Etienne Blanchard, Kenneth Dykema

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algebras

Etienne Blanchard, Kenneth Dykema

To cite this version:

Etienne Blanchard, Kenneth Dykema. Embeddings of reduced free products of operator algebras.

Pacific J. Math., 2001, 199, pp.1–19. �hal-00922864�

PRODUCTS OF OPERATOR ALGEBRAS

Etienne F. Blanchard, Kenneth J. Dykema¹

2 June, 1999

Abstract. Given reduced amalgamated free products of C^∗–algebras (A, φ) = ∗

ι∈I(A^ι, φι) and (D, ψ) = ∗

ι∈I(D^ι, ψ^ι), an embedding A ֒→ D is shown to exist assuming there are conditional expectation preserving embeddingsAι֒→Dι. This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, the reduced amalgamated free product of von Neumann algebras is defined in the general case and analogues of the above mentioned results are proved for von Neumann algebras.

Introduction.

We begin with some standard facts about freeness in groups, analogues of which we will consider in C^∗–algebras. If H is a subgroup of a group G and if G_ι is a subgroup of G containingH for everyιin some index setI, let us say that the family (G_ι)_ι∈I isfree overH (or free with amalgamation over H) if g1g2· · ·g_n ∈/ H whenever g_j ∈ G_ιj\H for some ι_j ∈ I withι16=ι2, ι26=ι3, . . . , ι_n−16=ι_n. For example, if

G= (∗_H)_ι∈IG_ι (1)

is the amalgamated free product of groups, then the family (G_ι)_ι∈I of subgroups is free over their common subgroup H. The amalgamated free product of groups (1) has the universal property that ifK is any group and ifπ_ι:G_ι→Kare group homomorphisms (ι∈I) such that the restriction π_ι↾_H is the same for all ι∈I then there is a group homomorphismπ:G→K such that π↾_Gι =π_ι for every ι∈I. If, moreover, each of the homomorphisms π_ι is injective and if the family π_ι(G_ι)

ι∈I of images is free overπ(H) then the homomorphismπis injective onG.

1991Mathematics Subject Classification. .

1Partially supported as an invited researcher funded by CNRS of France.

Typeset byAMS-TEX

1

Given a unital C^∗–algebra B and for each ι in some index set I a unital C^∗–algebra A_ι containing B as a unital C^∗–subalgebra,² there is a universal amalgamated free product C^∗–algebra [3] A= (∗_B)_ι∈IA_ι of the A_ι overB, satisfying a suitable universal property like that for the amalgamated free product of groups. If G is the amalgamated free product of groups (1) taken with discrete topology, then the full group C^∗–algebraC^∗(G) is the universal amalgamated free product of full group C^∗–algebrasC^∗(G_ι) over their common C^∗–subalgebra C^∗(H).

There is areducedamalgamated free product of C^∗–algebras, invented by Voiculescu [27], (and independently and less generally by Avitzour [2]), which is related to the amalgamated free product of groups via the reduced group C^∗–algebra instead of via the full group C^∗– algebra. This reduced amalgamated free product construction is natural in the context of Voiculescu’s noncommutative probabilistic notion of freeness, which is in turn an abstraction in the context of operator algebras of the phenomenon of freeness in groups. IfA is a unital C^∗–algebra having a unital C^∗–subalgebra B and a conditional expectation φ : A → B, and if A_ι is an intermediate C^∗–subalgebra, B ⊆ A_ι ⊆ A, for every ι in some index set I, then the family (A_ι)_ι∈I is said to be free with respect to φ if φ(a1a2· · ·a_n) = 0 whenever a_j ∈A_ιj∩kerφfor some ι_j ∈I withι16=ι2, ι26=ι3, . . . , ι_n−16=ι_n. The motivating example is for the reduced C^∗–algebraA=C_r^∗(G) of a groupGarising as an amalgamated free product of groups as in (1) and taken with discrete topology; thus A is the closed linear span of the image of the left regular representationλ:G→L(ℓ²(G)); lettingB = spanλ(H)⊆A, letting A_ι = spanλ(G_ι) ⊆ A, so that B ∼= C_r^∗(H) and A_ι ∼= C_r^∗(G_ι), and taking the conditional expectationτ_H^G :A→B given by, forg∈G,

τ_H^G(λ(g)) =

λ(g) ifg∈H 0 ifg /∈H, the family (A_ι)_ι∈I is free with respect to τ_H^G.

Let us now describe Voiculescu’s reduced amalgamated free product construction. IfB is a unital C^∗–algebra, if I is a set and if, for everyι∈I,A_ι is a unital C^∗–algebra containing B as a unital C^∗–subalgebra and having a conditional expectation φ_ι :A_ι → B whose GNS representation³ is faithful, then there is a unital C^∗–algebra A containing B as a unital C^∗– subalgebra, having a conditional expectationφ:A→B and having embeddings (i.e. injective

2By a unital C^∗–subalgebra of a C^∗–algebra A we will mean a C^∗–subalgebra B of A containing the identity element ofA.

