for all n, then f_{n,∞}^{−1}(U_{n}) are dense open sets in X_{∞}.

In particular, ⋂_{n=0}^{∞} f_{n,∞}^{−1}(U_{n}) is dense in X_{∞}. Thus, there exists a dense collection of x_{0} ∈ X_{0} for which there is a sequence x_{n} ∈ U_{n} n = 1, 2... with f_{n,m}(x_{m}) = x_{n} for all n < m.

Proof. The only thing which might not seem immediate is that f_{n,∞}^{−1}(U_{n}) are dense. So, we need to show that any non-empty open set in X_{∞}, has non-empty intersection with f_{n,∞}^{−1}(U_{n}). It suffices to do so for basic open sets: let V_{k} ⊆ X_{k}, k = 0, ..., m (for an arbitrary m, which we may assume is ≥ n) be open sets such that ⋂_{k=0}^{m} f_{k,m}^{−1}(V_{k}) ≠ ∅. It suffices to show that f_{n,∞}^{−1}(U_{n}) ∩ ⋂_{k=0}^{m} f_{k,∞}^{−1}(V_{k}) ≠ ∅. So,

f_{n,∞}^{−1}(U_{n})∩ ⋂_{k=0}^{m} f_{k,∞}^{−1}(V_{k}) ⊇ f_{m,∞}^{−1}(U_{m})∩ ⋂_{k=0}^{m} f_{m,∞}^{−1} (f_{k,m}^{−1}(V_{k})) = f_{m,∞}^{−1} (U_{m} ∩ ⋂_{k=0}^{m} f_{k,m}^{−1}(V_{k}))

which is nonempty since U_{m} is dense in X_{m} and ⋂_{k=0}^{m} f_{k,m}^{−1}(V_{k}) is open and nonempty. □

Lemma 3.3. Let B be a C^{∗}-algebra. The strictly continuous states on M(B) are a co-meager (dense G_{δ}) set in the state space of M(B).

Proof. Consider the natural embedding of B in B^{∗∗} (the double dual). The states of B (which we identify with the strictly continuous states on M(B), as well as the normal states on B^{∗∗}) are dense in the state space of B^{∗∗}, and therefore also in the state space of M(B). Let {e_{λ}}_{λ∈Λ} be an increasing approximate identity for B. A state φ on M(B) is strictly continuous if and only if sup_{λ∈Λ} φ(e_{λ}) = 1. So, U_{m} = {φ ∈ S(M(B)) | ∃λ ∈ Λ s.t. φ(e_{λ}) > 1 − 1/m} is open (and dense, since it contains the strictly continuous states), so the strictly continuous states are the intersection of open dense sets ⋂_{m} U_{m}, as required. □

Lemma 3.4. Suppose A is a subalgebra of M(B) which is non-degenerate, in the sense that AB = B. If φ is a state on M(B), then ∥φ|_{A}∥ ≥ ∥φ|_{B}∥. In particular, if φ is a strictly continuous state of M(B), then the restriction of φ to A is a state of A.

Proof. Let a_{λ} be an approximate unit for A, then non-degeneracy implies that for any b ∈ B, ∥a_{λ}b − b∥ → 0. Let r = ∥φ|_{B}∥. Given ϵ > 0, choose b ∈ B positive of norm 1 such that φ(b) > r − ϵ/2, and λ such that ∥a_{λ}b − b∥ < ϵ/2, so

r − ϵ < φ(b) − ϵ/2 < |φ(a_{λ}b)| ≤ φ(a_{λ}^{∗}a_{λ})^{1/2}φ(b^{∗}b)^{1/2} ≤ φ(a_{λ}^{∗}a_{λ})^{1/2}r^{1/2}.

It follows that ∥φ|_{A}∥ ≥ r, as required. □

Lemma 3.5. Let A_{0}, A_{1}, ... be C^{∗}-algebras, and suppose we are given non-degenerate injective homomorphisms A_{n} → M(A_{n+1}), n = 0, 1, ..., that is, we have a diagram as follows (where the map A_{n} → M(A_{n}) is the inclusion).

\xymatrix{M(A_{0}) \ar[r]^{φ_{1,0}} & M(A_{1}) \ar[r]^{φ_{2,1}} & M(A_{2}) \ar[r]^{φ_{3,2}} & ⋯ \\ A_{0} \ar[u] \ar[ur] & A_{1} \ar[u] \ar[ur] & A_{2} \ar[u] \ar[ur] & ⋯}

We use the same notation for a state on A_{n} and its strictly continuous extension to M(A_{n}). It follows that there is a dense set of states S_{0} of A_{0}, such that each ν_{0} ∈ S_{0} admits a sequence of strictly continuous states ν_{n} ∈ S(M(A_{n})) with ν_{n+1} ∘φ_{n+1,n} = ν_{n}.

