Reduced amalgamated free products of tracial algebras

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

Two tracial algebras that share a subalgebra \(D\), with trace-preserving conditional expectations onto \(D\), generate a free product over \(D\): an algebra in which products of elements alternating between the two factors and centered for the expectations have expectation zero. This lesson constructs it on a Hilbert module over \(D\), proves that it carries a faithful trace extending both traces, and constructs the conditional expectation onto one factor. The next lesson, Reduced HNN adjunctions, cuts a corner out of such a free product to adjoin a unitary. The construction follows [OpenAI-NA, Lemma 3.2]; the corresponding statements for reduced amalgamated free products are in [Blanchard–Dykema 2001, Lemma 1.1 and Theorem 1.3]. Notation is that of Pure states, excision and peaking projections and Expectations from peaking projections.

Throughout, Hilbert modules are right modules with inner products linear in the second variable, as in Hilbert C\(^*\)-modules; \(\mathcal L(X)\) denotes the C\(^*\)-algebra of adjointable operators on \(X\) (Theorem 2.2 of Adjointable operators).

Setting. \((F,\tau_F)\) and \((G,\tau_G)\) are tracial algebras containing a common unital C\(^*\)-subalgebra \(D\) on which the traces agree, with value \(\tau_D\); and \(E^F_D\colon F\to D\), \(E^G_D\colon G\to D\) are trace-preserving conditional expectations. The letter \(i\) stands for \(F\) or \(G\). An element \(a\in i\) is centered if \(E^i_D(a)=0\), and \(i^\circ=\ker E^i_D\).

1. Hilbert modules from expectations

On \(i\) put \(\langle x,y\rangle=E^i_D(x^*y)\). It is \(D\)-linear in \(y\), hermitian and positive, and definite: \(\tau_D(E^i_D(x^*x))=\tau_i(x^*x)>0\) for \(x\ne0\). So \(i\), with right multiplication by \(D\), is a pre-Hilbert \(D\)-module; let \(L_i\) be its completion (Theorem 2.3 of the Hilbert module lesson).

Lemma 1.1.

  1. For \(a\in i\), left multiplication \(x\mapsto ax\) extends to an adjointable operator \(\lambda_i(a)\) of \(L_i\) with adjoint \(\lambda_i(a^*)\), and \(\lambda_i\colon i\to\mathcal L(L_i)\) is an injective unital \(*\)-homomorphism.
  2. The map \(x\mapsto E^i_D(x)\) extends to a projection \(P\in\mathcal L(L_i)\) with range \(D\), on which the module norm is the C\(^*\)-norm. So \(L_i=D\oplus L_i^\circ\) with \(L_i^\circ=(1-P)L_i\), the closure of \(i^\circ\). Both summands are invariant under \(\lambda_i(D)\).

Proof. (1) Since \(x^*a^*ax\le\|a\|^2x^*x\) and \(E^i_D\) is positive, \(\langle ax,ax\rangle\le\|a\|^2\langle x,x\rangle\), so \(\|ax\|\le\|a\|\,\|x\|\). Also \(\langle ax,y\rangle=E^i_D(x^*a^*y)=\langle x,a^*y\rangle\). Multiplicativity and \(\lambda_i(1)=1\) are clear. As \(\lambda_i(a)1=a\) and \(L_i\) contains \(i\), \(\lambda_i\) is injective.

(2) A conditional expectation has norm one, so Tomiyama's theorem (Theorem 8.5(3) of The universal enveloping von Neumann algebra of a C\(^*\)-algebra and W\(^*\)-algebras) gives \[ \begin{aligned}\langle E^i_D(x),E^i_D(x)\rangle&=E^i_D(x)^*E^i_D(x)\\&\le E^i_D(x^*x).\end{aligned} \] So \(x\mapsto E^i_D(x)\) is bounded and extends to \(P\). Both \(\langle E^i_D(x),y\rangle\) and \(\langle x,E^i_D(y)\rangle\) equal \(E^i_D(x)^*E^i_D(y)\), so \(P\) is self-adjoint, and \(P^2=P\). On \(D\), \(\langle d,d\rangle=d^*d\), so the module norm is the C\(^*\)-norm, \(D\) is complete and closed in \(L_i\), and \(PL_i=D\). The range of \(1-P\) is the closure of \((1-P)(i)=i^\circ\). Finally \(E^i_D(dx)=dE^i_D(x)\), so \(\lambda_i(d)\) commutes with \(P\). \(\square\)