3The GNS representation of φιis the canonical representation of Aι as bounded adjointable operators on the HilbertB–moduleL²(A^ι, φ^ι)

∗–homomorphisms) A_ι ֒→ A that restrict to the identity map on B, this setup being unique such that the following hold:

(i) φ↾_Aι =φ_ι, for every ι∈I,

(ii) the family (A_ι)_ι∈I is free with respect to φ, (iii) A is the C^∗–algebra generated byS

ι∈IA_ι, (iv) the GNS representation ofφis faithful (on A).

We denote this reduced amalgamated free product by (A, φ) = ∗

ι∈I(A_ι, φ_ι).

If the C^∗–algebra over which one amalgamates is the scalars, B = C, then the φ_ι are states and the construction is called simply the reduced free product. For the actual construction, see [27]. Details of Voiculescu’s construction are reviewed in [16,§1], and we will in this note abide by the notation used there.

If we consider for the moment only the case when B = C, i.e. simply the reduced free product of C^∗–algebras, we may ask: to what extent does the reduced free product of C^∗– algebras have a universal property, analogous to those for the free product of groups and the universal free product of C^∗–algebras? Since the reduced free product of C^∗–algebras frequently gives rise to simple C^∗–algebras, (see [2], [19], [14], [15] and [10]), it is clear that any universal property for the reduced free product should be quite a bit more restrictive in character than for the universal free product. At first glance, it seems plausible that the reduced free product could have the universal property, which could be called the universal property for state preserving and freeness preserving∗–homomorphisms, that would be implied by a positive answer to the following question.

Question 1. If

(A, φ) = ∗

ι∈I(A_ι, φ_ι)

is a reduced free product of C^∗–algebras, where theφ_ι are states on the unital C^∗–algebras A_ι having faithful GNS representations, and ifD is a unital C^∗–algebra with a state ψ and with unital ∗–homomorphismsπ_ι:A_ι →D such that

(i) ψ◦π_ι=φ_ι for every ι∈I, (ii) the family π_ι(A_ι)

ι∈I is free with respect to ψ,

does it follow that there is a∗–homomorphism π:A→D such that, denoting by α_ι:A_ι→A the injective ∗–homomorphisms arising from the free product construction, π◦α_ι = π_ι for everyι∈I? (Note thatπ would necessarily be injective.)

As observed in [20, 1.3], the answer to Question 1 is “yes” if the stateψon Dis assumed to be faithful, (and a similar result holds in the amalgamated case). However, in [20, 1.4] an elementary example was given for which the answer to Question 1 is “no.” Unfortunately, in the printed version of [20], notational errors (for which the authors of [20] take full responsibility) were introduced into Example 1.4. We therefore repeat this example below.

Example 2([20]). If 0< γ <1 we denote by

p

Cγ ⊕ C

1−γ (2)

the pair (B, τ) where B is the two–dimensional C^∗–algebra with a minimal projection p and whereτ is the state on B satisfying τ(p) =γ. Let

(A1, φ1) = (C^p

3/4⊕ C

1/4), (A2, φ2) = (C^q

2/3⊕ C

1/3) and

(A, φ) = (A1, φ1)∗(A2, φ2).

Thenq is Murray–von Neumann equivalent in A to a proper subprojection of p; (see[14, 2.7]

or [1] for this result). Let D = M2(A) and let ψ be the state on D given by ψ(^b_b^11b12

21b22) = φ(b11). Although ψ is not faithful, clearly the GNS representation of ψ is faithful on D. Let π_j : A_j → D be the unital ^∗–homomorphisms such that π1(p) =

p0 0 0

and π2(q) =

q0 0 1

. Thenψ◦π_j =φ_j. Moreover, the pair π1(A1), π2(A2)

is clearly free with respect to ψ. There cannot, however, be a ^∗–homomorphism π :A→D such that π(p) =π₁(p) and π(q) =π₂(q), because, as can be seen from[14, 2.7], π2(q)is not equivalent in Dto a subprojection ofπ1(p).

The main result of this paper is an embedding result (Theorem 1.3) implying the following.

Property 3. Let

(A, φ) = ∗

ι∈I(A_ι, φ_ι)

be a reduced free product of C^∗–algebras, where the φ_ι are states on the unital C^∗–algebrasA_ι having faithful GNS representations. Denote by α_ι :A_ι →A the embeddings arising from the reduced free product construction. Suppose for every ι ∈ I there is a unital C^∗–algebra D_ι with a state ψ_ι having faithful GNS representation, and suppose there is a ∗–homomorphism π_ι:A_ι→D_ι such that ψ_ι◦π_ι =φ_ι. Let

(D, ψ) = ∗

ι∈I(D_ι, ψ_ι)

be the reduced free product of C^∗–algebras and denote by δ_ι :D_ι →D the embeddings arising from the reduced free product construction. Then there is a∗–homomorphism π:A→D such thatπ◦α_ι=δ_ι◦π_ι for every ι∈I.