Proof. Denote by φ_{m,n} : M(A_{n}) → M(A_{m}), n < m the map obtained by composition, and by f_{n,m} = φ_{m,n}^{∗} : S(M(A_{m})) → S(M(A_{n})) the induced map on the state space. As in the proof of Lemma 3.3, set

U_{m} = {φ ∈ S(M(A_{m})) | ∃λ ∈ Λ s.t. φ(e_{λ}) > 1 − 1/(m + 1)}

where {e_{λ}}_{λ∈Λ} is some approximate unit for A_{m}. U_{m} is open and dense in S(M(A_{m})), and we have f_{n,n+1}(ν) ∈ U_{n} (by Lemma 3.4) for all ν ∈ U_{n+1}. It follows now from Lemma 3.2 that there are states ν_{n} ∈ U_{n}, n = 0, 1, 2, ... such that f_{n,m}(ν_{m}) = ν_{n} for all n < m. Since ν_{n} ∈ ⋂_{m≥n} f_{n,m}(U_{m}), it follows that ν_{n} must be strictly continuous, as required. □

Corollary 3.6. There is a dense set of states ν_{0} of A which admit a sequence of strictly continuous exentsions ν_{n} ∈ S(B(E^{⊗n})), as described in the beginning of the section.

Proof. Apply the previous lemma to

\xymatrix{M(A) \ar[r] & B(E) \ar[r] & B(E^{⊗2}) \ar[r] & ⋯ \\ A \ar[u] \ar[ur] & K(E) \ar[u] \ar@{-->}[ur] & K(E^{⊗2}) \ar[u] \ar@{-->}[ur] & ⋯}

□

Let ψ be a representation of E on H. For z ∈ 𝕋, we can define a new representation ψ_{z} of E on H given by ψ_{z}(a) = ψ(a) (a ∈ A), and ψ_{z}(e) = zψ(e) (e ∈ E). Clearly, if ψ is essential then so is ψ_{z}. Recall the following theorem, from [4] (section 4). Let γ_{z} be the automorphism of O_{E} given by mapping e ↦ ze, and fixing A (where we view A and E as contained in O_{E}). Theorem 3.7 (Gauge invariant uniqueness theorem). Let π be a representation of O_{E}. If there is a point-norm continuous action α_{z} : 𝕋 → Aut(π(O_{E})) such that α_{z} ∘ π = π ∘ γ_{z} for all z ∈ 𝕋, and if π|_{A} is faithful, then π is faithful.

We can now prove the main theorem.

Proof of Theorem 1.2. If φ is a state of T_{E}, and ω is its corresponding decreasing functional, and a ∈ A, then φ(a) = ω(a), where A is thought of as a subalgebra of T_{E} and as J_{0}(E), respectively. Let S be the set of states ν_{0} of A which admit strictly continuous extensions ν_{1}, ν_{2}, ... as above. Since S is dense, it follows that for any a ∈ A_{+} and ϵ > 0 there is an essential representation ψ of E such that ∥ψ(a)∥ > ∥a∥−ϵ (we pick ν_{0} ∈ S such that ν_{0}(a) > ∥a∥−ϵ, let φ be the corresponding state of T_{E}, let π_{φ} be the GNS representation, and take ψ to be the representation of E associated to π_{φ}). Thus, if we let π_{ess} be the universal essential representation as above, we see that π_{ess}|_{A} is faithful. By the remarks preceeding Theorem 3.7, π_{ess} satisfies the hypothesis of the theorem, and therefore, π_{ess} is faithful, as required. □

4. Concluding remarks

Any representation ψ of E on H gives rise to an endomorphism α of ψ(A)′, by α(x)ψ(e)ξ = ψ(e)xξ (for x ∈ ψ(A)′, e ∈ E, ξ ∈ H, and defined to be 0 on ψ(E)H^{⊥}; the reader is referred to [7] for further details, including why this is well defined). α is unital if and only if ψ is essential.

