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On the universal property of Pimsner-Toeplitz C*-algebras and their continuous analogues

We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.


Introduction
Let E be a Hilbert module over A, equipped with a left action of A via adjointable operators. We shall refer to such an E as a Hilbert bimodule. We assume that E is full, i.e. E, E = A. We make no further assumptions on the left action of A. Let B be a C * -algebra. A covariant homomorphism ψ of E into B is a C-linear map ψ E : E → B along with a homomorphism ψ A : A → B such that for all e, f ∈ E, a, b ∈ A we have ψ E (aeb) = ψ A (a)ψ E (e)ψ A (b), ψ E (e) * ψ E (f ) = ψ A ( e, f ). In the sequel, we will write ψ for both ψ A and ψ E , when it causes no confusion. When B = B(H) for a Hilbert space H, we call a covariant homomorphism a representation.
Remark 1.1. In [MS] what we call a representation is called an isometric covariant representation. Since we won't deal with other kinds of representations considered in [MS], we shall use the shorter terminology.
We recall Pimsner's construction. Let E = ∞ n=0 E ⊗n (where we take E ⊗0 = A). For e ∈ E, let T e ∈ B(E) be given by T e (ξ) = e ⊗ ξ. The map sending e → T e , A ∋ a → left multiplication by a is a covariant homomorphism of E into B(E). We let T E be the C * -subalgebra of B(E) generated by {T e | e ∈ E}. We will lighten notation by identifying e ∈ E with T e , and a with multiplication by a in T E , when it does not lead to confusion.
Note that if π : T E → B(H) is a representation, then the restrictions of π to E and A form a representation of E. Our goal in the first section will be is to give a new proof of the following theorem -a restatement of a theorem of Pimsner ([P], Theorem 3.4) -which shows that any representation arises in this manner.
Theorem 1.2 (Pimsner). Let ψ be a representation of E on a Hilbert space H. The map e → ψ(e) extends to a homomorphism T E → B(H).
Remark 1.3. Pimsner's proof relies on the conditional expectation map of the algebra T E onto the fixed point subalgebra for the dual action of T, generalizing the proof for the Cuntz algebras from [Cu]. The motivation leading to the proof presented herein was to obtain the continuous analogue, Theorem 1.8 below. The continuous analogues, described below, admit an analogous action of R, rather than T. Thus, one cannot obtain a bounded expectation map by averaging the group action. We note that an unbounded expectation map has been used by Zacharias to study Arveson's spectral C *algebras in [Z]. We refer the reader to [A2] for more details on the spectral C * -algebras, and to [HZ] for a recent survey.
In [H], we considered a certain continuous analogue of the algebras T E , generalizing to the context of Hilbert modules Arveson's spectral C * -algebras (see [A2]). We recall the definitions.
Definition 1.4. Let A be a separable C * -algebra. A measurable bundle of Hilbert A-bimodules over Ω, E, is a collection {E x | x ∈ Ω} of right Hilbert A-modules with left actions via adjointable operators, along with a distinguished vector subspace Γ of Π x∈Ω E x (called the set of measurable sections) such that (1) For any ξ ∈ Γ, a ∈ A, the functions x → ξ(x), ξ(x) , x → aξ(x), ξ(x) are measurable (as functions Ω → A).
We refer the reader to the appendix of [H] for more details.
Definition 1.5. Let A be a separable C * -algebra. A product system of Abimodules E is a measurable bundle of A-bimodules over R + , along with a multiplication map E × E → E, which descends to an isomorphism E x ⊗ A E y → E x+y for all x, y ∈ R + (where E x is the fiber over x), and is measurable in the sense that if ξ is a measurable section and e ∈ E y then the sections The elements e ∈ E act on ⊕ R + E x dx on the left as adjointable operators, which we denote W e , or by abuse of notation, just e. Note that e Ex ≥ W e B( ⊕ R + Exdx) . Denote by L 1 (E) the space of measurable sections ξ that satisfy R + ξ(x) dx < ∞.
We refer the reader to [H] for examples, and a discussion of the K-theory of W E .
Definition 1.7. Let E be a product system of A-bimodules. A representation ψ of E on H is a map ψ E : E → B(H), along with a representation ψ A : A → B (H) such that (1) The restriction of ψ to each fiber of E is a representation of the fiber.