This property was previously known under the additional assumption that every ψ_ι is faithful, which by [13] implies that ψis faithful on D; as noted after Question 1, this in turn implies the existence ofπ. Theorem 1.3 actually proves more generally a version of Property 3 for reduced amalgamated free products of C^∗–algebras. Such an embedding result is frequently useful for understanding reduced free product C^∗–algebras; it has been used in [17] and several times in [16].

We should point out that M. Choda has in [9] stated a theorem about reduced free products of completely positive maps which is more general than Property 3. However, her proof is incomplete, as it implicitly uses the full generality of Property 3 without justifying its validity. Therefore, a proof of Property 3 is called for.

In§1, the main theorem about embeddings of reduced amalgamated free products of C^∗– algebras is proved. In§2, Choda’s argument proving the existence of reduced free products of state preserving completely positive maps is generalized to prove existence of reduced amal- gamated free products of certain sorts of completely positive maps. In §3, we consider the reduced free product with amalgamation of von Neumann algebras and prove analogues of the results in§1 and §2 for von Neumann algebras.

§1. Embeddings.

In the following lemma, with the reduced amalgamated free product of C^∗–algebras (A, φ) = ∗

ι∈I(A_ι, φ_ι) we view each A_ι as a C^∗–subalgebra of A via the canonical embedding arising from the free product construction.

Lemma 1.1. Let B be a unital C^∗–algebra, let I be a set and for every ι ∈ I let A_ι be a unital C^∗–algebra containing a copy of B as a unital C^∗–subalgebra and having a conditional expectation φ_ι :A_ι→B whose GNS representation is faithful. Let

(A, φ) = ∗

ι∈I(A_ι, φ_ι)

be the reduced amalgamated free product. Then for every ι0 ∈I there is a conditional expec- tation Φ_ι0 : A → A_ι0 such that Φ_ι0↾_Aι = φ_ι for every ι ∈ I\{ι0} and Φ_ι0(a1a2· · ·a_n) = 0 whenevern≥2 and a_j ∈A_ιj ∩kerφ withι16=ι2, . . . , ι_n−16=ι_n.

Proof. We use the same notation as in [16] for the free product construction. Thus E_ι = L²(A_ι, φ_ι), ξ_ι=d1_Aι∈E_ι,E_ι=ξ_ιB⊕E_ι^o and A acts on the HilbertB–module

E =ξB⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,ι26=ι3,... ,ιn−16=ιn

E_ι^o₁⊗_BE_ι^o₂⊗_B· · · ⊗_BE_ι^o_n.

Identify the submoduleξB⊕E_ι^o₀ of E with the Hilbert B–moduleE_ι0 and let Q_ι0 :E →E_ι0 be the projection. Then Φ_ι0(x) =Q_ι0xQ_ι0 has the desired properties.

Explication 1.2. Consider the GNS representation σ, L²(A,Φ_ι0), η

= GNS(A,Φ_ι0) associated with the conditional expectation Φ_ι0 : A → A_ι0 found in Lemma 1.1. Since A is the closed linear span of B and the set of reduced words of the form a1a2· · ·a_n where a_j ∈ A_ιj ∩kerφ and ι_j 6=ι_j+1, we see that the HilbertA_ι0–module in the GNS representation is

L²(A,Φ_ι0) =A_ι0⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

ιn6=ι0

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_n⊗_BA_ι0. (3)

Moreover, the action σ of A on L²(A,Φ_ι0) is determined by its restrictions σ↾_Aι, which are easily described.

Let ρ : A_ι0 → L(V) be a unital ∗–homomorphism, for some Hilbert space V. Then σ⊗1 :A→L L²(A,Φ_ι0)⊗_ρV

is a∗–homomorphism; it is the representation induced, in the sense of Rieffel [25], fromρ up to A, with respect to the conditional expectation Φ_ι0, and we will denote this induced representation byρ⇂^A. We have the following explicit description of ρ⇂^A, obtained by tensoring (3) with ⊗_ρVon the right. Writing H=L²(A,Φ_ι0)⊗_ρV we have

H=V⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

ιn6=ι0

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_n ⊗_ρV. (4)

Moreover, the∗–homomorphismσ⊗1 is determined by its restrictions

σ_ι^def= (σ⊗1)↾_Aι :A_ι→L(H),

given as follows. Consider the Hilbert spaces

H(ι) =





























(η_ιB⊗_ρ↾BV)⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

ιn6=ι0, ι16=ι

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_n⊗_ρV ifι6=ι0

M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

ιn6=ι0, ι16=ι0

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_n⊗_ρV ifι=ι0,

whereη_ιB is just the HilbertB–moduleB with identity element denoted by η_ι. Ifι∈I\{ι₀} let

W_ι:E_ι⊗_BH(ι)→H (5)

be the unitary defined, using the symbol ⊗^.. to denote the tensor product in (5), by W_ι:ξ_ι⊗^.. (η_ι⊗v)7→v

