# Normal tensor tests and tracial GNS identifications
*Tensor constructions and normal-map proofs by GPT-6 Astra (OpenAI). Original programme exposition: CC0-1.0.*
The two construction scopes below are different. TF1 concerns arbitrary concrete von Neumann algebras on Hilbert spaces of arbitrary cardinality, with no trace, separability or faithful-state assumption. TF2–TF3 construct and compare countable products equipped with specified faithful normal tracial states. The component algebras in that second construction need not be factors or finite-dimensional. Its applications here use full matrix algebras. TF4 supplies the predual limits, retaining an arbitrary nontracial complementary tensor leg. TF5 proves the representation-independent tensor theorem for normal star homomorphisms, and TF6 supplies normal completely positive tensor maps through a full normal dilation.
Inner products are linear in the first variable. Write \(\omega_{\xi,\eta}(x)=\langle x\xi,\eta\rangle\). The exact existing inputs are the real Hilbert tools, GP0, NP1–NP2, AB04–AB05, CP01–CP07, BK01–BK04, HA03 and HAP04–HAP05 in the [bounded-topology companion](bounded-topology-foundations.md), the faithful normal GNS proof in Lemma 3A.3 and the image/topology argument of Theorem 3.1 in [the bounded-topology lesson](../src/bounded-topology-and-tracial-representations.md), and TP1 of the [tracial-normality companion](tracial-normality-foundations.md). Their completed proofs are reused here. They supply Hilbert completion and Riesz representation, bounded calculus, concrete predual duality and norm closure, fixed-multiplication normality, compact dual balls and bounded strong* density. The first two companions retain the exact proof and source notices for those dependencies. No general modular or standard-form theorem is an input to this packet.
## TF0. The Hilbert tensor operations used below
Give the algebraic tensor space \(H\odot K\) the form
\[
\begin{aligned}
&\left\langle\sum_i h_i\otimes k_i,\sum_j h'_j\otimes k'_j\right\rangle\\
&\qquad=\sum_{i,j}\langle h_i,h'_j\rangle\langle k_i,k'_j\rangle.
\end{aligned}
\tag{TF.1}
\]
The bilinear relations respect this form. Choose an orthonormal basis of the finite span of the \(h_i\), and write a finite tensor as \(\sum_r e_r\otimes\ell_r\). Its squared norm is \(\sum_r\|\ell_r\|^2\). The coordinate maps on that finite span show that zero norm means the algebraic tensor itself is zero. The Hilbert completion is \(H\otimes K\); finite sums of elementary tensors are dense by construction. This is exactly the complete construction in OA-APPROX, *Infinite tensor products*, Section 10.3. That source uses the other inner-product convention; (TF.1) fixes the conversion explicitly.
For \(A\in B(H)\), expansion over finitely many orthonormal second factors proves
\(\|(A\otimes1)v\|\leq\|A\|\|v\|\). Expanding over orthonormal first factors gives the other leg. Thus \(A\otimes B\) extends boundedly, its adjoint is \(A^*\otimes B^*\), and its norm is \(\|A\|\|B\|\) when both spaces are nonzero: the reverse inequality is tested on unit elementary vectors approaching the two operator norms. Multiplication and adjoints are checked on the dense elementary span. Permutations and regroupings preserve (TF.1), have dense ranges, and therefore extend to unitaries. These are the finite tensor operations proved in Section 10.4 of that provider; all spans used in the proof are finite.
Amplification \(A\mapsto A\otimes1\) is normal on the whole of \(B(H)\). Indeed, a vector coefficient on the target pulls back to a norm limit of finite sums of vector coefficients on \(B(H)\): approximate its two vectors by finite elementary sums and use the coefficient estimate (TF.4) below and the norm bound for amplification. CP06 makes that norm limit a normal functional. Every target normal functional is a norm-convergent sum of vector coefficients by CP04–CP06; its pullback is therefore also normal, by the same norm bound and norm closure. This proves global ultraweak continuity by the defining predual tests. It makes no assertion that an arbitrary ultraweakly convergent net has a bounded tail.
Similarly, conjugation by any specified Hilbert unitary is a normal isomorphism with normal inverse: its pullback sends the vector-series test \(\sum_j\omega_{\xi_j,\eta_j}\) to \(\sum_j\omega_{U^*\xi_j,U^*\eta_j}\). These are square-summable sequences with exactly the original sums of squared norms.
## TF1. Product normal functionals are norm total
**Theorem.** Let \(M\subseteq B(H)\) and \(N\subseteq B(K)\) be concrete unital von Neumann algebras and let
\(P=(M\odot N)''\subseteq B(H\otimes K)\).
For every \(\varphi\in M_*\) and \(\psi\in N_*\) there is a unique normal functional \(\varphi\otimes\psi\in P_*\) satisfying
\[
\begin{aligned}
(\varphi\otimes\psi)(a\otimes b)&=\varphi(a)\psi(b),\\
\|\varphi\otimes\psi\|&=\|\varphi\|\|\psi\|.