(2) For any e, f ∈ E, ψ(e)ψ(f ) = ψ(ef ). ( Our goal in the second part of this paper will be to prove the following continuous analogue of Theorem 1.2. This theorem generalizes a theorem of Arveson ([A2], Theorem 4.6.6) from the case of product systems of Hilbert spaces. Specializing our proof below to the case of Hilbert spaces will give a simpler approach to Arveson's theorem.
is well defined and extends to an isometry If ψ is a representation of E, n > 0, then we can define a representation of E ⊗n by e 1 ⊗ · · · ⊗ e n → ψ(e 1 )ψ(e 2 ) · · · ψ(e n ). We will denote this representation by ψ as well.
We may assume without loss of generality that ψ A is non-degenerate, and we make this assumption throughout (i.e., we assume throughout that ψ(A)H = H).
If π is a representation of A on a Hilbert space H, then we can define a representation T ⊗ A 1 of E on E ⊗ π H (this is called an induced representation in [MS]). We note that clearly, T ≻ T ⊗ A 1. The following lemma generalizes the fact that any isometry S which satisfies S n S n * → 0 in the strong operator topology is unitarily equivalent to a direct sum of copies of the unilateral shift on ℓ 2 . The reader can find a proof in [MS].
for all e ∈ E, i.e. it implements a unitary equivalence between the covariant representations T ⊗ A 1 H 0 and ψ.
Corollary 2.4. Let ψ be as in Lemma 2.3, then T ≻ ψ. Now let ψ be any (non-degenerate) representation of E on H. By Corollary 2.4, to prove Theorem 1.2, it suffices to show that ψ is majorized by a representation which satisfies the condition of Lemma 2.3.
Lemma 2.6. If A is in the * -algebra generated by {ψ(e) | e ∈ E} then P + AP + = A Proof. Note that any operator of the formψ(e) commutes with all operators of the form 1 H ⊗ U n , n ∈ Z. Therefore A commutes with 1 ⊗ U n , n ∈ Z as well. Let P n denote the projection onto H ⊗ ℓ 2 ({n, n + 1, ...}) (so P 0 = P + ). We have (1 ⊗ U m )P n (1 ⊗ U m ) * = P n+m for all n, m ∈ Z, and therefore we have (1 ⊗ U n )P + AP + (1 ⊗ U n ) * = P n AP n so P + AP + = P n AP n for all n ∈ Z. Since P n → 1 ℓ 2 (Z) as n → −∞ in the strong operator topology, we have Corollary 2.7. ψ + ≻ψ.
Remark 2.8. The proof in this section was obtained in the course of the author's dissertation work under the supervision of W.B. Arveson, and is motivated by ideas from [A2].
3. The continuous case -proof of Theorem 1.8 The approach here will differ from the proof above for the discrete case, in that we do not have a continuous analogue of Lemma 2.3 (see Remark 3.9 below). Aside for that, we shall follow a similar path.
We begin by giving the analogue of Definition 2.1.
Definition 3.1. Let ψ, ρ be two representations of a product system E (over A) on H. We say that ψ majorizes ρ, and write ψ ≻ ρ, if for any f 1 , ..., f n ∈ L 1 (E) and any polynomial p in 2n non-commuting variables, we have In other words, ψ ≻ ρ if there is a (necessarily unique) homomorphism As in the discrete case, the relation ≻ is clearly transitive. Theorem 1.8 states that W ≻ ψ for any representation ψ of E.
As in the discrete case, we may assume without loss of generality that ψ A is non-degenerate, and we make this assumption throughout. Let ψ be a representation of E on H, and let H ′ be invariant for ψ, then we have a representation of E on H ′ by restriction. Let P be the projection onto H ′ , and let ψ ′ denote the restriction, then ψ ′ (f ) = ψ(f )P . Notice that if H ′ is furthermore reducing, then ψ ≻ ψ ′ .
We will make use of the following approximation lemma. The proof is straightforward, and we leave it to the reader. Lemma 3.3. Let ψ be a representation of E on H. Suppose that there is a sequence of projections P n → 1 in the strong operator toplogy, such that P n H is invariant for ψ for all n. Denote by ψ n the restricted representation of ψ to P n H. For any polynomial p(x 1 , ..., x 2k ) in 2k non-commuting variables and Consequently, if ρ is a representation of E such that ρ ≻ ψ n for all n then ρ ≻ ψ.