2. The free product module

A word is a finite sequence \(\iota=(i_1,\ldots,i_m)\), \(m\ge1\), of letters from \(\{F,G\}\) with \(i_t\ne i_{t+1}\). For a word put \[ L_\iota=L^\circ_{i_1}\otimes_DL^\circ_{i_2}\otimes_D\cdots\otimes_DL^\circ_{i_m}, \] the interior tensor product formed with the left actions \(\lambda_{i_t}|_D\) (Theorems 1.2 and 3.1 of Tensor products and C\(^*\)-correspondences). Let \[ K=D\oplus\bigoplus_\iota L_\iota , \] the completion of the algebraic direct sum for the sum of the inner products (Theorem 2.3 of the Hilbert module lesson). The vacuum vector is \(\Omega=1\in D\). The algebra \(D\) acts on \(K\) on the left: by multiplication on the summand \(D\), and on \(L_\iota\) through its first tensor factor (Theorem 2.1 of the tensor lesson). For \(i\in\{F,G\}\) let \(K^{(i)}\) be the closed span of \(D\) and the \(L_\iota\) with \(i_1\ne i\), a submodule invariant under the left action of \(D\).

Interior tensor products distribute over orthogonal direct sums, finite or countable: on finite sums of elementary tensors the inner products (1.2) of the tensor lesson add over the summands, and the images are dense. Together with the units \(D\otimes_DY\cong Y\) for a unital left action (Proposition 3.2 there), this gives the following.

Lemma 2.1. There is a unitary \(V_i\colon L_i\otimes_DK^{(i)}\to K\) of Hilbert \(D\)-modules with \[ \begin{gathered} V_i(d\otimes\eta)=d\eta\quad(d\in D,\ \eta\in K^{(i)}),\\ V_i(x\otimes d)=xd\in L_{(i)},\\ V_i(x\otimes\eta)=x\otimes\eta\in L_{(i,\iota)} \end{gathered} \] for \(x\in L_i^\circ\), \(d\in D\), and \(\eta\in L_\iota\) with \(\iota\) not starting with \(i\).

Proof. By Lemma 1.1(2) and distributivity, \[ \begin{aligned}&L_i\otimes_DK^{(i)}\\&\quad\cong\bigl(D\otimes_DK^{(i)}\bigr)\oplus\bigl(L_i^\circ\otimes_DD\bigr)\\&\qquad\oplus\bigoplus_{i_1\ne i}\bigl(L_i^\circ\otimes_DL_\iota\bigr).\end{aligned} \] The first summand is \(K^{(i)}\), the second is \(L_i^\circ=L_{(i)}\), and the remaining ones are the \(L_{(i,\iota)}\), which are exactly the summands of \(K\) indexed by words starting with \(i\) and having length at least two. Together these summands make up \(K\). \(\square\)

For \(a\in i\) put \[ \pi_i(a)=V_i\bigl(\lambda_i(a)\otimes1\bigr)V_i^*\in\mathcal L(K), \] an adjointable operator by Theorem 2.1 of the tensor lesson.

Lemma 2.2.

  1. \(\pi_F\) and \(\pi_G\) are injective unital \(*\)-homomorphisms. On \(D\) both equal the left action of \(D\) on \(K\).
  2. For \(a\in i\), \(\pi_i(a)\Omega\) is the vector \(E^i_D(a)+\bigl(a-E^i_D(a)\bigr)\) of \(D\oplus L_{(i)}\).
  3. For \(a\in i^\circ\) and \(\eta\in K^{(i)}\), \(\pi_i(a)\eta=V_i(a\otimes\eta)\) lies in the closed span of the \(L_\iota\) with \(\iota\) starting with \(i\). For \(\eta\in L_\iota\) with \(i_1\ne i\), \(\pi_i(a)\eta=a\otimes\eta\in L_{(i,\iota)}\); for \(d\in D\), \(\pi_i(a)d=ad\in L_{(i)}\).