ζ⊗^..(η_ι⊗v)7→ζ⊗v

ξ_ι⊗^.. (ζ1⊗ · · · ⊗ζ_n⊗v)7→ζ1⊗ · · · ⊗ζ_n⊗v ζ⊗^..(ζ1⊗ · · · ⊗ζ_n⊗v)7→ζ⊗ζ1⊗ · · · ⊗ζ_n⊗v

wheneverv∈V,ζ ∈E_ι^o,ζ_j ∈E_ι^o_j and ι6=ι1, ι16=ι2, . . . , ι_n−16=ι_n, ι_n 6=ι0. Then for every ι∈I\{ι0}and a∈A_ι, we have

σ_ι(a) =W_ι(a⊗1H(ι))W_ι^∗. Similarly, define the unitary

W_ι0 :V⊕ E_ι⊗_BH(ι0)

→H by

W_ι0:v⊕07→v 0⊕ ξ_ι0

⊗..(ζ1⊗ · · · ⊗ζ_n⊗v)

7→ζ1⊗ · · · ⊗ζ_n⊗v 0⊕ ζ⊗^..(ζ1⊗ · · · ⊗ζ_n⊗v)

7→ζ⊗ζ1⊗ · · · ⊗ζ_n⊗v.

Then

σ_ι0(a) =W_ι0 ρ(a)⊕(a⊗1H(ι0)) W_ι^∗₀.

Note that the above description is related to the construction of the conditionally free product, due to Bo˙zejko and Speicher [7], (see also [6]).

Theorem 1.3. Let Be be a unital C^∗–algebra. Let I be a set and for every ι∈I let Ae_ι be a unital C^∗–algebra containing a copy of Be as a unital C^∗–subalgebra and having a conditional expectation φ˜_ι : Ae_ι → B. Suppose thate B is a unital C^∗–algebra that is contained as a C^∗– subalgebra of Be (without necessarily containing the identity element of B) and suppose thate for every ι ∈ I A_ι is a unital C^∗–algebra that is contained as a C^∗–subalgebra of Ae_ι, that B ⊆A_ι, that B contains the identity element of A_ι and that φ˜_ι(A_ι) = B. Let φ_ι : A_ι → B be the restriction of φ˜_ι and suppose that φ˜_ι and φ_ι have faithful GNS representations. Let κ_ι : A_ι → Ae_ι denote the inclusion. Consider the reduced amalgamated free products of C^∗– algebras

(A,e φ) =˜ ∗

ι∈I(Ae_ι,φ˜_ι) (A, φ) = ∗

ι∈I(A_ι, φ_ι)

and denote the inclusions arising from the free product construction by e

α_ι:Ae_ι →Ae α_ι:A_ι →A.

Then there is a∗–homomorphismκ:A→Ae such that

∀ι∈I κ◦α_ι=αe_ι◦κ_ι. (6) Moreover, κ is necessarily injective and is the unique ∗–homomorphism satisfying (6).

Proof. Since A is generated by S

ι∈Iα_ι(A_ι), it is clear that κ will be unique if it exists. Let 1 denote the identity element of Be and let p be the identity element of B. If p 6= 1 then we may replace B by B+C(1−p) and each A_ι by A_ι+C(1−p); hence we may without loss of generality assume that B is a unital C^∗–subalgebra of Be and thus each A_ι is a unital C^∗–subalgebra ofAe_ι. Let

(eπ_ι,Ee_ι,ξ˜_ι) = GNS(Ae_ι,φ˜_ι), (π_ι, E_ι, ξ_ι) = GNS(A_ι, φ_ι) and

(E,e ξ) =˜ ∗

ι∈I(Ee_ι,ξ˜_ι), (E, ξ) = ∗

ι∈I(E_ι, ξ_ι).

The inclusionκ_ιgives an inner product preserving isometry of Banach spacesE_ι֒→Ee_ιsending ξ_ι to ˜ξ_ι, and we identifyE_ι with this subspace ofEe_ι and therebyE_ι^owith the subspace ofEe_ι^o. This allows canonical identification of the tensor product module

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_p−1⊗_BEe_ι^o_p⊗_Be · · · ⊗_BeEe_ι^o_n

with a closed subspace of Ee_ι^o₁ ⊗_Be · · · ⊗_Be Ee_ι^o_n. Hence, we may and do identify E with the subspace

ξB˜ ⊕ M

ι1,... ,ιn≥1n∈I ι16=ι2,... ,ιn−16=ιn

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_n

of E. Lete A = (∗_B)_ι∈IA_ι be the universal algebraic free product with amalgamation over B. Let σ : A → L(E), respectively eσ : A → L(E), be the homomorphism extending thee homomorphismsα_ι :A_ι →L(E), respectively αe_ι◦κ_ι :A_ι→ L(E), (ιe ∈I). In particular, we haveσ(A) =A. In order to show that κ exists, it will suffice to show that

∀x∈A kσ(x)k ≤ kσ(x)k.e

Note that the subspace E of Ee is invariant underσ(e A) and that the restriction of σ(·) toe E givesσ. This implies

∀x∈A kσ(x)k ≥ kσ(x)k,e

which will in turn imply that κ is injective, once it is known to exist. Let τ be a faithful representation of Be on a Hilbert space W. Consider the Hilbert space Ee⊗_τ W and let λe : L(E)e →L(Ee⊗_τW) be the∗–homomorphism given byλ(x) =e x⊗1^W. Theneλis faithful, and hence it will suffice to show that

∀x∈A keλ◦eσ(x)k ≤ kσ(x)k. (7) Our strategy will be to show that λe◦σe decomposes as a direct sum of subrepresentations, each of which is of the form (ν⇂^A)◦σ, where ν⇂^A is the∗–representation of A induced from a representationν of some A_ι.