\end{aligned}
\tag{TF.2}
\]
If the two functionals are positive, so is their product; a product of states is a state. Finite linear combinations of these functionals are norm dense in \(P_*\). Already the products of restricted vector functionals have norm-dense span. All Hilbert cardinalities are allowed. If either space is zero, every assertion reduces to the zero functional on the zero product algebra.
**Proof of existence, normality and norm.** For \(h\in H\), let \(S_h:K\to H\otimes K\) send \(k\) to \(h\otimes k\). For \(X\in P\), the compression
\(S_{h'}^*XS_h\) belongs to \(N\): it commutes with \(N'\), because \(X\) commutes with every \(1\otimes b'\), \(b'\in N'\), and the two elementary intertwining identities for \(S_h\) pass that commutation through the compression. BK02 gives \(N''=N\).
For a bounded linear \(\psi\) on \(N\), the formula
\[
\langle T_\psi(X)h,h'\rangle=\psi(S_{h'}^*XS_h)
\tag{TF.3}
\]
is a bounded sesquilinear form with bound \(\|\psi\|\|X\|\|h\|\|h'\|\). Riesz representation gives a linear map \(T_\psi:P\to B(H)\), with \(\|T_\psi(X)\|\leq\|\psi\|\|X\|\). If \(a'\in M'\), the relations \(S_{a'h}=(a'\otimes1)S_h\) and \(S_{h'}^*(a'\otimes1)=S_{a'^*h'}^*\), together with commutation of \(X\) and \(a'\otimes1\), show that \(T_\psi(X)a'=a'T_\psi(X)\). Thus \(T_\psi(X)\in M\). It sends \(a\otimes b\) to \(\psi(b)a\).
Choose CP06 vector-series representations
\(\varphi=\sum_i\omega_{h_i,h_i'}|_M\) and
\(\psi=\sum_j\omega_{k_j,k_j'}|_N\), with all four sequences square summable. Formula (TF.3) and absolute convergence give
\[
\varphi(T_\psi(X))
=\sum_{i,j}\langle X(h_i\otimes k_j),h_i'\otimes k_j'\rangle.
\]
The two tensor-vector families are square summable, since each double sum of squared norms factors into the two finite sums on its legs. Their index set is countable. CP06 therefore proves that this functional belongs to \(P_*\), globally. Its definition as \(\varphi T_\psi\) makes it independent of the chosen series. Its norm is at most \(\|\varphi\|\|\psi\|\). The reverse inequality follows by testing \(a\otimes b\) on the two unit balls and taking the two independent suprema. The zero-functional cases are immediate.
If \(X\geq0\) and \(\psi\geq0\), (TF.3) gives \(\langle T_\psi(X)h,h\rangle\geq0\), since \(S_h^*XS_h\geq0\). Thus positivity of \(\varphi\) proves positivity of the product. Evaluation at the identity proves the state assertion. The algebraic tensor algebra is ultraweakly dense in \(P\): BK02 identifies its bicommutant with its weak closure, HAP05 supplies bounded strong* approximants, and BK03 converts these specified bounded nets to ultraweak convergence. Two normal functionals agreeing there consequently agree on \(P\). This proves uniqueness, including for the restricted vector products.
**Proof of norm density.** CP04–CP06 represent every \(\rho\in P_*\) as
\(\rho=\sum_l\omega_{\zeta_l,\eta_l}|_P\), with \(\zeta,\eta\in\ell^2(H\otimes K)\). The norm of its tail is at most
\((\sum_{l>L}\|\zeta_l\|^2)^{1/2}(\sum_{l>L}\|\eta_l\|^2)^{1/2}\), and tends to zero. For each of the finitely many retained pairs, use TF0 to choose finite elementary sums \(\alpha,\beta\) close to \(\zeta,\eta\). Uniformly on the operator unit ball,
\[
\begin{aligned}
\|\omega_{\zeta,\eta}-\omega_{\alpha,\beta}\|
&\leq\|\zeta-\alpha\|\|\eta\|\\
&\quad+\|\alpha\|\|\eta-\beta\|.
\end{aligned}
\tag{TF.4}
\]
Expand the coefficient of those two finite sums. Each term is a product of restricted vector functionals, by (TF.2) and uniqueness. First truncate the series, then approximate its finitely many vectors closely enough that the sum of the errors in (TF.4) is as small as prescribed. This gives approximation in predual norm. No trace or faithful state has appeared. \(\square\)
This is the full mechanism in *Spatial tensor products of von Neumann algebras*, Theorems 9.2 and 10.1(1)–(2), with its concrete-predual dependencies supplied by the already inspected CP proofs. It also expands the brief density sentence in OA-APPROX, *Strong stability and tensor absorption*, Section 5.
## TF2. The countable tracial product and its GNS space
Let \((A_j,\tau_j)_{j\geq1}\) be nonzero unital von Neumann algebras with specified faithful normal tracial states. Write \(H_j=L^2(A_j,\tau_j)\), \(\Lambda_j(a)\) for the GNS vector and \(\Omega_j=\Lambda_j(1)\). Lemma 3A.3 gives the faithful normal left representation \(\lambda_j\).
