A NOTE ON AMALGAMATED BOOLEAN, ORTHOGONAL AND CONDITIONALLY MONOTONE OR ANTI-MONOTONE PRODUCTS OF OPERATOR-VALUED C*-ALGEBRAIC PROBABILITY SPACES

ORTHOGONAL AND CONDITIONALLY MONOTONE OR ANTI-MONOTONE PRODUCTS

OF OPERATOR-VALUED C *-ALGEBRAIC PROBABILITY SPACES

VALENTIN IONESCU

We directly prove that the amalgamated Boolean product (in M. Sch¨urmann [27], or R. Speicher and R. Worudi’s sense [30], for the scalar-valued case), or the amal- gamated orthogonal product (in R. Lenczewski’s sense [19], for the scalar-valued case) of some bimodule maps (in particular, conditional expectations), and the amalgamated conditionally monotone or anti-monotone products (in T. Hasebe’s sense [11] for the scalar-valued case; see also, M. Popa’s preprint [25], for an operator-valued case) of pairs of some bimodule maps preserve the (complete) positivity in C*-algebraic setting. Our statements are formulated in terms of Schwarz maps. The proofs follow the method in [18], inspired by the scalar case technique due to M. Bo˙zejko, M. Leinert, and R. Speicher from [7] concerning the conditionally free product of states (see also [15], [17]).

AMS 2010 Subject Classification:Primary 46L09, 46L53, 46L54; Secondary 46L60, 46N50, 81S25.

Key words: (*-,C*-)algebra with adjoined algebra, complete positivity, condi- tional expectation, Schwarz map, Stinespring dilation, amalgamated universal free product (*-,C*-)algebra, amalgamated conditionally- free, Boolean, orthogonal, conditionally-monotone or conditionally- anti-monotone product maps.

1. INTRODUCTION

There are five non-commutative probability theories (R.L. Hudson’s Bo- son or Fermion probability theory, D.V. Voiculescu’s free probability theory, R. Speicher and W. von Waldenfels’ Boolean probability theory, and N. Mu- raki or Y.G. Lu’s monotone and anti-monotone probability theories) arised from an associative product which fulfills a quasi-universal rule for mixed mo- ments (due to Muraki’s work [20] on the quasi-universal products of algebraic

REV. ROUMAINE MATH. PURES APPL.,57(2012),4, 341-370

probability spaces and U. Franz’s axiomatic study in [8]) or even from a nat- ural product in A. Ben Ghorbal and M. Sch¨urmann’s spirit [1] (according to Muraki’s classification of his natural products [21]).

The Boolean product of linear functionals on algebras (originated in M. Bo˙zejko’s investigations concerning the positive definite functions on the free groups [5]) is defined on the associated universal free product algebra with- out unit, is commutative, and on involutive algebras it preserves the positivity ([14], [27], [30]). We should mention that M. Popa’s statements in [24, Theo- rem 6.5] and [23, Theorem 2.3] can also be considered as results on Boolean products in the operator-valued case; they are formulated in more general set- ting, but, in fact, their (incomplete) proofs work, in the most general case, for this version of the Boolean product ([18], Remark 3.9).

The conditionally monotone (c-monotone for short) product and the conditionally monotone (c-anti-monotone for short) product (which is a c- monotone product with respect to the opposite order) of pairs of linear func- tionals on algebras (indexed by a totally ordered set) are defined on the same corresponding universal free product algebra without unit, and on involutive algebras they preserve the positivity ([11], [12]).

These products were introduced by T. Hasebe in [11] as a parallel concept to Bo˙zejko and Speicher’s c-free product from [6] (see also [7]); although they may be introduced as parts of c-free products. Popa extended in [25] this Hasebe’s approach to the more general frame involving some bimodule maps on algebras over a common subalgebra; then, by using F. Boca’s main result [4], he stated a theorem about the complete positivity of the amalgamated c-monotone product involving unital maps betweenC*-algebras.

The c-monotone or c-anti-monotone products naturally generalize the Boolean product, the monotone or anti-monotone products, and the so-called orthogonal product or its dual object. The mentioned products of non-com- mutative probability spaces and the associated stochastic independences are basic in the corresponding Boolean,c-monotone andc-anti-monotone quantum probability theories and related topics ([2], [3], [9], [11], [12], [13], [23], [25], [26], [30]) (see also [14], [15], [16], [17]).

The orthogonal product of linear functionals on algebras may be defined on the corresponding universal free product algebra without unit, is neither commutative, nor associative, and on involutive algebras it also preserves the positivity. From this product derives R. Lenczewski’s orthogonal convolution which is essential in his work in [19] on some alternating decompositions of Voiculescu’s free additive convolution.

In this Note, we consider amalgamated Boolean and orthogonal products of conditional expectations, and amalgamatedc-monotone orc-anti-monotone products of pairs of conditional expectations (but also, more generally, of some

bimodule maps), as parts of appropriate amalgamated c-free product maps in F. Boca’s sense in [4] defined on algebras over (respectively, possessing a compatible bimodule structure with respect to) a common algebra and directly prove they preserve the (complete) positivity in C*-algebraic setting. Our approach to positivity is made in terms of (possibly non-unital) Schwarz maps, being different of Popa’s approach in [23], [24], [25] and more elementary than it. We illustrate this by our statement on the complete positivity of the amalgamated c-monotone product which is obtained without any use of Boca’s main result in [4]. The proofs follow the method in [18] being inspired by the scalar case technique due to Bo˙zejko, Leinert, and Speicher from [7]

concerning the c-free product of states on unital involutive algebras (see also [15], [17]).

2. (*-,C*-)ALGEBRA WITH ADJOINED ALGEBRA, SCHWARZ MAP, COMPLETE POSITIVITY, AMALGAMATED UNIVERSAL FREE PRODUCT

(*-,C*-)ALGEBRA We remind some preliminaries from [18].

Let B be an (associative) algebra. Let A be an algebra (over the same field) endowed with a compatibleB-B-bimodule structure, such that: b(a1a2)

= (ba₁)a₂, (a₁b)a₂ =a₁(ba₂), (a₁a₂)b=a₁(a₂b), for all a₁, a₂ ∈A and b∈B. Then, the direct sum ˜A := B ⊕A is an (associative) algebra over B (i.e., A includes B as a subalgebra) with the multiplication

(b₁⊕a₁)(b₂⊕a₂) :=b₁b₂⊕(b₁a₂+a₁b₂+a₁a₂), b_i∈B, a_i∈A.