Given n ≥ 1 and ι1, . . . , ι_n with ι1 6= ι2, . . . , ι_n−1 6= ι_n, and given p ∈ {1,2, . . . , n}, consider the Hilbert space

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_p−1⊗_BK_ιp⊗_BeEe_ι^o_p+1 ⊗_Be· · · ⊗_Be Ee_ι^o_n⊗_τ W=

def=



 E_ι^o₁⊗_B· · · ⊗_BE_ι^o_p−1⊗_BEe_ι^o_p ⊗_Be Ee_ιp+1⊗_Be · · · ⊗_BeEe_ι^o_n ⊗_τ W

⊖ E_ι^o₁⊗_B· · · ⊗_BE_ι^o_p−1⊗_BE_ι^o_p ⊗_BEe_ιp+1⊗_Be · · · ⊗_BeEe_ι^o_n ⊗_τ W



.

Heuristically,K_ι takes the place ofEe_ι⊖E_ι, even when the latter does not make sense. Then Ee⊗_τW= (E⊗_τ↾BW)⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

p∈{1,2,... ,n}

E_ι^o₁⊗_B· · ·⊗_BE_ι^o_p−1⊗_BK_ιp⊗_BeEe_ι^o_p+1⊗_Be· · ·⊗_BeEe_ι^o_n⊗_τW.

As we mentioned earlier, eσ(A)E ⊆ E and eσ(·)↾_E = σ(·), so E ⊗_τ↾_B W is invariant under eλ◦eσ(A), and

∀x∈A kλe◦σ(x)↾e _E⊗τWk=kσ(x)k.

Sinceπe_ι(A_ι)E_ι⊆E_ι, it is not difficult to check that for everyn≥1 and for everyι1, . . . , ι_n∈I withι16=ι2, . . . , ι_n−16=ι_n,

e

W(ι1, . . . , ι_n)^def= eλ◦σ(e A)(K_ι1⊗_Be Ee_ι^o₂⊗_Be· · · ⊗_BeEe_ι^o_n⊗_τ W) =

= (K_ι1⊗_Be Ee_ι^o₂⊗_Be· · · ⊗_BeEe_ι^o_n ⊗_τ W) ⊕

⊕ M

q≥1 ι^′1,... ,ι^′q∈I ι^′16=ι^′2,... ,ι^′q−16=ι^′q

ι^′q6=ι1

E_ι^o′

1⊗_B· · · ⊗_BE_ι^o′

q ⊗_BK_ι1⊗_BeEe_ι^o₂⊗_Be · · · ⊗_BeEe_ι^o_n ⊗_τW.

Thus

Ee⊗_τ W= (E⊗_τ↾BW)⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

e

W(ι1, . . . , ι_n);

hence in order to prove the theorem it will suffice to show that for every choice ofι1, . . . ι_n,

∀x∈A keλ◦σ(x)↾e _fW(ι

1,... ,ιn)k ≤ kσ(x)k. (8)

But lettingV=K_ι1⊗_BeEe_ι^o₂⊗_Be· · ·⊗_BeEe_ιn⊗_τW, lettingν :A_ι1 →L(V) be the∗–homomorphism ν(a) = (eπ_ι1(a)⊗1_Eeo

ι2⊗B···⊗BEe_ιn^o ⊗τW)↾V, and appealing to Explication 1.2, it is straightforward to check that

eλ◦eσ(·)↾_fW(ι

1,... ,ιn)= (ν⇂^A)◦σ,

whereν⇂^Ais the representation ofAinduced fromνwith respect to the conditional expectation Φ_ι1 :A→A_ι1 found in Lemma 1.1; this in turn implies (8).

Remark 1.4. Let us consider for a moment Theorem 1.3 when the subalgebraB over which we amalgamate is the scalars, C. When taking the reduced free product (A, φ) = ∗

ι∈I(A_ι, φ_ι) of

C^∗–algebras, the statesφ_ι are required to have faithful GNS representation in order that the C^∗–algebrasA_ι are embedded in the free product C^∗–algebra A. However, upon relaxing this condition to the case of completely general statesφ_ι and performing the reduced free product construction, one obtains