Complete the increasing Hilbert union
\[
\begin{gathered}
\mathcal H_n=H_1\otimes\cdots\otimes H_n,\\
\mathcal H_n\longrightarrow\mathcal H_{n+1},\\
\xi\longmapsto\xi\otimes\Omega_{n+1},
\end{gathered}
\]
and call the completion \(H\). The common reference vector is \(\Omega\). Each inclusion is isometric by (TF.1). Finite-support elementary vectors are dense. Splitting at any finite head, or into disjoint sets of coordinates, preserves their inner products and has dense range, and so gives the corresponding Hilbert unitary. This is the pointed-product construction of OA-APPROX, Section 2, now used only at its stated countable tracial scope.
Let \(\mathcal A\) be the unital *-algebra of finite sums of local operator tensors \(\lambda_1(a_1)\otimes\cdots\otimes\lambda_n(a_n)\), acting as identity on the tail, and put \(P=\mathcal A''\). Their actions are bounded by TF0, compatible with unit padding, and faithful on each specified local factor. Finite-head amplification is normal by TF0. Each \(\Lambda_j(A_j)\) is dense in \(H_j\); approximating the finitely many factors of an elementary vector and using \(\|h\otimes k\|=\|h\|\|k\|\) therefore proves that \(\mathcal A\Omega\) is dense in \(H\). Hence this constructs the concrete countable tracial product in the specified GNS representations.
The vector functional \(\tau(X)=\langle X\Omega,\Omega\rangle\) is normal by CP06, and its value on a local elementary tensor is \(\prod_j\tau_j(a_j)\). It is faithful. For clarity, on each \(H_j\) right multiplication
\(r_j(b)\Lambda_j(a)=\Lambda_j(ab)\) is bounded by \(\|b\|\): traciality gives
\[
\|\Lambda_j(ab)\|^2
=\tau_j(a^*a\,bb^*)
\leq\|b\|^2\tau_j(a^*a).
\]
For the inequality, use traciality to replace the middle expression by \(\tau_j((a^*a)^{1/2}bb^*(a^*a)^{1/2})\), and apply \(bb^*\leq\|b\|^2 1\) between these two square roots.
Its adjoint is \(r_j(b^*)\), by the same trace identity. The local right actions commute with all local left actions, hence with \(P\), and their orbit of \(\Omega\) has the same dense span as the local left orbit. If \(X\in P\) and \(X\Omega=0\), it annihilates that dense right orbit and is zero. If \(X\geq0\) and \(\tau(X)=0\), then \(X^{1/2}\Omega=0\), and the preceding argument gives \(X=0\).
The state is tracial on \(\mathcal A\), by the finite product formula. For fixed local \(a\), the identity \(\tau(Xa)=\tau(aX)\) passes to all \(X\in P\), using ultraweak density of \(\mathcal A\), normality of \(\tau\), and BK03's fixed-multiplication normality. Now fix such an arbitrary \(X\) and use the same argument in the other variable. Thus \(\tau\) is tracial on \(P\). This proves the full faithful normal product trace without a tensor-commutant theorem.
The map \(\Lambda_\tau(X)\mapsto X\Omega\) is isometric, since its squared norm is \(\tau(X^*X)\), and its range is dense because it includes all local vectors. It extends to a unitary from the trace GNS space of \(P\) onto the constructed \(H\), intertwining left multiplication. Thus the identification with the tracial GNS product is proved, not inferred merely from agreement of scalar traces. No type or hyperfinite uniqueness assertion is needed here.
## TF3. Compatible finite identifications extend normally
**Theorem.** Let \((P,\tau)\), \((Q,\sigma)\) be von Neumann algebras with faithful normal tracial states. Let \(\mathcal A\subset P\), \(\mathcal B\subset Q\) be ultraweakly dense unital *-subalgebras. Every bijective unital *-algebra map \(\theta_0:\mathcal A\to\mathcal B\) satisfying \(\sigma\theta_0=\tau\) extends uniquely to a trace-preserving normal *-isomorphism \(\theta:P\to Q\), with normal inverse.
**Proof.** The map
\[
U_0:\Lambda_\tau(a)\longmapsto\Lambda_\sigma(\theta_0(a))
\quad(a\in\mathcal A)
\tag{TF.5}
\]
is complex linear and preserves inner products, since
\(\sigma(\theta_0(b)^*\theta_0(a))=\tau(b^*a)\).
Its domain and range are dense in their GNS spaces. Indeed HAP05 supplies bounded strong* approximation from the two subalgebras, and Theorem 3.1 transports it to their faithful normal GNS representations; applying it to the cyclic vectors gives the required density. Surjectivity onto \(\mathcal B\) proves dense range on the second side. Thus \(U_0\) extends to an onto Hilbert unitary \(U:H_\tau\to H_\sigma\), with \(U\Omega_\tau=\Omega_\sigma\).
For \(a,b\in\mathcal A\), multiplicativity gives
\(U\lambda_\tau(a)\Lambda_\tau(b)=\lambda_\sigma(\theta_0(a))U\Lambda_\tau(b)\).