If B has a unit 1_B, and 1_Ba=a1_B =a, for all a∈A, then ˜A has the unit 1B⊕0.

In [18], ˜A is called the algebra with adjoined algebra B, corresponding toA.

IfBandAare (complex) *-algebras (i.e., complex algebras endowed with conjugate linear involutions *, which are anti-isomorphisms), andAis endowed with a corresponding compatible B-B-bimodule structure (i.e., besides of the above properties, (ab)^∗ = b^∗a^∗, (ba)^∗ = a^∗b^∗, for all a∈ A and b∈ B), then A˜:=B⊕A becomesa*-algebra by

(b⊕a)^∗ =b^∗⊕a^∗, b∈B, a∈A.

The following fact is from [18].

Proposition 2.1. Let B be a C*-algebra, and A be a C*-algebra en- dowed with a corresponding compatible B-B-bimodule structure.

Then, the *-algebra A˜:=B⊕A endowed with the norm kb⊕ak:= sup

kxk≤1, x∈A

kbx+axk, b∈B, a∈A

becomes aC*-algebra overB, andA˜containesAas a closed two-sided ideal.

Let A be a (complex) *-algebra (i.e., a complex algebra endowed with a conjugate linear involution *, which is an anti-isomorphism). We consider the cone A₊ of positive elements in A consisting of finite sums P

a^∗_ia_i, with ai ∈A.Thus,A+ determines a preorder structure on the real linear subspace of self-adjoints elements in A.

For any positive integern, let M_n(A) be the *-algebra ofn×nmatrices [aij] with entries fromA. Every positive elementM∗M inMn(A) can be ex- pressed as

n

P

k=1

[a^(k)∗_i a^(k)_j ]i,j=1,...,n with somea^(k)_i ∈A.WhenA is aC*-algebra, A+={a∗a;a∈A}determines an order structure on the real linear subspace of self-adjoint elements in A, andM_n(A) becomes a C*-algebra.

LetB be another *-algebra andϕ:A→B be a linear map. We say :ϕ is Hermitian if ϕ(a^∗) =ϕ(a)^∗, for all a∈A;ϕis positive ifϕ(A₊)⊂B₊; and ϕ is a Schwarz map ifϕ(a^∗a)≥ϕ(a)^∗ϕ(a), for alla∈A.

For any positive integern, letϕ_n:M_n(A)→M_n(B) be the inflation map given byϕ_n([a_ij]) = [ϕ(a_ij)], for [a_ij]∈M_n(A). Then,ϕis calledn-positive if the map ϕn induced by ϕ is positive. The mapϕ is completely positive if it is n-positive, for all positive integern.

IfB is a subalgebra ofA, a map ϕ:A→B is a conditional expectation of Aonto B, if ϕis a B-B-bimodule map (i.e.,ϕ(ab) =ϕ(a)b,ϕ(ba) =bϕ(a), for a ∈ A and b ∈ B) which is a projection on B (i.e., ϕ | B = idB); and we view (A, ϕ) as a quantum probability space overB, according to [33], [34].

More generally, if B, D and A are algebras, D and A being endowed with compatible B-B-bimodule structures, and ϕ: A → D, ψ :A → B are B-B- bimodule maps, we regard (A, ϕ) as aD-valued quantumB-probability space, and (A, ϕ, ψ) as a D,B-valued quantum B-probability space.

IfB and A are algebras, Abeing also endowed with a compatible B-B- bimodule structure, every B-B-bimodule map φof A inB naturally extends to a conditional expectation φe of ˜A := B ⊕A, the algebra with algebra B adjoined to A, ontoB:

φ(b˜ ⊕a) :=b+φ(a), for all b∈B, a∈A.

When B and A as above are *-algebras, A := B ⊕A becomes a *- algebra as above and if φ is a Hermitian B-B-bimodule Schwarz map, then the conditional expectation ˜φ is a Hermitian Schwarz map, too.

For the following fact one can consult, e.g., [31, II, 9.2–9.3].

Proposition 2.2. Let B ⊂ A be an inclusion of C*-algebras, and ϕ : A → B be a positive conditional expectation of A onto B. Then ϕ is a Her- mitian completely positive Schwarz map, and a projection of norm one.

In this frame, a projectionϕ:A→ B ⊂A, which is a Schwarz map, is a positive, conditional expectation.

Conversely, by a famous theorem due to J. Tomiyama [32] (see also [31]), every projection of norm one of a C*-algebraA onto aC*-subalgebra B ⊂A is a positive conditional expectation.

The celebrated Stinespring dilation theorem characterizes the completely positive maps defined onC*-algebras, and extends the GNS theorem concern- ing the (unital positive functionals, i.e.,) states on this kind of *-algebras.

F. Boca remarked [4] that the classical Stinespring dilation theorem remains true for unital completely positive maps defined on unital *-algebras A veri- fying the Combes axiom (i.e., for every a∈A, there exists a scalar λ(a) >0 with x^∗a^∗ax≤λ(a)x^∗x, for all x∈A).More exactly:

Theorem 2.3. Let A be a unital *-algebra verifying the Combes axiom, L(H)be the bounded linear operators on a Hilbert space H, and ϕ:A→L(H) be a unital completely positive map.

Then, there exists a unique (up to a unitary equivalence) Stinespring dilation (K, π) of ϕ, where K⊃ H is a Hilbert space, and π :A →L(K) is a unital *-representation, such that ϕ(a) = P_H^Kπ(a) |H, for all a ∈ A; and sp⁻π(A)H=K.

Throughout, B will denote a (non-necessary unital) fixed (complex) (*-)algebra. Let us recall other preliminaries as in [18].

IfA is a (complex) (*-,C*-)algebra endowed with a (*-)homomorphism ε :B →A, we say that A is a (*-, C*-) B-algebra, and call εthe structural morphism of theB-algebraA. (WhenAandBare unital, consider, as usually, unital structural morphism; whenever it is not null.) When the structural morphism is injective, we say that the B-algebra A is an algebra over B. The same algebra can have different B-algebra structures. AB-algebra is an algebra naturally endowed with a compatible B-B-bimodule structure. Every algebra endowed with a compatibleB-B-bimodule structure may be regarded as a B-algebra with respect to the null structural morphism.