Proof. For \(\eta\in K^{(i)}\), \(V_i^*\eta=1\otimes\eta\). So \(\pi_i(a)\eta=V_i(a\otimes\eta)\), and with \(a=E^i_D(a)+(a-E^i_D(a))\) the formulas of Lemma 2.1 give (2) and (3). The map \(a\mapsto\lambda_i(a)\otimes1\) is a unital \(*\)-homomorphism (Theorem 2.1 of the tensor lesson), so \(\pi_i\) is one. By (2), \(\pi_i(a)\Omega\ne0\) for \(a\ne0\). For \(d\in D\), \(\pi_i(d)\) acts on \(V_i(d'\otimes\eta)=d'\eta\) as \(dd'\eta\), and on \(V_i(x\otimes\eta)\) as \(dx\otimes\eta\), with \(dx\in L_i^\circ\) by Lemma 1.1(2): this is the left action of \(d\) on \(K\). \(\square\)

3. The algebra and its trace

Let \(Z\) be the C\(^*\)-subalgebra of \(\mathcal L(K)\) generated by \(\pi_F(F)\cup\pi_G(G)\). We identify \(F\) and \(G\) with their images; they intersect in \(D\) as required. A centered alternating product of type \(\iota=(i_1,\ldots,i_m)\) is a product \(a_1a_2\cdots a_m\) with \(a_t\in i_t^\circ\).

Lemma 3.1.

  1. A centered alternating product of type \(\iota\) maps \(\Omega\) to \(a_1\otimes a_2\otimes\cdots\otimes a_m\in L_\iota\).
  2. \(D\) and the centered alternating products span a dense \(*\)-subalgebra \(S\) of \(Z\).

Proof. (1) By Lemma 2.2(3), \(a_m\Omega=a_m\in L_{(i_m)}\), and if \(\eta\in L_{(i_{t+1},\ldots,i_m)}\), a word not starting with \(i_t\), then \(a_t\eta=a_t\otimes\eta\).

(2) The adjoint of a centered alternating product is one, since \(E^i_D(a^*)=E^i_D(a)^*\). Multiplying by elements of \(D\) keeps the product centered alternating, as \(E^i_D(da)=dE^i_D(a)\). We show by induction on \(m+n\) that the product of centered alternating products \(w=a_1\cdots a_m\) and \(w'=b_1\cdots b_n\) lies in \(S\). If the letters of \(a_m\) and \(b_1\) differ, \(ww'\) is centered alternating. Otherwise \(c=a_mb_1\) lies in one factor; write \(c=E(c)+c^\circ\) with \(E(c)\in D\) and \(c^\circ\) centered. Then \(a_1\cdots a_{m-1}c^\circ b_2\cdots b_n\) is centered alternating, and in the remaining term \(a_1\cdots a_{m-1}\cdot E(c)b_2\cdots b_n\) the second factor is centered alternating of length \(n-1\) (or lies in \(D\)), so the induction applies. Hence \(S\) is a \(*\)-algebra. It contains \(F\) and \(G\), because \(a=E(a)+(a-E(a))\), so it is dense in \(Z\). \(\square\)

For \(z\in Z\) put \(E_D(z)=\langle\Omega,z\Omega\rangle\in D\), the vacuum component of \(z\Omega\).

Proposition 3.2. \(E_D\) is a conditional expectation of \(Z\) onto \(D\). It extends \(E^F_D\) and \(E^G_D\) and vanishes on every centered alternating product. The functional \(\tau_Z=\tau_D\circ E_D\) is a faithful tracial state of \(Z\) extending \(\tau_F\) and \(\tau_G\).

Proof. Expectation. \(E_D(1)=1\) and \(E_D(d)=\langle1,d\rangle=d\). For \(d,d'\in D\), \(d'\Omega=\Omega d'\), so \[ E_D(dzd')=\langle\Omega d^*,z\Omega d'\rangle=d\,E_D(z)\,d' . \] For \(z_1,\ldots,z_n\in Z\) and \(d_1,\ldots,d_n\in D\), \[ \begin{aligned}&\sum_{j,k}d_j^*E_D(z_j^*z_k)d_k\\&\quad=\Bigl\langle\sum_jz_j\Omega d_j,\sum_kz_k\Omega d_k\Bigr\rangle\ge0,\end{aligned} \] so \(E_D\) is completely positive (Proposition 3.2(1) of Completely positive maps). By Lemma 2.2(2), \(E_D(a)=E^i_D(a)\) for \(a\in i\), and by Lemma 3.1(1), \(E_D\) vanishes on centered alternating products.