The operators are bounded and these vectors are dense, so
\(U\lambda_\tau(a)U^*=\lambda_\sigma(\theta_0(a))\).
The faithful normal represented algebras are von Neumann algebras by Theorem 3.1; the same bounded approximation shows that they are generated by these two local subalgebras. Taking bicommutants therefore gives equality
\[
U\lambda_\tau(P)U^*=\lambda_\sigma(Q).
\tag{TF.6}
\]
Define \(\theta=\lambda_\sigma^{-1}\circ\operatorname{Ad}U\circ\lambda_\tau\). These maps are typed respectively from \(P\) to \(\lambda_\tau(P)\), to \(\lambda_\sigma(Q)\), and to \(Q\). The GNS maps are faithful unital *-isomorphisms onto their images, and (TF.6) proves surjectivity of the middle comparison.
Their inverses are normal at the required global scope. A *-isomorphism and its inverse preserve positive order, hence preserve every bounded increasing positive supremum: the defining least-upper-bound property transfers in both directions. Composing any positive normal functional with either inverse therefore gives an order-normal bounded positive functional. TP1 makes it ultraweakly continuous, and CP07's positive spanning theorem gives this for every predual test. This proves global normality of the inverses. TF0 proves normality of unitary conjugation; Lemma 3A.3 supplies normality of the forward GNS maps. Consequently \(\theta\) and \(\theta^{-1}\) are normal. Preservation of the reference vector proves \(\sigma\theta=\tau\). Uniqueness follows from normality and the specified ultraweak density of \(\mathcal A\). \(\square\)
There is also a bounded proof of the normal-inverse step: ST-C and ST1–2 prove whole-domain coefficient normality of the faithful trace representations, their von Neumann ranges and their normal inverses. Together with TF0, this proves the same normal identification. The order-normality argument above remains a second proof.
For the countable products of TF2, take their local unions. Compatible trace-preserving *-isomorphisms of finite heads, or a bijective trace-preserving identification of their local tensor unions obtained by blocking and reindexing, give exactly \(\theta_0\). Every finite support must be covered in each direction; that condition proves surjectivity of the union map. TF3 then supplies the normal extension and its inverse. This is a reference-tracial product theorem, not a claim that the reference product is the universal inductive limit in the category of all von Neumann algebras.
If the second product is an already constructed algebra \(K\) generated by commuting matrix blocks and carrying a faithful normal state whose finite-head restrictions are the normalized traces, that state is tracial on \(K\) by the two density arguments of TF2. The finite-head multiplication maps must be injective and preserve those traces; under these explicit hypotheses TF3 applies to \(P\to K\). These are exactly the data proved in the tensor-splitting prerequisite. Commutation alone would not suffice for this conclusion.
If local maps intertwine specified trace-preserving product actions, their extensions intertwine those actions, because the two normal composites agree on the ultraweakly dense local union. One can also leave any fixed concrete von Neumann algebra \(C\subseteq B(L)\) unchanged: in the displayed GNS representations, \(U\otimes1_L\) conjugates \(\lambda_\tau(P)\overline\otimes C\) onto \(\lambda_\sigma(Q)\overline\otimes C\), by its action on generators and (TF.6), and is normal in both directions by TF0.
## TF4. The predual limits actually used by the consumers
First, the finite-head density used in the matrix model has a direct proof. By CP04–CP06, vector coefficients have norm-dense span in \(P_*\). Approximate their two vectors by local vectors \(a\Omega,b\Omega\), using TF2 and (TF.4). Traciality gives
\[
\begin{gathered}
\langle X a\Omega,b\Omega\rangle
=\tau(b^*Xa)=\tau(ab^*X),\\
a,b\in\mathcal A.
\end{gathered}
\tag{TF.7}
\]
The element \(ab^*\) lies in a finite tensor head. Thus the functionals \(\psi_y(X)=\tau(yX)\), with \(y\) in a finite head, have norm-dense span in \(P_*\). If \((x_v)\) is norm bounded and each \(x_v\) lies entirely beyond every fixed head for all sufficiently large \(v\), then it eventually commutes with such a \(y\). For every \(X\in P\), traciality and \(x_vy=yx_v\) give \(\tau(yXx_v)=\tau(yx_vX)\). Hence \([x_v,\psi_y]=0\) eventually. The bound \(\|[x_v,\rho]\|\leq2\sup_v\|x_v\|\|\rho\|\) extends the limit to all of \(P_*\). This proves exactly the finite-head normal-functional test used for the model's far-tail matrix units; it does not assert the entire noncommutative \(L^1\) representation theorem.
Let \(\alpha_j\) be trace-preserving normal automorphisms of the component algebras of TF2. The product of all the \(\alpha_j\), and the action \(\alpha^{(n)}\) which uses them only on the first \(n\) coordinates, are supplied by TF3. Their GNS unitaries and inverse unitaries agree eventually with the full product unitary and its inverse on every local vector. They therefore converge strongly on all vectors, since all are unitaries and local vectors are dense.