Consider the category of the (*-, C*-) B-algebras, with morphisms be- ing the (∗-)algebraic homomorphisms π : A₁ → A₂ such that the following diagrams commute

A1 π

−→ A2

ε1 - %_ε2

B

The amalgamated universal free product of a family of (*-, C*-) B- algebras (A_i)i∈I of structural morphisms ε_i : B → A_i, i ∈ I, or the uni- versal free product of a family of (*-, C*-) algebras (Ai)i∈I, endowed with (*-)homomorphisms εi : B → Ai, i ∈ I, with B amalgamated, denoted A=?i∈I(A_i, ε_i, B),is the direct sum in this category.

This (*-, C*-)algebra A is endowed with a structural morphism ε : B → A, and canonical (∗-)algebraic homomorphisms j_i : A_i → A such that ji ◦εi = ε, for all i ∈ I, A is generated by S

i∈Iji(Ai), and satisfies the following universality property: whenever D is a (*-, C*-) B-algebra en- dowed with a structural morphism η : B → D, and λ_i : A_i → D, i ∈ I, are (*-)homomorphisms satisfying λi ◦εi = η, for all i ∈ I, there exists a (*-)homomorphism λ:A→D such thatλ◦j_i =λ_i,for all i∈I.

When the structural morphisms are embeddings, i.e., Ai are algebras over B, this universal object A is the universal free product of (A_i)i∈I with amalgamation over B (see, e.g., [4], [22], [29], [34]), and B identifies to a subalgebra of A; A is commonly denoted by ∗_BA_i, although this notation does not reveal the dependence of A on the embeddingsε_i,i∈I.

Moreover, as B-B-bimodule, a realization of the universal free product

∗_BA_i associated to a family of (*-,C*-)algebras (A_i)i∈I over B, such that Ai = B ⊕A^◦_i are direct sums of B-B-bimodules, A^◦_i being endowed with a compatible scalar multiplication, is (see, e.g., [4])

A=B⊕n≥1⊕_{i16=···6=in}A^o_i1 ⊗_B. . .⊗_BA^o_in =:B⊕A^o.

When (*-, C*-)algebras Ai, i ∈ I, are endowed with compatible B-B- bimodule structures, a realization (as B-B-bimodule) of the universal free productA^o =?i∈I(Ai, εi = 0, B) withB amalgamated isA^o =⊕n≥1⊕_{i16=···6=in} A_i1 ⊗_B. . .⊗_BA_in; and B does not identify to a (*-, C*-) subalgebra of A^o, whenever B is not {0}.

By natural operations, the aboveB-B-bimodulesAandA^oare organized as (*-)algebras.

In particular, ifA_i andB areC*-algebras, AandA^o satisfy the Combes axiom.

After separation and completion of the corresponding universal free pro- duct *-algebra A, respectivelyA^o,in its enveloping C*-seminorm

kak= sup{kπ(a)k; π ∗-representation ofA(resp.A^o) as bounded linear operators on a Hilbert space},

one can realize the universal (or full) amalgamated free product ∗_BA_i,respec- tively?i∈I(Ai, εi = 0B), in the category ofC*-algebras overB, respectively, of C*-algebras endowed with (corresponding) compatible B-B-bimodule struc- tures.

When Ai, i∈ I, are (*-)algebras over B, and, thus, they naturally are B-algebras, we may also consider A_i as B-algebras with respect to the null structural morphisms. IfA is their universal free product with amalgamation over B (hence with the identification of B as a subalgebra of A), and A^o denotes their universal free product withB amalgamated (but, generally, with the non-identification of B to a subalgebra of A^o), there exists a canonical epimorphism of A^o onto A arising from the embeddings of A_i into A, i∈ I, via the universality property.

3. AMALGAMATED INVOLVING PAIRS OF MAPS VALUED IN THE SAME C*-ALGEBRA

Let remind the framework in [18].

LetAi,i∈I,be (*-) algebras over B, andϕi,ψi,i∈I, be (Hermitian) conditional expectations of A_i ontoB. Then,A_i=B⊕A^o_i are direct sums of B-B-bimodules, with A^o_i = kerψi,the kernel of ψi. In particular, when ψi is a homomorphism (which is the identity on B), then A^o_i is an algebra.

The amalgamatedc-free productϕ=∗^{ψi}

B ϕi, in Boca’s sense, of (ϕi)i∈I

with respect toψ_i,i∈I,is the unique linear map well defined on the universal free product A=∗_BAi (see, e.g.,[4]) such that:

1)ϕ|A_i =ϕ_i, for each i∈I;

2) ϕ(a1. . . an) = ϕi1(a1). . . ϕin(an), for all n ≥ 2, i1 6= · · · 6= in, and a_k∈A_ik, withψ_ik(a_k) = 0, ifk= 1, . . . , n;

relatively to the natural embeddings ofAiintoAarising from the free product construction.

Therefore, ϕ = ∗^{ψi}

B ϕ_i is a (Hermitian) conditional expectation of A onto B, and∗^{ψi}

B ψi is Voiculescu’s amalgamated free product [34]ψ=∗_Bψi. WhenI ={1,2}, we denote∗^{ψi}

B ϕ_ibyϕ_1ψ1∗_ψ2 ϕ₂, adopting Franz’s no- tation in [10]; and∗_Bψ_i byψ₁∗_Bψ₂; moreover, ifA_i are adequate (complex)*- algebras, we denote by ?0Ai,A1?0A2, and?1Ai,A1?1A2, the non-unital, and respectively, the unital free product *-algebra.

The following definition comes from [1], [14], [15], [18] [27], [28].

Definition 3.1. LetB be an algebra, and I be a set. Let Ai be algebras endowed with compatible B-B-bimodule structures, and ϕ_i : A_i → B be B- B-bimodule maps ofAi inB,i∈I.

The amalgamated Boolean product ϕ:= _Bϕ_i of (ϕ_i)i∈I is the unique linear map well-defined on the universal free product A of (A_i)i∈I with B amalgamated, such that:

1)ϕ|A_i =ϕ_i,for each i∈I;

2) forn≥2, i1 6=· · · 6=in, and ak∈Aik,withk= 1, . . . , n,

ϕ(a1. . . an) = ϕ_i

1(a1). . . ϕ_in(an); with respect to the natural embeddings of A_i intoA arising from the free product construction.

Thus,ϕ:=_Bϕi is aB-B-bimodule map of A inB.

The next statement describes the amalgamated Boolean product as a part of an amalgamated c-free product (Compare with [13] and [30]; or [18]

and [14]).