If π is a representation of A on H, we can form a representation W ⊗ A 1 of W E on ⊕ R + E x dx⊗ π H, as in the discrete case. We clearly have W ≻ W ⊗ A 1. Let S x : L 2 (R + ) → L 2 (R + ) denote the unilateral shift semigroup. We form a representation ψ + on H ⊗ L 2 (R + ) by ψ + (e) = ψ(e) ⊗ S x (e ∈ E x ), ψ + (a) = ψ(a) ⊗ 1. The following Lemma is a straightforward generalization, with a small improvement, of an argument in the proof of [A2], Theorem 4.4.3. We include a full proof for the reader's convenience.
(1) Let ψ be a (non-degenerate) representation of E on a Hilbert space H. There is a unique isometry C : (2) H # is invariant for the representation ψ + . Denote the restriction of ψ + to H # by ψ # . C implements a unitary equivalence between ψ # and W ⊗ A 1 of E on Proof. C is well defined on the given domain, which is total in To show that C extends to an isometry, it suffices to check inner products on those vectors. So, for f, g ∈ L 1 (E) such that R + f (x) 2 dx < ∞, R + g(x) 2 dx < ∞, and ξ, η ∈ H, we have: ψ(g(x)) ⊗ A η Ex⊗ A H and now integrating both sides dx gives the required identity.
To prove that C has the required range, we first note that the range of C is clearly contained in H # . For the converse, suppose g ∈ H # is orthogonal to the range of C. We must prove that g = 0.
Let e 1 (x), e 2 (x), ... be a sequence of measurable sections of E such that for all x, {e 1 (x), e 2 (x), ...} are total in E x (we are guaranteed the existence of such sequence by the definition of a product system). We may assume that all those sections are bounded, without loss of generality. Let u 1 , u 2 , ... be a dense sequence in L 1 (R + ) ∩ L 2 (R + ). Let ξ 1 , ξ 2 , ... be a dense sequence in H. So, for all m, n, p, we have: Since u 1 , u 2 , ... are dense in L 2 (R + ), we see that ψ(e n (x))ξ p , g(x) = 0 a.e.
Finally, we need to check that C(W e ⊗ A 1) = (ψ(e) ⊗ S x )C for all e ∈ E x , x ∈ R + . It suffices to check this for vectors of the form f ⊗ A ξ as in the statement. Indeed, Lemma 3.5. Let ψ be a representation of E on H. Let ψ + , ψ # , H # be as in Lemma 3.4, then ψ + ≈ ψ # .
Let U x be the bilateral shift group on L 2 (R). We form a representationψ of E on H ⊗ L 2 (R) byψ(e) = ψ(e) ⊗ U x (e ∈ E x ),ψ(a) = a ⊗ 1. Using the Fourier transform, we see thatψ andψ are unitarily equivalent, soψ ≻ ψ.
Let V x = 1 H ⊗S x , and let P + the projection of H ⊗L 2 (R) onto H ⊗L 2 (R + ) The following are immediate analogues of Observation 2.5 and Lemma 2.6 above (and immediate generalizations of 4.5.3 and 4.5.4 in [A2]). We leave the simple proofs to the reader.
Remark 3.9. There is a continuous analogue of the Wold decomposition, due to Cooper ([Co]), which states that if S x (x > 0) is a strongly continuous semigroup of isometries on a Hilbert space, then S x is unitarily equivalent to a direct sum of a one-parameter unitary group and copies of the unilateral shift semigroup on L 2 (R + ). Unlike the case of a single bimodule, this does not quite generalize to product systems. There is an approximate version, due to Arveson ([A1,A2] for product systems of Hilbert spaces. Arveson's theorem states the following. Let E is a product system of Hilbert spaces, and let ψ be a representation of E on H such that x>0 ψ(E x )H = {0}. For any ε > 0, let H ε = ψ(E ε )H, and let ψ ε be the restriction of ψ to H ε , then ψ ε is unitarily equivalent to a direct sum of the regular representation of E (on ⊕ R + E x dx, which is a Hilbert space here). However, Arveson showed in [A1] that H ε cannot be replaced by H in the theorem. This theorem was used by Arveson to prove the special case of Theorem 1.8 for Hilbert spaces. We do not know if the generalization of Arveson's theorem to the case of Hilbert modules holds.