Trace. We show \(\tau_Z(aw)=\tau_Z(wa)\) for \(a\in F\cup G\) and \(w\) in \(S\). For \(a=d\in D\), \(\tau_D(dE_D(w))=\tau_D(E_D(w)d)\) because \(\tau_D\) is tracial. So let \(a\in i^\circ\). If \(w\in D\), both sides vanish. If \(w=b\) is a single centered letter, both sides vanish when \(b\) lies in the other factor, and \(\tau_Z(ab)=\tau_i(ab)=\tau_i(ba)=\tau_Z(ba)\) when \(b\in i\). If \(w=b_1\cdots b_n\) with \(n\ge2\) and \(b_1\notin i\), then \(aw\) is centered alternating, and if \(b_1\in i\), then \(aw=E(ab_1)b_2\cdots b_n+(ab_1)^\circ b_2\cdots b_n\) is a sum of two centered alternating products of positive length. In all cases \(\tau_Z(aw)=0\); by the same argument at the right end, \(\tau_Z(wa)=0\). Products of elements of \(F\cup G\) span \(S\), and moving one letter at a time from the front to the back gives \(\tau_Z(xw)=\tau_Z(wx)\) for \(x,w\in S\); by continuity \(\tau_Z\) is tracial. It extends \(\tau_i\), since \(\tau_D\circ E^i_D=\tau_i\).

Faithfulness. Let \(\tau_Z(z^*z)=0\). For \(t\in Z\), \[ \begin{aligned}0\le\tau_Z(t^*z^*zt)&=\tau_Z(ztt^*z^*)\\&\le\|t\|^2\tau_Z(zz^*)\\&=\|t\|^2\tau_Z(z^*z)=0 .\end{aligned} \] As \(\tau_D\) is faithful, \(E_D((zt)^*(zt))=\langle zt\Omega,zt\Omega\rangle=0\), so \(zt\Omega=0\). By Lemma 3.1(1), \(Z\Omega\) contains the elementary tensors \(a_1\otimes\cdots\otimes a_m\) with \(a_t\in i_t^\circ\), which span a dense subspace of each \(L_\iota\) because \(i_t^\circ\) is dense in \(L^\circ_{i_t}\); it also contains \(D\). So \(Z\Omega\) is dense in \(K\), \(z\) vanishes on a dense subspace, and \(z=0\). \(\square\)

4. The expectation onto a factor

Let \(P_F\) be the projection of \(K\) onto \(D\oplus L_{(F)}=V_F(L_F\otimes\Omega)\), and \(P_G\) the projection onto the closed span of the \(L_\iota\) with \(\iota\) starting with \(G\). A core is a centered alternating product \(w=b_1a_1b_2\cdots a_{m-1}b_m\), \(m\ge1\), that starts and ends with a letter from \(G\): \(b_t\in G^\circ\), \(a_t\in F^\circ\).

Lemma 4.1.

  1. \(P_F\) commutes with \(F\), and \(P_FwP_F=0\) for every core \(w\).
  2. For every core \(w\), \((1-P_G)w(1-P_G)=0\); hence \(w=P_Gw+(1-P_G)wP_G\).
  3. For \(d\in F\), \(\|dP_G\|\le\|E^F_D(d^*d)\|^{1/2}\).

Proof. (1) \(\lambda_F(a)\otimes1\) maps \(L_F\otimes\Omega\) into itself and preserves its orthogonal complement \(L_F\otimes(K^{(F)}\ominus D)\), so \(\pi_F(a)\) commutes with \(P_F\). The range of \(1-P_G\) is \(D\) together with the words starting with \(F\), which lies in \(K^{(G)}\). By Lemma 2.2(3), \(b_m\) maps \(K^{(G)}\) into the words starting with \(G\), which lie in \(K^{(F)}\); then \(a_{m-1}\) maps these into the words starting with \(F\), and so on; finally \(b_1\) maps into the words starting with \(G\). So \(w(1-P_G)K\subseteq P_GK\). Since \(P_FK\subseteq(1-P_G)K\) is orthogonal to \(P_GK\), \(P_FwP_F=0\).

(2) This is the inclusion \(w(1-P_G)K\subseteq P_GK\) just proved.

(3) \(P_GK\subseteq K^{(F)}\), and for \(\eta\in K^{(F)}\), \(d\eta=V_F(d\otimes\eta)\). By (1.2) of the tensor lesson, \[ \begin{aligned}\langle d\otimes\eta,d\otimes\eta\rangle&=\langle\eta,E^F_D(d^*d)\eta\rangle\\&\le\|E^F_D(d^*d)\|\,\langle\eta,\eta\rangle,\end{aligned} \] since \(c\le\|c\|\) for \(c\ge0\) in \(D\) and the left action of \(D\) is a \(*\)-homomorphism into \(\mathcal L(K)\). \(\square\)

Proposition 4.2. There is a faithful trace-preserving conditional expectation \(E_F\colon Z\to F\), determined by \(P_FzP_F=E_F(z)P_F\). It vanishes on every product \(a_0wa_m\) with \(a_0,a_m\in F\) and \(w\) a core, and \(E_D=E^F_D\circ E_F\). The space \(F\) together with these products spans a dense subspace of \(Z\).