Proposition 3.2. Let B be an algebra andI be a set. LetAi be algebras endowed with compatible B-B-bimodule structures, and ϕi :Ai →B be B-B- bimodule maps of A_i in B, i∈I.

Consider the algebras fA_i := B⊕A_i with adjoined algebra B, define the conditional expectations ϕei, and the homomorphisms δi of fAi onto B by ϕe_i(b⊕a) :=b+ϕ_i(a),respectively δ_i(b⊕a) :=b;if b⊕a∈B⊕A_i=fA_i,i∈I.

LetA be the universal free product of (A_i)i∈I withB amalgamated,Ae:=

∗_BfA_i,be the universal free product of (Ae_i)i∈I with amalgamation over B, and δ :=∗_Bδi.

Then A = kerδ, Ae = B ⊕A is the algebra with adjoined algebra B, corresponding to A, and _B ϕi =∗^{δ_B^i} ϕei |A.

The proposition below presents the link between the amalgamated Boo- lean product and the amalgamatedc-free product map considered, whenB= C, in [13] and [30], as the Boolean product of unital functionals (compare with Proposition 3.1 in [10] or Theorem 6.6 in [24]).

Proposition 3.3. Let B be an algebra andI be a set. LetAi be algebras over B, such thatA^._i =B⊕A^◦_i,A^◦_i being algebras, endowed with compatibleB- B-bimodule structures andϕi :Ai →B be conditional expectations of Ai onto B. Let δ_i be the unique homomorphisms of A^._i onto B such that A^◦_i = kerδ_i, and consider the B-B-bimodule maps ϕ^◦_i :=ϕi|A^◦_i,i∈I.

Let A :=∗_BAi be the universal free product of (Ai)i∈I, with amalgama- tion over B, and A^◦ be the universal free product of (A^◦_i)i∈I with B amalga- mated.

Let ϕ:=∗^{δ_B^i}ϕ_i :A→ B be the amalgamated c-free product conditional expectation, and ϕ^◦ :=_Bϕ^◦_i :A^◦ → B be the amalgamated Boolean product B-B-bimodule map.

Thenϕ^◦=ϕ◦j,j being the canonical homomorphism of A^◦inA, arising from the embeddings A^◦_i ,→ A^._i ,→Avia the universality property.

IfB and A_i are *-algebras andϕ_i, δ_i are Hermitian, then the above maps ϕ, ϕ^◦ and j preserve the natural involutions.

Adopting the terminology from [10], [13], [23], [30], we consider, in the sequel,∗^{δ_B^i}ϕ_ias the amalgamated Boolean product conditional expectation of

(ϕi)i∈I, ifϕi are conditional expectations as above; thus, in the scalar-valued case B =C, we call this map theunital Boolean product.

The following definition comes from [11], [18], [20], [25].

Definition 3.4. Let B be an algebra, and I be a totally ordered set.

Let A_i be algebras endowed with compatible B-B-bimodule structures, and ϕi, ψi :Ai →B be B-B-bimodule maps ofAi inB,i∈I.

The amalgamatedc-monotone productϕ:=B^{ψ_B^i}ϕ_i of (ϕ_i)i∈I with res- pect to (ψ_i)i∈I is the unique linear map well-defined on the universal free product A of (Ai)i∈I withB amalgamated, such that:

1)ϕ|A_i =ϕ_ii,for each i∈I;

2) forn≥2, i1 6=· · · 6=in, and ak∈Aik,withk= 1, . . . , n;

i)ϕ_i(a₁. . . a_n) =ϕ₁(a₁)ϕ(a₂. . . a_n),ifi₁ > i₂; ii) ϕ(a1. . . an) =ϕ(a1. . . an−1)ϕin(an),ifin−1< in;

iii)ϕ(a₁. . . a_n) =ϕ(a₁. . . ak−1ψ_ik(a_k)a_k+1. . . a_n)+ϕ(a₁. . . ak−1)[ϕ_ik(a_k)−

ψ_ik(a_k)]ϕ(a_k+1. . . a_n),ifik−1< i_k > i_k+1, for 2≤k≤n−1;

with respect to the natural embeddings of Ai into A arising from the free product construction.

Thus,ϕ:=B^{ψ_B^i}ϕi is aB-B-bimodule map of A inB.

In particular,B^{ψ_B^i}ψi =B_Bψi is the amalgamated monotone product of (ψ_i)i∈I (see Definition 3.4 in [18]).

When I ={1,2}, we denote B^{ψ_B^i}ϕ_i by ϕ₁B^{ψ_B^2} ϕ₂. By reversing the order structure, one can define the amalgamated c-anti-monotone product, denoted C^{ψ_B^i}ϕ_i,respectively, for two maps,ϕ₁C^{ψ_B^1}ϕ₂,and derive from the two propositions below the assertions involving it.

The next statement extended Proposition 3.2 in [18] describes the amal- gamated c-monotone product as a part of an amalgamated c-free product (compare with [10] and [25]).

Proposition 3.5. LetB be an algebra,A_i be two algebras endowed with compatible B-B-bimodule structures,and ϕi :Ai →B be B-B-bimodule maps of A_i in B;let ψ₂ :A₂ →B be a B-B-bimodule map of A₂ in B.

Consider the algebras fA_i :=B⊕A_i, with adjoined algebra B, define the conditional expectations ϕe_i of fA_i onto B,ψf₂ of Af₂ onto B and the homomor- phism δ1 of Af1 onto B by ϕei(b⊕a) := b+ϕi(a), fψ2(b⊕a) := b+ψ2(a), respectively δ1(b⊕a) :=b;if b⊕a∈B⊕A_i =Afi.

Let A be the universal free product of A₁ and A₂ with B amalgamated, Ae:=Af₁∗_BAf₂ be the universal free product of Af₁ andAf₂ with amalgamation over B, and δ:=δ1∗_Bδ2.

Then, A = kerδ, Ae = B ⊕A is the algebra with adjoined algebra B, corresponding to A, and ϕ1B^{ψ_B^2}ϕ2 =ϕf1δ₁ ∗

ψf2 fϕ2 |A.

The proposition below presents the link between the amalgamated c- monotone product and the amalgamatedc-free product map, by analogy with the monotone case (see, e.g. Proposition 3.3 in [18]) (compare with Proposi- tion 3.1 in [10] or Theorem 6.6 in [24]).