For any restricted vector functional, pullback by \(\alpha^{(n)}=\operatorname{Ad}U_n\) is the coefficient with vectors \(U_n^*\xi,U_n^*\eta\). Estimate (TF.4) proves convergence in functional norm. CP04–CP06 and the predual isometry of automorphisms extend it to every \(\rho\in P_*\). Thus the partial products converge in the \(u\)-topology. This supplies the finite-coordinate approximation invoked in the aperiodic comparison proof, without importing an unproved \(L^1\)-density theorem.
On \(C\overline\otimes P\), pullback of \(\eta\otimes\rho\) by \(\operatorname{id}\otimes\alpha^{(n)}\) is \(\eta\otimes(\rho\circ\alpha^{(n)})\). TF1 gives convergence in norm, first for these products and then for finite sums. The two predual operators are isometries, so the error on an arbitrary functional differs from its error on a chosen finite sum by at most twice the approximation error. TF1 proves the full \(u\)-limit on this possibly nontracial ambient algebra.
The same density argument transfers centralizing sequences. Define \([x,\rho](y)=\rho(yx-xy)\). If \(\sup_n\|x_n\|=B<\infty\) in \(P\) and \(\|[x_n,\rho]\|\to0\) for every \(\rho\in P_*\), then
\[
[1\otimes x_n,\eta\otimes\rho]
=\eta\otimes[x_n,\rho].
\tag{TF.8}
\]
This identity holds first on elementary operators and then everywhere by normality and density. Its norm is \(\|\eta\|\|[x_n,\rho]\|\) by TF1. Finite sums and the uniform bound \(\|[1\otimes x_n,\chi]\|\leq2B\|\chi\|\) prove the assertion for every \(\chi\in(C\overline\otimes P)_*\). There is no trace assumption on \(C\).
Finally let \(D,E\) be arbitrary concrete von Neumann algebras, and suppose the normal tensor automorphism \(\alpha\otimes\beta\) on \(D\overline\otimes E\) is specified. If \(\operatorname{Ad}u_i\to\alpha\) and \(\operatorname{Ad}v_j\to\beta\) in the \(u\)-topology, use the product directed set for \((i,j)\). The implementers \(u_i\otimes v_j\) are unitaries. For a product functional their pullback error has norm at most
\[
\begin{aligned}
&\|\varphi\circ\operatorname{Ad}u_i-\varphi\circ\alpha\|\,\|\psi\|\\
&\quad+\|\varphi\|\,\|\psi\circ\operatorname{Ad}v_j-\psi\circ\beta\|.
\end{aligned}
\tag{TF.9}
\]
Both terms tend to zero. TF1 and the predual isometry bound extend the limit to all normal functionals, proving the required tensor approximate innerness. The existence and normal inverse of this tensor automorphism are supplied by Theorem TF5.1; Corollary TF5.2 applies the same estimate to arbitrary nets of automorphisms.
## The precise classification interfaces
The matrix absorption lesson, Section 5 and Corollary 6.2, uses TF3 for its coordinate bijections, regroupings and the map \(J_p\), tensored with the unchanged complement. The aperiodic comparison lesson, Section 3, uses TF3 for \(\iota:K_{\mathrm{mod}}\to K\), and TF4 for the limit of the finite product actions. The finite cyclic lesson, Lemma 4.1, uses TF3 for the finite trace-preserving basis permutations and their regrouped infinite product; its tensor approximate-innerness and central-sequence passages use TF1 and TF4.
The finite matrix identities, the existence and trace independence of the embedded commuting blocks, and the normal tensor splitting of the ambient algebra are still the actual hypotheses at these uses; this packet does not manufacture them. Hyperfinite uniqueness, type classification and centralizer/lifting/cohomology results retain their own hypotheses. Theorems TF5.1 and TF6.1 supply the normal homomorphism and completely positive tensor maps in arbitrary faithful normal representations. The general theorem TF1 has not been reduced to the countable tracial setting.
## TF5. Normal tensor homomorphisms in arbitrary representations
Normality in this section means ultraweak continuity on the entire algebra. No trace, faithful state, separable predual or restriction on Hilbert dimension is assumed.
**Theorem TF5.1.** Let \(A,B,C,D\) be concrete von Neumann algebras and let \(\alpha:A\to C\), \(\beta:B\to D\) be normal star homomorphisms, possibly nonunital or zero. There is a unique normal star homomorphism
\[
\begin{gathered}
\alpha\overline\otimes\beta:
A\overline\otimes B\longrightarrow C\overline\otimes D,\\
(\alpha\overline\otimes\beta)(a\otimes b)
=\alpha(a)\otimes\beta(b).
\end{gathered}
\tag{TF.10}
\]
Its range is \(\alpha(A)\overline\otimes\beta(B)\), represented on the range of \(\alpha(1)\otimes\beta(1)\). If
\[
\ker\alpha=zA,\qquad \ker\beta=wB
\]
for the corresponding central projections, then
\[
\begin{gathered}
q=(1-z)\otimes(1-w),\\
\ker(\alpha\overline\otimes\beta)
=(1-q)(A\overline\otimes B).