ι∈I∗ (A_ι, φ_ι) = ∗

ι∈I((A_ι/kerπ_ι),φ^._ι),

where π_ι is the GNS representation of φ_ι and where φ^._ι is the state induced on the quotient A_ι/kerπ_ιbyφ_ι. Thus the canonical∗–homomorphismα_ι:A_ι→Ahas the same kernel as π_ι. As a caveat, we would like to point out that with this relaxed definition of reduced free product, the statement of Theorem 1.3 does not in general hold if one tolerates φ_ι with nonfaithful GNS representations. Indeed, if for someι∈I A_ι=C⊕Cwith φ_ι non–faithful, ifAe_ι=M2(C) with a unital embeddingκ_ι:A_ι ֒→Ae_ι and if ˜φ_ι is a state onM2(C) such that φ˜_ι ◦κ_ι = φ_ι, then α_ι : A_ι → A is not injective, while ˜α_ι◦κ_ι is injective. This shows that there can be no ∗–homomorphism κ : A → Ae satisfying equation (6). However, there is no problem allowing the ˜φ_ι to have nonfaithful GNS representations, as long as the restrictions φ_ι are taken with faithful GNS representations.

§2. Completely positive maps.

M. Choda [9] gave an argument which, when combined with an embedding result like Property 3, proves that if θ_ι : A_ι → D_ι is a unital completely positive map between unital C^∗–algebras for every ι ∈ I, if φ_ι and ψ_ι are states on A_ι and respectively D_ι, each having faithful GNS representation, and ifψ_ι◦θ_ι=φ_ι then letting

(A, φ) = ∗

ι∈I(A_ι, φ_ι) (D, ψ) = ∗

ι∈I(D_ι, ψ_ι)

be the reduced free products of C^∗–algebras and denoting by α_ι : A_ι → A and δ_ι :D_ι → D the injective ∗–homomorphisms arising from the free product constructions, there is a unital completely positive map θ : A → D such that θ◦α_ι = θ_ι ◦δ_ι for every ι ∈ I, and such thatθ(a1a2· · ·a_n) =θ(a1)θ(a2)· · ·θ(a_n) whenever a_j ∈α_ιj(A_ιj)∩kerφ for some ι_j ∈I with ι₁6=ι₂, . . . , ι_n−16=ι_n.

In this section, we generalize this argument of Choda’s to the case of reduced amalgamated free products of C^∗–algebras. The generalization consists of, in essence, replacing Stinespring’s dilation theorem for completely positive maps into bounded operators on a Hilbert space by

Kasparov’s generalization [21] to the case of completely positive maps into the algebra of bounded adjointable operators on a HilbertB–module (see alternatively the book [22]). We would like to point out that Theorem 2.2 is quite similar in appearance to analogous results of F. Boca [4], [5] about completely positive maps on universal amalgamated free products of C^∗–algebras. However, the universal and reduced free products of C^∗–algebras are quite different in character, and we do not believe that Boca’s results can be used directly to prove Theorem 2.2.

Lemma 2.1. Let A and B be C^∗–algebras, let E and Ee be Hilbert A–modules, let F and F˜ be Hilbert B–modules and let v ∈ L(E,Ee), w ∈ L(F,F˜). Suppose π : A → L(F) and e

π:A→L( ˜F) are ∗–homomorphisms and suppose that

∀a∈A ∀ξ∈F w(π(a)ξ) =π(a)w(ξ).e (9) Let E⊗_πF and Ee⊗_eπF˜ be the interior tensor products. Then there is an element v⊗w ∈ L(E⊗_πF,Ee⊗_eπF˜) such that

∀ζ ∈E ∀ξ ∈F (v⊗w)(ζ⊗ξ) = (vζ)⊗(wξ).

If, moreover, hv(ζ), v(ζ)i = hζ, ζi for every ζ ∈E and hw(ξ), w(ξ)i = hξ, ξi for every ξ ∈F thenhv⊗w(η), v⊗w(η)i=hη, ηi for every η∈E⊗_πF.

Proof. Thatv⊗wis bounded is a standard argument (compare p. 42 of [22]). Then one sees (v⊗w)^∗=v^∗⊗w^∗. The final statement follows using the polarization identity.

Theorem 2.2. Let B be a unital C^∗–algebra, letI be a set and for every ι∈I let A_ι and D_ι be unital C^∗–algebras containing copies of B as unital C^∗–subalgebras and having conditional expectations φ_ι :A_ι → B, respectively ψ_ι :D_ι →B, whose GNS representations are faithful.

Suppose that for eachι∈I there is a unital completely positive map θ_ι:A_ι→D_ι that is also a B–B bimodule map and satisfies ψ_ι◦θ_ι=φ_ι. Let

(A, φ) = ∗

ι∈I(A_ι, φ_ι) (D, φ) = ∗

ι∈I(D_ι, ψ_ι)

be the reduced amalgamated free products of C^∗–algebras and denote by α_ι : A_ι → A and δ_ι:D_ι→Dthe embeddings arising from the free product constructions. Then there is a unital completely positive map θ:A→Dsatisfying

∀ι∈I θ◦α_ι =δ_ι◦θ_ι (10)

and

θ(a1a2· · ·a_n) =θ(a1)θ(a2)· · ·θ(a_n) (11) whenevera_j ∈α_ιj(A_ιj∩kerφ_ιj) and ι16=ι2, ι26=ι3, . . . , ι_n−16=ι_n.