Proposition 3.6. Let B be an algebra. Let A_i be two algebras over B, such that A^.₁=B⊕A^◦₁,A^◦₁being an algebra, endowed compatibleB-B-bimodule structure and ϕ_i be conditional expectations ofA_i onto B.

Let ψ2 : A2 → B be a conditional expectation of A2 onto B, δ1 be the unique homomorphism of A^.₁ onto B such that A^◦₁ = kerδ₁,and consider the B-B-bimodule map ϕ^◦₁ :=ϕ₁ |A^◦₁.

Let A := A₁ ∗_B A₂ be the universal free product of A^.₁ and A^.₂, with amalgamation over B, and A^◦ be the universal free product of A^◦₁ and A^.₂ with B amalgamated.

Let ϕ := ϕ_{1 δ1} ∗_ψ2 ϕ₂ : A → B be the amalgamated c-free product con- ditional expectation, and ϕ^◦ := ϕ^◦₁ B^{ψ_B^2}ϕ₂ : A^◦ → B be the amalgamated c-monotone productB-B-bimodule map.

Then ϕ^◦ = ϕ◦j, j being the canonical homomorphism of A^◦ in A, arising from the embeddings A^◦₁ ,→A^.₁,→Aand A^.₂ ,→A,via the universality property.

IfB andA_i are *-algebras, andϕ_i,δ₁,ψ₂ are Hermitian, then the above maps ϕ, ϕ^◦ and j preserve the natural involutions.

Adopting/adapting the terminology from [10], [13], [24], we consider, in the sequel, ϕ1 δ1 ∗_ψ2ϕ2 as the amalgamated c-monotone product conditional expectation of (ϕ_i)_i with respect to ψ₂, if ϕ_i are conditional expectations as above; thus, in the scalar-valued case B = C, we consider this map as the unital c-monotone product.

By duality, we may also consider unital versions of the c-anti-monotone products.

Let B be a *-algebra, as above, let A_i be *-algebras over B, let ψ_i be (Hermitian) conditional expectation of Ai onto B, and A^o_i = kerψi, i ∈ I.

Now, I is an arbitrary set having at bast two elements. The Lemma below was restates Lema 3.4 in [8].

Denote by W = {a₁. . . an; n ≥ 1, a_k ∈ A^o_i

k, i1 6= · · · 6= in} the set of reduced words inA=∗_BA_i.

For w = a1. . . an ∈ W, call n the length of w and a1 the first letter of w. If x = P

k

w^(k) ∈ A^o call the length of x the maximal length in this representation of x.

Lemma 3.7. Let B be a *-algebra, I be a set; let A_i be *-algebras over B, and ϕi, ψi : Ai → B be Hermitian conditional expectations, i ∈ I. Let

ϕ = ∗^{ψi}

B ϕi be the amalgamated c-free product of (ϕi)i∈I with respect to ψi, i∈I, defined on the *-algebraic free product A=∗_BA_i.

Consider two words x1 =y1. . . yn and x2 =z1. . . zm in W. (1)If y₁ and z₁ do not belong to the same A^o_i, then

ϕ(x^∗₁x₂) =ϕ(x₁)^∗ϕ(x₂).

(2)Let a∈A_i,for somei∈I. Ify₁, z₁∈/ A^o_i, thenϕ(x^∗₁a) =ϕ(x₁)^∗ϕ_i(a), ϕ(ax2) =ϕ1(a)ϕ(x2), and

ϕ(x^∗₁ax2) =ϕ(x^∗₁ψi(a)x2)−ϕ(x1)^∗ψi(a)ϕ(x2) +ϕ(x1)^∗ϕi(a)ϕ(x2).

By this lemma, we get a Boolean analogue of Proposition 3.5 in [18] (see also Theorem 6.5 in [24]).

Proposition 3.8. Let B be a C*-algebra, and I by a set. Let A_i be C*-algebras over B, and ϕi, δi be a positive conditional expectation, and re- spectively a *-homomorphism of A_i onto B such thatδ_i|B =id_B, i∈I;and A:=∗_BAi be the *-algebraic free product.

Then, the *-algebraic amalgamated Boolean product ϕ=_Bϕi :=∗^{δ_B^i}ϕi

is a Schwarz map.

Thus,ϕis a positive conditional expectation ofA onto B.

As quantum probability spaces over B,_B (A_i, ϕ_i) = (A,_Bϕ_i).

Proof. Let A^o_i = kerδ_i. In view of Lemma 3.7, it suffices to prove the asserted property for every x(i) in A^o represented as P

k

w^(k) with w^(k) ∈W having the first letter in a same A^o_i.

Assuming that x(i) has p terms of length one, that is x(i) =

p

P

k=1

a^(k)+

N

P

k=p+1

a^(k)y^(k), with all a^(k) ∈ A^o_i; and y^(k) ∈ W, but the first letter of y^(k) does not belong to A^o_i (else, the argument is similar); [since ϕi =ϕ |Ai are B-B-bimodule maps], we infer, by the same Lemma 3.7,

ϕ(x(i)^∗x(i))−ϕ(x(i))^∗ϕ(x(i)) =

=ϕ(y(i)^∗y(i))−ϕ(y(i))^∗ϕ(y(i)) +ϕ_i(x◦(i)^∗x◦(i))−ϕ_i(x◦(i))^∗ϕ_i(x◦(i)), with

x◦(i) :=

p

X

k=1

a^(k)+

N

X

k=p+1

a^(k)ϕ(y^(k))∈Ai, where y(i) =

N

P

k=p+1

δi(a^(k))y^(k) ∈ A^◦ has the length less than the length of x(i).

In consequence, the proof completes by induction; because everyϕi is a Schwarz map, by the above Proposition 2.2 (or Proposition 2.3 in [18]).

Corollary 3.9. Let A_i be unital(complex)C*-algebras, such that A_i= C⊕A^◦_i is a direct sum of linear spaces, A^◦_i being algebras, too; letϕi be states of A_i,i∈I.

Let ϕ=_Bϕibe the *-algebraic Boolean product, defined on the *-algebra A:=∗₁A_i.Then,ϕ(a^∗a)≥ |ϕ(a)|², for all a∈A.Thus,ϕis positive.

We obtain the next statement via Proposition 3.2.

Corollary 3.10. Let B be a C*-algebra and I be a set. Let Ai be C*- algebras endowed with compatible B-B-bimodule structures, and ϕ_i :A_i → B be B-B-bimodule Schwarz maps of Ai in B, i∈I.

Let A be the universal free product of (A_i)i∈I withB amalgamated.