\end{gathered}
\tag{TF.11}
\]
In particular, faithful normal representations yield a faithful tensor map; normal isomorphisms yield a normal isomorphism with the tensor of their inverses as inverse. Tensor maps preserve composition and identities and commute with the canonical finite permutations and regroupings.
*Proof.* We first supply the representation and range facts needed to define the map beyond algebraic tensors.
**A normal representation is an amplified subrepresentation.** Fix \(A\subseteq B(H)\). The complete proof [NA3–NA4](../../OA-APPROX/normal-representations-and-finite-trace-corners.html#na3) applies to arbitrary normal unital representations, including nonfaithful ones. It uses the concrete predual CP1–6, the Hilbert and positive-calculus proofs CF1–10, and the maximal principle in CF1. Its construction can be described explicitly. A positive normal functional \(f\), expressed by square-summable coefficient lists \(u_j,v_j\), is dominated on positive elements by
\[
\nu(a)=\frac12\sum_j
\bigl(\langle au_j,u_j\rangle+
\langle av_j,v_j\rangle\bigr).
\tag{TF.12}
\]
The vector \(\Xi=2^{-1/2}(u_1,v_1,u_2,v_2,\ldots)\) in a countable amplification represents \(\nu\). On the closed orbit of \(\Xi\), the form taking \((a\Xi,b\Xi)\) to \(f(b^*a)\) is bounded by the product of the two vector norms. The positive-functional Schwarz inequality proves both this bound and independence of orbit representatives. Its representing positive contraction commutes with the restricted algebra; its positive root sends \(\Xi\) to a vector representing \(f\). The corresponding cyclic representation is therefore an isometric intertwining copy of a reducing subspace of that amplification.
For a normal unital representation \(\pi:A\to B(K)\), choose a maximal orthogonal family of cyclic reducing subspaces of \(K\). Their orthogonal complement is zero, since otherwise a vector orbit in it would enlarge the family. Apply the preceding construction to each cyclic functional and take the full Hilbert direct sum. Thus for some Hilbert space \(L\) there is an isometry
\[
\begin{gathered}
V:K\longrightarrow H\otimes L,\\
V\pi(a)=(a\otimes1_L)V,\\
VV^*\in(A\otimes1_L)'.
\end{gathered}
\tag{TF.13}
\]
The family and \(L\) may have arbitrary cardinality. This is exactly the dominated-form and orbit-isometry argument of NA3–NA4; it does not use a modular commutation theorem.
**Every normal homomorphism has a von Neumann range.** Let \(\gamma:A\to B(K)\) be normal. Its kernel \(I\) is an ultraweakly closed two-sided star ideal. For \(x\in I\), continuous calculus and the ideal property put
\[
h(h+\varepsilon1)^{-1}\in I,
\qquad h=x^*x,\quad\varepsilon>0.
\]
These positive contractions converge strongly to the projection onto \(\overline{\operatorname{ran}h}\). Indeed they vanish on \(\ker h\), and on vectors \(h\xi\) the error from the identity has norm at most \(\varepsilon\|\xi\|\); density and the bound one give the full strong limit. Bounded strong convergence is ultraweak convergence by the CP vector-series estimate. Thus the support of \(x^*x\) belongs to \(I\). Finite joins of projections in \(I\) belong to \(I\) by applying the same construction to their sum. The net of all these finite joins converges strongly to their supremum \(z\), so \(z\in I\). Conjugation by any unitary of \(A\) preserves this supremum, making \(z\) central because every element is a linear combination of unitaries by the bounded calculus in CF. Every \(x\in I\) has right support at most \(z\), hence \(I=zA\).
The restriction of \(\gamma\) to \((1-z)A\) is faithful and has the same range. On \(\gamma(1)K\) it is unital. ST2 gives a von Neumann range and a normal inverse onto that range. Extension by zero on the perpendicular Hilbert space preserves ultraweak closedness. The zero-map case has \(z=1\) and range zero. This proves the range assertion for every normal star homomorphism, before any tensor product is formed.
**Construct the tensor map by compression.** First assume \(\alpha\) and \(\beta\) are unital into their acting Hilbert spaces \(K_C,K_D\). Apply (TF.13) to obtain isometries \(V_A,V_B\) into amplifications of the given source spaces \(H_A,H_B\). Regrouping elementary Hilbert tensors gives a unitary
\[
\begin{gathered}
J:(H_A\otimes L_A)\otimes(H_B\otimes L_B)\\
\longrightarrow (H_A\otimes H_B)\otimes(L_A\otimes L_B).
\end{gathered}
\]
Put \(W=J(V_A\otimes V_B)\). Its range projection commutes with every amplified elementary source operator, and hence with the amplified spatial product. Define
\[
\Phi(X)=W^*(X\otimes1)W,
\qquad X\in A\overline\otimes B.
\tag{TF.14}
\]
The commutation of the range projection and \(W^*W=1\) prove multiplicativity; adjoints and the unit are preserved. TF0 proves normality of amplification. Compression is normal by replacing the two lists in each square-summable vector-series test by their \(W\)-images. Thus \(\Phi\) is normal on its entire domain.