Proof. Note first that the assumptions imply that each θ_ι is the identity map on B. Let (π_ι, E_ι, ξ_ι) = GNS(D_ι, ψ_ι), (E, ξ) = ∗

ι∈I(E_ι, ξ_ι).

(We will usually write simplydζ instead ofπ_ι(d)ζ, whend∈D_ι andζ ∈E_ι.) Recall that then E_ι=ξ_ιB⊕E_ι^o, that the actionπ_ι↾_B leavesE_ι^o globally invariant, and that

E =ξB⊕ M

n≥1 ι1,... ,ιn∈I ι16=ι2,... ,ιn−16=ιn

E_ι^o₁⊗_B· · · ⊗_BE_ι^o_n.

Consider the HilbertB–module F_ι=A_ι⊗_πι◦θιE_ι and the specified elementη_ι= 1⊗ξ_ι∈F_ι. Sinceθ_ι restricts to the identity map onB, inF_ι we haveb⊗ζ = 1⊗(bζ) for everyb∈B and ζ ∈E. Consider the unital∗–homomorphismσ_ι :A_ι→L(F_ι) given by

∀a^′, a∈A_ι ∀ζ ∈E_ι σ_ι(a^′)(a⊗ζ) = (a^′a)⊗ζ,

(see for example page 48 of [22]). Consider the mapρ_ι :L(F_ι)→B given byρ_ι(x) =hη_ι, xη_ιi.

If x∈ L(F_ι) and if b1, b2 ∈ B then ρ_ι σ_ι(b1)xσ_ι(b2)

= b1ρ_ι(x)b2. If we use σ_ι to identify B with σ_ι(B) ⊆ L(F_ι) then we have that ρ_ι : L(F_ι) → B is a conditional expectation. Clearly L²(L(F_ι), ρ_ι) ∼= F_ι and the GNS representation of ρ_ι is faithful on L(F_ι). We have that ρ_ι◦σ_ι=φ_ι since

ρ_ι◦σ_ι(a) =h1⊗ξ_ι, a⊗ξ_ιi=hξ_ι, θ_ι(a)ξ_ιi=ψ_ι◦θ_ι(a) =φ_ι(a). (12) Let

(M, ρ) = ∗

ι∈I(L(F_ι), ρ_ι)

be the reduced amalgamated free product of C^∗–algebras and let λ_ι : L(F_ι) → M be the embedding arising from the free product construction. Note thatM⊆L(F) where

(F, η) = ∗

ι∈I(F_ι, η_ι).

By Theorem 1.3 there is a∗–homomorphismσ :A→Msuch that

∀ι∈I σ◦α_ι=λ_ι◦σ_ι.

Consider the operator v_ι:E_ι →F_ι given by ζ → 1⊗ζ, and note thathv_ιζ, v_ιζi=hζ, ζi for every ζ ∈E_ι, hencev_ι(E_ι^o)⊆F_ι^o. A calculation using e.g. Lemma 5.4 of [22] shows that there is a bounded operator F_ι →E_ι sending a⊗ζ toθ_ι(a)ζ, which is then the adjoint ofv_ι. Hencev_ι∈L(E_ι, F_ι), and clearlyv_ι^∗v_ι= 1. Sinceθ_ιis a leftB–module map, we have for every b∈B and ζ ∈E_ι thatv_ι(bζ) = 1⊗(bζ) = b⊗ζ =b(v_ι(ζ)). Therefore, taking direct sums of operators v_ι1 ⊗ · · · ⊗v_ιn given by Lemma 2.1, we get v ∈L(E, F) such thathvζ, vζi=hζ, ζi for everyζ ∈E,vξ=η and

v(ζ₁⊗ζ₂⊗ · · · ⊗ζ_n) = (v_ι1ζ₁)⊗(v_ι2ζ₂)⊗ · · · ⊗(v_ιnζ_n)

wheneverζ_j ∈E_ι^o_j,ι1, . . . , ι_n ∈I and ι_j 6=ι_j+1. Let θ :A→ L(E) be the unital completely positive mapθ(x) =v^∗σ(x)v.

We will show that (10) and (11) hold, which will furthermore imply that θ(A) ⊆D. In order to show (10), let w_ι :E → E_ι⊗_BE(ι) and y_ι :F →F_ι⊗_B F(ι) be the unitaries used in the free product constructions to define the embeddingsα_ι and, respectively,λ_ι. Note that v_ι E(ι)

⊆ F(ι) and that y_ιv = (v_ι⊗v↾_E(ι))w_ι. Furthermore, observe that for a ∈ A_ι and ζ ∈E_ι,

v^∗_ισ_ι(a)v_ι)ζ =v^∗_ι(a⊗ζ) =θ_ι(a)ζ.

Hence fora∈A_ι,

θ◦α_ι(a) =v^∗σ◦α_ι(a)v=v^∗λ_ι◦σ_ι(a)v=v^∗y^∗_ι σ_ι(a)⊗1_F(ι) y_ιv =

=w_ι^∗ v_ι^∗σ_ι(a)v_ι⊗(v↾_E(ι))^∗v↾_E(ι)

w_ι =w^∗_ι(θ_ι(a)⊗1_E(ι))w_ι=α_ι◦θ_ι(a).