Then, the *-algebraic amalgamated Boolean product ϕ := _Bϕi is a Schwarz map of A in B.

Thus,ϕ is a(completely) positive B-B-bimodule map.

As quantum B-probability spaces, _B(Ai, ϕi) = (A,_Bϕi).

The assertion involving the Boolean product according to Speicher’s study in [28] is as follows (see also Corollary 3.3 in [15]).

Corollary 3.11.Let A_ibe(complex)C*-algebras, and ϕ_ibe functionals on Ai,such that their unitizations are states, i∈I.

Letϕ:=ϕ_i be their *-algebraic Boolean product defined on the *-algebra A:=∗₀A_i.

Then,ϕ(a^∗a)≥ |ϕ(a)|²,for all a∈A. Thus,ϕis positive, too.

The next result extends the previous Proposition 3.8 and Proposition 3.5 in [18].

Theorem 3.12. Let B be a C*-algebra. Let A_i be two C*-algebras over B, and A:=A1∗_BA2 be the ∗-algebraic free product.

Let ϕ1, δ1 be a positive conditional expectation, and respectively a *- homomorphism of A₁ onto B such that δ₁ |B =id_B. Let ϕ₂, ψ₂ be positive conditional expectations of A2 onto B.

Then, the *-algebraic amalgamatedc-free product ϕ:=ϕ_{1 δ1}∗_ψ2 ϕ₂ is a Schwarz map.

Thus,ϕis a positive conditional expectation ofA onto B.

Proof. Denote now A^o₁ = kerδ₁, andA^o₂ = kerψ₂.By Lemma 3.7 again, it is enough to check the necessary condition for every x(i) in A^o represented as P

k

w^(k) withw^(k)∈W having the first letter in a sameA^o_i; ifi∈ {1,2}.

Suppose, for example, that x(i) =

p

P

k=1

a^(k)+ PN

k=p+1

a^(k)y^(k), every a^(k) ∈ A^o_i; and y^(k)∈W, but the first letter ofy^(k) does not belong toA^o_i. Then, as above, we obtain, through the mentioned Lemma 3.7, since ϕi = ϕ | Ai are B-B-bimodule maps,

ϕ(x(i)^∗x(i))−ϕ(x(i))^∗ϕ(x(i)) =b(i) +ϕ_i(x◦(i)^∗x◦(i))−ϕ_i(x◦(i))^∗ϕ_i(x◦(i)), with the same x◦(i) :=

p

P

k=1

a^(k)+

N

P

k=p+1

a^(k) ϕ(y^(k)) ∈A_i; but now b(1) :=ϕ(y(1)^∗y(1))−ϕ(y(1))^∗ϕ(y(1)),

where y(1) =

N

P

k=p+1

δ1(a^(k))y^(k) ∈ A^◦ has the length less than the length of x(1); and respectively,

b(2) :=

N

X

k,l=p+1

[ϕ(y^(k)∗ψ2(a^(k)∗a^(l))y^(l))−ϕ(y^(k))^∗ψ2(a^(k)∗a^(l))ϕ(y^(l))].

And now, by the complete-positivity ofψ₂,we may expressψ₂(a^(k)∗a^(l)) = P

r

b^(k)∗_r b^(l)_r , with someb^(k)_r ∈B (according to Proposition 2.2), and thusb(2) = P

r

[ϕ(x^(r)∗x^(r))−ϕ(x^(r))^∗ϕ(x^(r))], where x^(r) =

N

P

k=p+1

b^(k)r y^(k) ∈ A^o has the length less than the length of x(2).

Therefore, we conclude by induction on the length of thex(i)⁰s, because ϕ_i are Schwarz maps; due to the same Proposition 2.2 (or Proposition 2.3 in [18]).

Thus, we derive the following facts involving operator-valued versions of Lenczewski’s orthogonal product in [19].

Corollary 3.13. Let B be a C*-algebra. Let A_i be two C*-algebras over B, and A:=A₁∗_BA₂ be their ∗-algebraic free product.

Let ϕ1, δ1 be a positive conditional expectation, and respectively a *- homomorphism of A₁ onto B such that δ₁|B =id_B.Let δ₂, ψ₂ be *-homomor- phism, and respectively a positive conditional expectation of A2 onto B such that δ₂|B =id_B.

Then the *-algebraic amalgamated orthogonal product ϕ1 `_B ψ2 :=

ϕ_{1 δ1} ∗_ψ2 δ₂ is a Schwarz map.

Thus, this is a positive conditional expectation ofA onto B.

As quantum probability spaces over B,

(A1, ϕ1)`<sub>B</sub>(A2, ψ2) = (A, ϕ1`_B ψ2).

We get by Proposition 2.1.

Corollary 3.14. Let B be aC*-algebra. Let A_ibe twoC*-algebras en- dowed with compatible B-B-bimodule structures. Let ϕ₁psi₂ beB-B-bimodule Schwarz maps of A1, respectively A2 inB.

Consider the C*-algebras Afi :=B⊕A_i with adjoined algebra B, define the conditional expectations fϕ1 ψf2 of Af1, respectively Af2 onto B, and the *- homomorphismsδiof AfiontoBby ϕei(b⊕a) :=b+ϕi(a),ψf2(b⊕a) :=b+ψ2(a), respectively δ1(b⊕a) :=b;if b⊕a∈B⊕A_i =Afi

Let A be the *-algebraic universal free product of (A_i)_i with B amalga- mated.

Then, the *-algebraic amalgamated orthogonal product ϕ₁ `_B ψ₂ :=

ϕ₁B^{ψ_B^2}0₂ =ϕf_{1 δ1} ∗

ψf2 δ₂ |A is a Schwarz map. As quantum B-probability spaces,

(A₁, ϕ1)`<sub>B</sub>(A2, ψ2) = (A, ϕ1`_B ψ2).

(Here 02 is the null map ofA2 inB.)

The next two facts concern the amalgamated c-monotone product, ex- tending Proposition 3.5 (and Corollary 3.7 in [18]; concerning operator-valued versions of Muraki’s monotone product).

Corollary 3.15. Let B be a C*-algebra. Let Ai be two C*-algebras over B, endowed with pairs of positive conditional expectations ϕi, ψi of Ai

onto B. Let δ₁ be a *-homomorphism of A₁ onto B such that δ₁ |B = id_B, and A:=A1∗_BA2 be the ∗-algebraic free product.