Proof. A centered alternating product either lies in \(F\) or contains a letter from \(G\); in the second case it is \(a_0wa_m\), where \(w\) runs from the first to the last letter from \(G\), and \(a_0,a_m\) are the remaining \(F\)-letters or \(1\). By Lemma 3.1(2), \(F\) and the products \(a_0wa_m\) span a dense subspace. For such products, Lemma 4.1(1) gives \(P_Fa_0wa_mP_F=a_0P_FwP_Fa_m=0\). So \(P_FzP_F\) lies in \(\Theta(F)\) for \(z\) in this span, where \(\Theta(f)=fP_F\) restricted to \(P_FK\). The map \(\Theta\) is an injective \(*\)-homomorphism (it sends \(f\) to an operator mapping \(\Omega\) to \(f\in L_F\)), so it is isometric with closed range. As \(z\mapsto P_FzP_F\) is continuous, \(P_FzP_F\in\Theta(F)\) for every \(z\in Z\), and \(E_F(z)=\Theta^{-1}(P_FzP_F)\) is well defined. It is unital, completely positive, fixes \(F\), and is \(F\)-bimodular because \(P_F\) commutes with \(F\). Since \(\Omega\in P_FK\), \(E_D(z)=\langle\Omega,P_FzP_F\Omega\rangle=E^F_D(E_F(z))\), and \(\tau_F\circ E_F=\tau_D\circ E_D=\tau_Z\). Faithfulness follows from that of \(\tau_Z\). \(\square\)

We call \(Z\) the reduced amalgamated free product of \(F\) and \(G\) over \(D\), and write \(Z=F*_DG\).

5. Exercises

5.1. Show that \(E_F(g)=E^G_D(g)\) for \(g\in G\).

5.2. For centered \(a_1,a_2\in F^\circ\) and \(b_1,b_2\in G^\circ\), compute \(\tau_Z(a_1b_1a_2b_2)\) and \(\tau_Z(a_1b_1b_2a_2)\) in terms of \(\tau_F\), \(\tau_G\) and the expectations.

5.3. Let \(D=\mathbb C\), \(F=G=\mathbb C^2\) with the uniform traces, and \(u=(1,-1)\in F\), \(v=(1,-1)\in G\). Show that \(\tau_Z((uv)^n)=0\) for every integer \(n\ne0\).

5.4. Show that \(K=D\) and \(Z=D\) when \(F=G=D\).

5.5. Show that the vacuum vector \(\Omega\) is cyclic for \(Z\): the closed span of \(Z\Omega\) is \(K\).

6. Solutions

5.1. Write \(g=E^G_D(g)+g^\circ\). Here \(g^\circ\) is a core of length one, so \(E_F(g^\circ)=0\) by Proposition 4.2, and \(E_F\) fixes \(D\subset F\).

5.2. \(a_1b_1a_2b_2\) is centered alternating, so its trace is \(0\). In \(a_1(b_1b_2)a_2\), write \(b_1b_2=E^G_D(b_1b_2)+(b_1b_2)^\circ\). The second part gives the centered alternating product \(a_1(b_1b_2)^\circ a_2\) of trace \(0\). The first part gives \(a_1E^G_D(b_1b_2)a_2\in F\), so \[ \tau_Z(a_1b_1b_2a_2)=\tau_F\bigl(a_1E^G_D(b_1b_2)a_2\bigr). \]

5.3. Here \(E^F_{\mathbb C}=\tau_F\), and \(\tau_F(u)=0\), so \(u\) and \(v\) are centered. For \(n\ge1\), \((uv)^n=uvuv\cdots uv\) is a centered alternating product, so its trace is \(0\). For \(n\le-1\), \((uv)^n=(v^{-1}u^{-1})^{|n|}=(vu)^{|n|}\), again centered alternating.

5.4. Then \(i^\circ=0\) and \(L_i^\circ=0\), so every \(L_\iota\) is zero and \(K=D\). Both representations are left multiplication on \(D\), so \(Z=D\).

5.5. This was shown in the proof of Proposition 3.2: \(Z\Omega\) contains \(D\) and the elementary tensors \(a_1\otimes\cdots\otimes a_m\) with centered entries, which span a dense subspace of every \(L_\iota\).

References