On elementary operators it equals the right side of (TF.10). Bounded density BD1–5 makes algebraic tensors ultraweakly dense. Normality therefore puts \(\Phi(X)\) in \(\alpha(A)\overline\otimes\beta(B)\). Conversely its range is a von Neumann algebra by the preceding range proof and contains every elementary image, so it is exactly that product. Ultraweak density also proves uniqueness, independent of the selected amplifications and isometries.
For nonunital maps restrict the target spaces to the ranges of their represented units, perform the same construction, and include the result by zero on the complementary tensor subspace. Both restriction and inclusion are normal fixed compressions. If either represented unit is zero, the tensor map is zero. This proves (TF.10) at the full stated scope.
If both maps are faithful, their inverses from their von Neumann ranges are normal by ST2. Apply the construction to those inverses. Both composites fix all algebraic tensors, so normality and density make them the identity everywhere. Thus the tensor map is faithful. In general the central corner \(q(A\overline\otimes B)\) is the spatial product \(((1-z)A)\overline\otimes((1-w)B)\): bounded approximation by elementary tensors, followed by multiplication by \(q\), proves this equality in the corresponding Hilbert corner. The tensor map annihilates \(1-q\), and on this corner is the tensor of two faithful maps. This proves (TF.11). Finally composition, identities, permutations and regroupings agree on elementary tensors. Their maps are normal, so the same density argument proves all the asserted identities. \(\square\)
The theorem makes the spatial product independent of the particular faithful normal representations: apply it in both directions to the normal identifications of each leg. This conclusion concerns normal representations; a merely faithful C*-representation does not supply the needed ultraweak identification.
**Corollary TF5.2.** Tensoring automorphisms is jointly continuous for the \(u\)-topology, on arbitrary von Neumann algebras. It preserves approximate innerness. Tensoring a normal conjugating isomorphism and its correcting unitary transports outer conjugacy.
*Proof.* Theorem TF5.1 supplies the tensor automorphisms and their inverses. If \(\alpha_i\to\alpha\) and \(\beta_j\to\beta\) in the \(u\)-topology, use the product directed set. The pullback difference of a product normal functional is bounded by the two terms in (TF.9). They tend to zero. TF1 supplies norm-dense finite sums of those functionals; the norm-one predual actions bound the remaining error by twice the approximation error. This proves joint continuity. Apply it to inner implementers; their tensor is implemented by the tensor of their unitaries. For outer conjugacy, tensor the equality defining it and use the composition identity in TF5.1. \(\square\)
## TF6. Normal completely positive tensor maps
**Theorem TF6.1.** If \(\phi:A\to C\) and \(\psi:B\to D\) are bounded normal completely positive maps, they have a unique bounded normal completely positive tensor map between the spatial products. It has the elementary values \(\phi(a)\otimes\psi(b)\), and
\[
\begin{gathered}
\|\phi\overline\otimes\psi\|=\|\phi\|\,\|\psi\|,\\
(\phi\overline\otimes\psi)(1)
=\phi(1)\otimes\psi(1).
\end{gathered}
\tag{TF.15}
\]
It respects composition. In particular normal ucp maps, normal cpc maps and normal conditional expectations can be tensored, with the corresponding property retained. The result is independent of faithful normal representations.
*Proof.* We prove the needed normal dilation rather than presuming normality of a C*-dilation. Regard \(C\subseteq B(K)\). On the algebraic space \(A\odot K\) put
\[
\begin{aligned}
&\left\langle\sum_i a_i\otimes\xi_i,
\sum_j b_j\otimes\eta_j\right\rangle_\phi\\
&\qquad=\sum_{i,j}
\langle\phi(b_j^*a_i)\xi_i,\eta_j\rangle.
\end{aligned}
\tag{TF.16}
\]
The algebraic tensor relations respect the form. Complete positivity makes it nonnegative: the matrix of products is a Gram matrix, and its image is positive on the finite Hilbert direct sum. Applying nonnegativity to two vectors with an arbitrary scalar coefficient gives Cauchy–Schwarz, including the zero diagonal case. Therefore the null vectors form a linear subspace orthogonal to every vector. Quotient by them and complete to a Hilbert space \(H_\phi\).
Left multiplication defines a representation \(\pi\). For \(c\in A\), the matrix
\[
[a_i^*(\|c\|^2 1-c^*c)a_j]_{i,j}
\]
is positive, being a Gram matrix after multiplication by the positive square root of the middle element. Complete positivity applied to this matrix gives
\[
\|\pi(c)v\|_\phi\leq\|c\|\|v\|_\phi.
\tag{TF.17}
\]
Thus left multiplication respects the null space and extends boundedly. The product and adjoint identities follow on elementary vectors and extend by density; the unit acts as the identity. Define \(V:K\to H_\phi\) by \(V\xi=[1\otimes\xi]\). Then
\[
\begin{gathered}
V^*\pi(a)V=\phi(a),\qquad
V^*V=\phi(1),\\
\|V\|^2=\|\phi(1)\|=\|\phi\|.