Then, the *-algebraic amalgamated c-monotone product (ϕ, ψ) :=

(ϕ1 δ1∗_ψ2ϕ2 ψ1 δ1∗ _ψ2 ψ2) consists of Schwarz maps; i.e., of positive condi- tional expectations of A onto B.

As quantum probability spaces over B,

(A₁, ϕ₁, ψ₁)BB(A₂, ϕ₂, ψ₂) = (A, ϕ, ψ).

We get by Proposition 3.5 (and Proposition 2.1).

Corollary 3.16. Let B be a C*-algebra. Let A_i be two C*-algebras endowed with compatible B-B-bimodule structures, and with pairs of B-B- bimodule Schwarz maps ϕi, ψi of Ai in B.

Consider the C*-algebras fA_i :=B⊕A_i with adjoined algebra B, define the conditional expectations ϕei ψei offAi onto B, and the *-homomorphisms δ1

of Af1 onto B by ϕei(b⊕a) := b+ϕi(a), ψei(b⊕a) := b+ψi(a), respectively δ1(b⊕a) :=b;if b⊕a∈B⊕A_i=fAi.

Let A be the *-algebraic universal free product of (Ai)i with B amalga- mated. Then, the *-algebraic amalgamated c-monotone product

(ϕ, ψ) := (ϕ1B^{ψ_B^2}ϕ2, ψ1BBψ2) := (ϕf1 δ1 ∗

ψf2ϕf2 |A,ψf1 δ1 ∗

ψf2 ψf2|A) consists of B-B-bimodule Schwarz maps. As quantum B-probability spaces, (A1, ϕ1, ψ1)BB (A2, ϕ2, ψ2) = (A, ϕ, ψ).

The statement involving the unital c-monotone product (see [13]) is as follows.

Corollary 3.17. Let A_i be two unital(complex)C*-algebras, such that A₁=C⊕A^◦₁ is a direct sum of linear spaces, A^◦₁ being an algebra, and ϕ_i, ψ_i be states of Ai.

Let (ϕ, ψ) := (ϕ1B^{ψ2}ϕ2, ψ1Bψ2) be their unital c-monotone product defined on the *-algebra A:=A1∗₁A2.

Then,ϕ(a^∗a)≥ |ϕ(a)|², and ψ(a^∗a)≥ |ψ(a)|²,for all a∈A.

Thus,ϕand ψ are states, too.

The statement involving the c-monotone product in Hasebe’s sense in [11] is the next

Corollary 3.18. Let A_i be two (complex) C*-algebras endowed with pairs of linear functionals ϕi ψi,such that their unitizations are states of Ai. Let (ϕ, ψ) := (ϕ1 B^{ψ2} ϕ2, ψ1 Bψ2) be their *-algebraic c-monotone product defined on the *-algebra A:=A1∗₀A2.

Then,ϕ(a^∗a)≥ |ϕ(a)|²,and ψ(a^∗a)≥ |ψ(a)|²,for all a∈A.

Thus,ϕand ψ are positive, too.

By analogy, one shows the statements going to the amalgamated c-anti- monotone product (including the dual of the orthogonal product), and we omit the proofs. These facts concerning operator-valued versions of Muraki’s anti-monotone product extend Theorem 3.14, Corollary 3.16, Corollary 3.18, Corollary 3.19 from [18].

Theorem 3.19. Let B be a C*-algebra. Let Ai be two C*-algebras over B, and A := A₁∗_BA₂ be the ∗-algebraic free product. Let ϕ₁, ψ₁ be positive conditional expectations of A1 onto B.

Let ϕ₂, δ₂be a positive conditional expectation, and, respectively, a *- homomorphism of A₂ onto B such that δ₂|B=id_B.

Then the *-algebraic amalgamatedc-free product ϕ:=ϕ1 ψ1 ∗_δ2 ϕ2 is a Schwarz map.

Thus,ϕis a positive conditional expectation ofA onto B.

Corollary 3.20. Let B be a C*-algebra. Let A_i be two C*-algebras over B, endowed with pairs of positive conditional expectations ϕi, ψi of Ai

onto B; i = 1,2. Let δ2 be a *-homomorphism of A2 onto B such that δ2 | B =id_B,and A:=A₁∗_BA₂ be the ∗-algebraic free product.

Then, the *-algebraic amalgamated c-anti-monotone product (ϕ1 ψ1 ∗_δ2 ϕ2, ψ1 ψ1∗_δ2ψ2)consists of Schwarz maps; i.e., of positive conditional expec- tations of A onto B.

As quantum probability spaces over B, (A1, ϕ1, ψ1) CB (A2, ϕ2, ψ2) = (A, ϕ, ψ).

This fact can be deduced via Proposition 2.1 and the dual of Proposi- tion 3.5.

Corollary 3.21. Let B be a C*-algebra. Let A_i be two C*-algebras endowed with compatible B-B-bimodule structures, and with pairs of B-B- bimodule Schwarz maps ϕ_i, ψ_i of A_i in B.

Consider the *-algebras fA_i :=B⊕A_i with adjoined algebraB, define the conditional expectations ϕei, ψei of fAi onto B, and the *-homomorphisms δ2

of Af2 onto B by ϕei(b⊕a) := b+ϕi(a), ψei(b⊕a) := b+ψi(a), respectively δ2(b⊕a) :=b;if b⊕a∈B⊕A_i=fAi.

Let A be the *-algebraic universal free product of (A_i)_i with B amalga- mated.

Then, the *-algebraic amalgamated c-anti-monotone product

(ϕ, ψ) := (ϕ1C^{ψ_B^1}ϕ2, ψ1CBψ2) := (ϕf1 ψf1 ∗_δ2 ϕf2|A,fψ1 ψf1 ∗ _δ2 ψf2 |A) consists of B-B-bimodule Schwarz maps. As quantum B-probability spaces,

(A1, ϕ1, ψ1)CB(A2, ϕ2, ψ2) = (A, ϕ, ψ).

Corollary 3.22. Let A_i be two (complex) C*-algebras endowed with pairs of linear functionals ϕ_i ψ_i,such that their unitizations are states of A_i. Let(ϕ, ψ) := (ϕ₁C^{ψ1}ϕ₂, ψ₁Cψ₂)be their *-algebraic c-anti-monotone product defined on the *-algebra A:=A₁∗₀A₂.

Thenϕ(a^∗a)≥ |ϕ(a)|², and ψ(a^∗a)≥ |ψ(a)|²,for all a∈A.