Now, if M is a von-Neumann subalgebra of B(H) and α is a normal endomorphism of M′, then E_{α} = {T ∈ B(H) | α(x)T = T x} is naturally a W^{∗}-correspondence over M, where by a W^{∗}-correspondence we mean a C^{∗}-correspondence E over a von-Neumann algebra M, where E is self-dual (see [9]), and the left action M → B(E) is normal (recall that in this case, B(E) is naturally a von-Neumann algebra). α is unital if and only if this concrete representatation of E_{α} is essential. It is thus interesting to know whether any W^{∗}-correspondence arises as the intertwining space of a unital normal endomorphism. Recall that if E is a self-dual Hilbert module over M, then E has a predual, which gives it a weak^{∗}-topology (which we’ll call the ultraweak topology). Thus, the concrete question is the following: does any W^{∗}-correspondence E admit an essential representation ψ, such that ψ_{M} and ψ_{E} are continuous when E, M and B(H) are endowed with their ultraweak topologies?

Our construction guarantees the existence of essential representations, however it is unclear if they are ultraweakly continuous. It is worth noting that in this setting, if ν_{0} is a normal state of M, then we can obtain a sequence of strictly continuous states ν_{n} ∈ B(E^{⊗n}) as discussed in section 3. Thus, the corresponding state φ of the C^{∗}-algebra T_{E} will satisfty that its restriction to M ⊂ T_{E} be ultraweakly continuous. However, this in its own does not appear to guarantee that the restriction of the associated GNS representation to M be ultraweakly continuous. As mentioned in the introduction, Skeide ([12]) very recently settled this question in the affirmative.

Much of the motivation for this present work comes from Arveson’s work on E_{0}-semigroups, i.e. continuous one-parameter semigroups of unital endomorphisms of B(H) (see [2]). If α is an E_{0}-semigroup of B(H), the intertwiners of each α_{x}, x > 0, form a Hilbert space in B(H), and together, they form a so-called product system of Hilbert spaces. Much of this generalizes to E_{0}-semigroups of more general von-Neumann algebras (and certain C^{∗}-algebras), by replacing Hilbert spaces by correspondences. In [1], Arveson showed that every product system of Hilbert spaces is associated to an E_{0}-semigroup in this fashion, by constructing essential representations of product systems. There is hope that at least for the finite dimensional case, it might be possible to extend Arveson’s result, by constructing a continuous analogue of the argument herein for certain continuous analogues of Pimsner’s Toeplitz algebras (considered in [5, 6]), however this goes beyond the scope of this paper. We refer the reader to [8, 11] and references therein for more information on product systems of Hilbert modules in the context of E_{0}-semigroups.

References

[1] Arveson, W.B., Continuous analogues of Fock space. IV. Essential states. Acta Math. 164 (1990), no. 3-4, 265–300.

[2] Arveson, W.B., Noncommutative Dynamics and E-semigroups, Springer Monographs in Mathematics, Springer-Verlag, 2003.

[3] Fowler, N.J., States of Toeplitz-Cuntz algebras. J. Operator Theory 42 (1999), no. 1, 121– 144.

[4] Fowler, N.J, Muhly, P.S., Raeburn, I., Representations of Cuntz-Pimsner algebras. Indiana Univ. Math. J. 52 (2003), no. 3, 569–605.

[5] Hirshberg, I., C^{∗}-algebras of Hilbert module product systems, J. Reine. Angew. Math. 570 (2004), 131–142.

[6] Hirshberg, I., On the universal property of Pimsner-Toeplitz C^{∗}-algebras and their continuous analogues, J. Funct. Anal. 219 (2005), no. 1, 21–33.

[7] Muhly, P., and Solel, B., Tensor algebras, induced representations and the Wold decomposition, Can. J. Math. 51 (1999), no. 4, 850–880.

[8] Muhly, P., and Solel, B., Quantum Markov processes (correspondences and dilations). Internat. J. Math. 13 (2002), no. 8, 863–906.

[9] Paschke, W.L., Inner product modules over B^{∗}-algebras. Trans. Amer. Math. Soc. 182 (1973), 443–468.

[10] Pimsner, M.V., A class of C^{∗}-algebras generalizing both Cuntz-Krieger algebras and crossed products by ℤ. Free probability theory (Waterloo, ON, 1995), 189–212, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997.

[11] Skeide, M., Dilation theory and continuous tensor product systems of Hilbert modules. Quantum probability and infinite dimensional analysis (Burg, 2001), 215–242, QP–PQ: Quantum Probab. White Noise Anal., 15, World Sci. Publishing, River Edge, NJ, 2003.

[12] Skeide, M., Unit vectors, Morita equivalence and endomorphisms, preprint (2004) arXiv:math.OA/0412231.

[13] Sz.-Nagy, B., and Foias, C., Analyse harmonique des opérateurs de l’espace de Hilbert, Akadémiai Kiadó, Budapest, 1967.

Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

E-mail address: ilan@imada.sdu.dk