Now to show that (11) holds, consider a_j ∈ A_ιj ∩kerφ_ιj for some ι_j ∈ I (1 ≤ j ≤ n) with ι_j 6=ι_j+1. Henceforth we will omit to write the inclusions α_ι and δ_ι, thinking instead of each A_ι as a subalgebra of A and of each D_ι as a subalgebra of D. It is easy to see that

θ_ι1(a1)· · ·θ_ιn(a_n)ξ =θ\_ι1(a1)⊗ · · · ⊗θ\_ιn(a_n) =

=v^∗ (a₁⊗ξ_ι1)⊗ · · · ⊗(a_n⊗ξ_ιn)

=θ(a₁· · ·a_n)ξ.

(13)

Now consider an elementζ1⊗ · · · ⊗ζ_p∈E, whereζ_j ∈E_k^o_j for somek_j ∈I withk_j 6=k_j+1. Let P0:E→ξB be the projection and for ℓ∈Nlet

P_ℓ:E → M

ι1,... ,ιℓ∈I ι16=ι2,... ,ιℓ−16=ιℓ

E_ι^o₁⊗ · · · ⊗E_ιℓ

be the projection. Taking adjoints and using (13), we see that

P0θ_ι1(a1)· · ·θ_ιn(a_n)(ζ1⊗ · · · ⊗ζ_p) =P0θ(a1· · ·a_n)(ζ1⊗ · · · ⊗ζ_p).

Now letting ℓ∈N we will use standard techniques (see, for example, [18] and [16]) to show that

P_ℓθ_ι1(a1)· · ·θ_ιn(a_n)(ζ1⊗ · · · ⊗ζ_p) =P_ℓθ(a1· · ·a_n)(ζ1⊗ · · · ⊗ζ_p). (14) Ifℓ > n+porℓ <|n−p|then it is clear that both sides of (14) are zero. Ifℓ=n+pthen both sides of (14) are zero unlessι_n6=k1, in which case a calculation similar to (13) shows that (14) holds. Let Q^o_ι :E_ι → E_ι^o and R^o_ι :F_ι → F_ι^o be the projections, and note that R^o_ιv_ι = v_ιQ^o_ι. Consider whenn+p−ℓ= 1. Then both sides of (14) are zero unless ι_n=k1, in which case

P_ℓθ_ι1(a1)· · ·θ_ιn(a_n)(ζ1⊗ · · · ⊗ζ_p) =

=θ\_ι1(a₁)⊗ · · · ⊗θ_ιn−1\(a_n−1)⊗Q^o_ιn(θ_ιn(a_n)ζ₁)⊗ζ₂⊗ · · · ⊗ζ_p

=v^∗_ι (a1⊗ξ_ι1)⊗ · · · ⊗(a_n−1⊗ξ_ιn−1)⊗R^o_ιn(a_n⊗ζ1)⊗(1⊗ζ2)⊗ · · · ⊗(1⊗ζ_p)

=P_ℓθ(a1· · ·a_n)(ζ1⊗ · · · ⊗ζ_p)

Ifp+n−ℓ= 2r+ 1 for r ∈ {1,2, . . . ,min(p, n)−2} then both sides of (14) are zero unless ι_n =k1, ι_n−1=k2, . . . , ι_n−r+1=k_r, in which case

P_ℓθ_ι1(a1)· · ·θ_ιn(a_n)(ζ1⊗ · · · ⊗ζ_p) =

= θ\_ι1(a₁)⊗ · · · ⊗θ_ιn−r−1\(a_n−r−1)⊗

⊗Q^o_ιn−r θ_ιn−r(a_n−r)hξ, θ_ιn−r+1(a_n−r+1)· · ·θ_ιn(a_n)ζ1⊗ · · · ⊗ζ_riζ_r+1

⊗

⊗ζ_r+2⊗ · · · ⊗ζ_p

= θ\_ι1(a1)⊗ · · · ⊗θ_ιn−r−1\(a_n−r−1)⊗

⊗Q^o_ιn−r θ_ιn−r(a_n−r)hθ\_ιn(a^∗_n)⊗ · · · ⊗θ_ιn−r+1\(a^∗_n−r+1), ζ1⊗ · · · ⊗ζ_riζ_r+1

⊗

⊗ζ_r+2⊗ · · · ⊗ζ_p

= v^∗

(a₁⊗ξ_ι1)⊗ · · · ⊗(a_n−r−1⊗ξ_ιn−r−1)⊗

⊗ R^o_ιn−r

θ_ιn−r(a_n−r)·

·

σ_ιn(a^∗_n)· · ·σ_ιn−r+1(a^∗_n−r+1)η, (1⊗ζ1)⊗ · · · ⊗(1⊗ζ_r)

(1⊗ζ_r+1)

⊗

⊗(1⊗ζ_r+2)⊗ · · · ⊗(1⊗ζ_p)