\end{gathered}
\tag{TF.18}
\]
The first two equalities follow by pairing vectors in \(K\). They and (TF.17) give \(\|\phi\|\leq\|\phi(1)\|\); evaluation at the unit gives the reverse inequality. If \(\phi=0\), the constructed Hilbert space and \(V\) are zero. For a unital map \(V\) is an isometry. The span of \(\pi(A)VK\) is dense by construction. In any other dilation, sending \([a\otimes\xi]\) to the corresponding vector \(\pi'(a)V'\xi\) preserves (TF.16), so extends to the unique intertwining isometry onto its closed cyclic span. This also proves the minimality and uniqueness of the constructed dilation.
**Its representation is normal.** A coefficient of \(\pi(x)\) between algebraic vectors is a finite sum of functionals
\[
x\longmapsto
\langle\phi(b^*xa)\xi,\eta\rangle.
\tag{TF.19}
\]
Fixed multiplication and \(\phi\) are normal, so these functionals belong to \(A_*\). Approximate any two Hilbert vectors by algebraic vectors. The bound (TF.17) gives functional-norm convergence of the coefficients by (TF.4); CP proves that \(A_*\) is norm closed. Every vector coefficient of \(\pi\) is therefore normal. Pullback of a square-summable coefficient series is a norm-convergent series of these normal functionals, since the sum of their norm bounds is finite. Completeness of \(A_*\) proves ultraweak continuity on the whole algebra. This proves the full normal dilation without an order-normality conversion.
Apply (TF.13) to this normal representation. Composing its intertwining isometry with \(V\) gives a bounded operator \(T_\phi:K\to H_A\otimes L_\phi\) such that
\[
\begin{gathered}
\phi(a)=T_\phi^*(a\otimes1)T_\phi,\\
\|T_\phi\|^2=\|\phi\|.
\end{gathered}
\tag{TF.20}
\]
Do the same for \(\psi\), and regroup the two amplified source spaces as in TF5. With \(T=J(T_\phi\otimes T_\psi)\), define
\[
\Theta(X)=T^*(X\otimes1)T.
\tag{TF.21}
\]
Amplification followed by this compression is normal and completely positive: apply compression on each finite Hilbert direct sum to verify every matrix cone. Its elementary values are the stated ones. Normality and algebraic density put the whole range in \(C\overline\otimes D\), and give uniqueness. The norm is at most \(\|T\|^2=\|\phi\|\|\psi\|\). Its value at the unit, together with TF0's elementary operator-norm identity and (TF.18), gives the reverse bound and proves (TF.15).
Composition and independence of faithful normal representations follow from uniqueness on the ultraweakly dense algebraic product and TF5. If both maps are normal conditional expectations, the tensor map has range in the spatial product of their ranges, fixes that product, and is unital and idempotent. Its bimodule identity holds on elementary operators by the two original identities; extend each of its three variables in turn by normality and fixed-multiplication continuity. Thus it is a normal conditional expectation. This completes the proof. \(\square\)
These theorems supply normal tensor maps and the tracial GNS changes of representation used in the classification lessons. They do not assert that every pair of commuting subalgebras in a common algebra is its spatial tensor decomposition. The summable-splitting theorem supplies that additional hypothesis where it is used.
## Sources and authorship
The Hilbert tensor operations, product normal functionals and tracial product identifications use the earlier programme expositions *Spatial tensor products of von Neumann algebras*, Theorems 9.2 and 10.1(1)–(2), by Claude Opus 5.5 (Anthropic), and *Infinite tensor products*, Sections 2, 4–5 and 10.3–10.4. The latter programme text uses Masamichi Takesaki, *Theory of Operator Algebras III*, XIV.1.1–XIV.1.14 and XIV.4.10. The summable-splitting hypotheses come from *Strong stability and tensor absorption*, Theorem 3.1.
The general amplification argument is given in GPT-6.1 Sol's [Normal representations and finite corners of tracial factors, NA3–NA4](../../OA-APPROX/normal-representations-and-finite-trace-corners.html#na3). The concrete predual, bounded calculus, density and normal range proofs are CF, CP, BD and ST. TF5–TF6 assemble these bounded constructions into normal tensor maps, prove their exact kernel and range, and give the normal completely positive dilation directly.
For human mathematical context, Bruce Blackadar's [*Operator Algebras*, revised author edition of 8 February 2017](https://bruceblackadar.com/Mathematics/Cycr.pdf), III.2.2.4–III.2.2.8, printed pages 266–268, gives normal dilations, tensor products of normal completely positive maps and amplified representations. The source's compact account refers back to its C*-dilation proof; TF6 writes out the quotient form, all matrix-positivity estimates and the entire predual normality argument.
The exposition of TF0–TF6 is by GPT-6 Astra (OpenAI) and is dedicated to CC0-1.0 to the extent of rights held. Human sources retain their own rights. The linked programme components retain their stated authorship and terms. This text incorporates no human expression from the separate Lebl or Daws adaptations in the infinite-product provider. The [source record](normal-tensor-and-tracial-product-foundations.provenance.json) identifies the sources and mathematical scope.