Thus,ϕand ψ are positive.

The above *-algebraic amalgamated c-free, Boolean, orthogonal, c-mo- notone or c-anti-monotone product maps extend to corresponding completely positive Schwarz maps on the amalgamated universal (full) free product C*- algebra∗_BAi, or the involvedC*-algebra?i∈I(Ai, εi = 0, B), (see, e.g., Remark 3.23 in [18]).

4. AMALGAMATED INVOLVING PAIRS OF MAPS VALUED IN DIFFERENT C*-ALGEBRAS

In the sequel, we extend the results from the previous section.

Consider an inclusion of (*-)algebrasB ⊂D.

LetA_i,i∈I,be (*-)algebras overD(therefore,A_iare naturally endowed with compatible B-B-bimodule structures, too); letϕi be (Hermitian) condi- tional expectations of Ai onto D (therefore, ϕi are also B-B-bimodule maps and their restrictions toB coincide with the embedding ofB intoD), and let ψi be (Hermitian) conditional expectations of Ai ontoB. Thus, Ai =B⊕A^◦_i, are direct sums of B-B-bimodules, withA^o_i = kerψ_i,the kernel ofψ_i,i∈I.

The amalgamated c-free product ϕ = ∗^{ψi}

B ϕ_i, in Boca’s sense [4], of (ϕ_i)i∈I with respect to ψ_i,i∈I,is the unique linear map well-defined on the amalgamated universal free product A= ∗_BA_i, and taking values in D (see, e.g., [4]) such that:

1)ϕ|A_i =ϕ_i, for each i∈I;

2) ϕ(a1. . . an) = ϕi1(a1). . . ϕin(an), for all n ≥ 2, i1 6= · · · 6= in, and a_k∈A_ik, withψ_ik(a_k) = 0, ifk= 1, . . . , n;

relatively to the natural embeddings ofAiintoAarising from the free product construction.

Therefore, ϕ = ∗^{ψi}

B ϕ_i : A → D is a (Hermitian) B-B-bimodule map (which is not a conditional expectation anymore) and its restriction to B coincides with the embedding ofB intoD.

WhenD=B, we recover the case in the previous section.

Throughtout, as before,I is a set having at least two elements.

The following definition naturally extend Definition 3.1.

Definition 4.1. Let B ⊂ D be an inclusion of algebras, and I be a set.

LetAi be algebras overD, andϕi :Ai →Dbe conditional expectations ofAi

onto D,i∈I.

The amalgamated Boolean product ϕ := _Bϕi of (ϕi)i∈I is the unique linear map well-defined on the universal free product A of (A_i)i∈I with B amalgamated, such that:

1)ϕ|Ai =ϕi,for each i∈I;

2) forn≥2, i₁ 6=· · · 6=i_n, and a_k∈A_ik,withk= 1, . . . , n;

ϕ(a1. . . an) = ϕ_i

1(a1). . . ϕ_in(an); with respect to the natural embeddings of Ai intoA arising from the free product construction.

Thus, ϕ := _Bϕ_i is a B-B-bimodule map of A in D (which is not a conditional expectation anymore).

WhenD=B, we recover the situation in the previous section.

The next statement describes the amalgamated Boolean product as a part of an amalgamated c-free product, and extends Proposition 3.2.

Proposition 4.2. Let B ⊂ D be an inclusion of algebras and I be a set. Let Ai be algebras over D, and ϕi :Ai → D be conditional expectations of Ai onto D, i∈I.

Consider the algebras Af_i :=B⊕A_i with adjoined algebra B, define the conditional expectations ϕei of Afi →D,and the homomorphismsδiof fAi onto B by ϕei(b⊕a) :=b+ϕi(a),respectively δi(b⊕a) :=b;if b⊕a∈B⊕A_i =Afi, i∈I.

LetA be the universal free product of (Ai)i∈I withB amalgamated,Ae:=

∗_BfAi be the universal free product of (Aei)i∈I with amalgamation overB, and δ :=∗_Bδi.

Then, A = kerδ, Ae = B ⊕A is the algebra with adjoined algebra B, corresponding to A,and _Bϕ_i =∗^{δ_B^i}ϕe_i|A.

The proposition below presents the connection between the amalgamated Boolean product and the amalgamatedc-free product map, extending Propo- sition 3.3.

Proposition 4.3.Let B ⊂Dbe an inclusion of algebras andI be a set.

Let A_i be algebras overD, such that A^._i=B⊕A^◦_i, A^◦_i being algebras endowed with compatible B-B-bimodule structures, and ϕi : Ai → D be conditional expectations of A_i onto D. Let δ_i be the unique homomorphisms of A^._i onto B such that A^◦_i = kerδi, and consider the B-B-bimodule maps ϕ^◦_i :=ϕi |A^◦_i, i∈I.

Let A:= ∗_BA_i be the universal free product of (A_i)i∈I, with amalgama- tion over B, and A^◦ be the universal free product of (A^◦_i)i∈I with B amalga- mated.

Let ϕ := ∗^{δ_B^i}ϕi : A → D be the amalgamated c-free product B-B- bimodule map andϕ^◦ :=_Bϕ^◦_i :A^◦→Dbe the amalgamated Boolean product B-B-bimodule map.

Then, ϕ^◦ = ϕ◦j, j being the canonical homomorphism of A^◦ in A, arising from the embeddings A^◦_i ,→A^._i,→Avia the universality property.

IfB,D and Ai are *-algebras and ϕi, δi are Hermitian, then the above maps ϕ, ϕ^◦ andj preserve the natural involutions.

Adopting the terminology before from ([10], [13], [23], [30]), we consider, in the sequel, ∗^{δ_B^i}ϕi as the amalgamated Boolean product of (ϕi)i∈I, if ϕi

are conditional expectations as above.

This definition naturally extends Definition 3.4.

Definition 4.4. LetB ⊂Dbe an inclusion of algebras, andI be a totally ordered set. Let Ai be algebras over D. Let ϕi : Ai → D be conditional expectations of Ai ontoD, and ψi:Ai→B beB-B-bimodule maps,i∈I.

The amalgamatedc-monotone productϕ:=B^{ψ_B^i}ϕ_i of (ϕ_i)i∈I with res- pect to (ψi)i∈I is the unique linear map well-defined on the universal free product A of (A_i)i∈I withB amalgamated, such that:

1)ϕ|A_i =ϕ_ii,for each i∈I;