Proof of Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings

Each result cited is universally quantified over the data in its own statement.

Conventions. Write $V_{k}=V_{\pi_{k}}$ and $\tau_{k}=\tau_{M_{k}}$. By Tracial W*-Probability Spaces §space each $(H_{k},M_{k},\Omega_{k})$ is a cyclic tracial operator algebra, so $M_{k}\subseteq\mathcal{L}(H_{k})$ contains $I$ and is closed under sums, complex scalar multiples, products and adjoints (Cyclic Tracial Operator Algebras and Their Traces §star-algebra), $\lVert\Omega_{k}\rVert=1$ and $M_{k}\Omega_{k}$ is dense in $H_{k}$ (Cyclic Tracial Operator Algebras and Their Traces §cyclic). A trace-preserving embedding $\pi$ satisfies the identities of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding, which are the hypotheses of A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry, and its implementing isometry $V_{\pi}$ is the operator $V$ of A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry (Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry). In particular, for every $k$ and $S\in M_{k}$: $V_{k}^{*}V_{k}=I$ and $V_{k}\Omega_{k}=\Omega_{k+1}$ by A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry, and $\pi_{k}(S)V_{k}=V_{k}S$ by A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §intertwining. Here $\mathbb{N}=\{1,2,3,\dots\}$ with successor $n\mapsto n+1$ (Natural Numbers); an "induction on $n\ge k$" is an application of Principle of Induction for the Natural Numbers to the set of $m\in\mathbb{N}$ for which the assertion holds at $n=k+m-1$. Two sequences $(x_{n})$ and $(y_{n})$ agree from $m$ on if $x_{n}=y_{n}$ for every $n\ge m$.

Step 1. (Composite embeddings.) Fix $k\in\mathbb{N}$. Let $U$ be the union of the sets $M_{n}$, $n\in\mathbb{N}$, let $X$ be the set of maps $M_{k}\to U$, and define $f:\mathbb{N}\times X\to X$ by $f(m,\varphi)=\pi_{k+m-1}\circ\varphi$ if $\varphi(M_{k})\subseteq M_{k+m-1}$ and $f(m,\varphi)=\varphi$ otherwise. By Definition of Sequences by Recursion on the Natural Numbers §recursion there is a sequence $\sigma$ in $X$ with $\sigma(1)=\mathrm{id}_{M_{k}}$ and $\sigma(m+1)=f(m,\sigma(m))$ for every $m$. By induction on $m$, $\sigma(m)(M_{k})\subseteq M_{k+m-1}$ and hence $\sigma(m+1)=\pi_{k+m-1}\circ\sigma(m)$. Set $\pi_{k,n}=\sigma(n-k+1):M_{k}\to M_{n}$ for $n\ge k$. Then

The identity map $\mathrm{id}_{M_{k}}$ trivially satisfies the six identities of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding, so it is a trace-preserving embedding of $(H_{k},M_{k},\Omega_{k})$ into itself; since $I_{H_{k}}\in\mathcal{L}(H_{k})$ (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations) and $I_{H_{k}}S\Omega_{k}=\mathrm{id}_{M_{k}}(S)\Omega_{k}$, the uniqueness in Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry shows that its implementing isometry is $I_{H_{k}}$. By induction on $n\ge k$, using Trace-Preserving Embeddings Preserve Operator Norms and Compose §composite with $\pi=\pi_{k,n}$ and $\pi'=\pi_{n}$ in the induction step: each $\pi_{k,n}$ is a trace-preserving embedding of $(H_{k},M_{k},\Omega_{k})$ into $(H_{n},M_{n},\Omega_{n})$, and its implementing isometry $V_{k,n}=V_{\pi_{k,n}}\in\mathcal{L}(H_{k},H_{n})$ satisfies

Next, $\pi_{k,n}=\pi_{k+1,n}\circ\pi_{k}$ for every $n\ge k+1$, by induction on $n\ge k+1$: for $n=k+1$ both sides equal $\pi_{k}$, and if it holds at $n$ then $\pi_{k,n+1}=\pi_{n}\circ\pi_{k+1,n}\circ\pi_{k}=\pi_{k+1,n+1}\circ\pi_{k}$. Hence Trace-Preserving Embeddings Preserve Operator Norms and Compose §composite, applied with $\pi=\pi_{k}$ and $\pi'=\pi_{k+1,n}$, gives $V_{k,n}=V_{k+1,n}V_{k}$ for $n\ge k+1$. Finally, for $S\in M_{k}$ and $n\ge k$, A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §intertwining and Trace-Preserving Embeddings Preserve Operator Norms and Compose §norm applied to $\pi_{k,n}$ give

Step 2. (The pre-Hilbert space.) The sequences $\xi=(\xi_{n})_{n\in\mathbb{N}}$ with $\xi_{n}\in H_{n}$ for every $n$ form a complex vector space $\mathcal{W}$ (Vector Space over a Field) under termwise addition and multiplication by complex scalars, each axiom holding termwise because every $H_{n}$ is one. Call $\xi\in\mathcal{W}$ coherent from $k$ if $\xi_{n+1}=V_{n}\xi_{n}$ for every $n\ge k$; then $\xi$ is coherent from every $l\ge k$, and by induction on $n\ge k$ (using $V_{k,n+1}=V_{n}V_{k,n}$) $\xi_{n}=V_{k,n}\xi_{k}$ for every $n\ge k$. Let $\mathcal{V}$ be the set of $\xi\in\mathcal{W}$ that are coherent from some $k$. It is a linear subspace of $\mathcal{W}$ (Linear Subspace): the zero sequence is coherent from $1$, and if $\xi$ is coherent from $k$ and $\eta$ from $l$, then $\xi+\eta$ and $c\xi$ ($c\in\mathbb{C}$) are coherent from the larger of $k,l$ because each $V_{n}$ is linear. So $\mathcal{V}$ is a complex vector space, by claim 1 of A Linear Subspace is a Vector Space and Inherits an Inner Product.

For $\xi,\eta\in\mathcal{V}$ call $n$ admissible for $(\xi,\eta)$ if both are coherent from $n$; admissible indices exist, and every index beyond an admissible one is admissible. If $n$ is admissible, then by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint and $V_{n}^{*}V_{n}=I$,

By induction, $\langle\xi_{n},\eta_{n}\rangle$ therefore has the same value at all admissible $n$ (any two admissible indices lie below a common one), and we define $h(\xi,\eta)$ to be this value. Given $\xi,\eta,\zeta\in\mathcal{V}$ and $c\in\mathbb{C}$, choose $n$ admissible for all pairs formed from $\xi,\eta,\zeta,\eta+\zeta,c\eta$; the properties of the inner product of $H_{n}$ give $h(\xi,\eta+\zeta)=h(\xi,\eta)+h(\xi,\zeta)$, $h(\xi,c\eta)=c\,h(\xi,\eta)$, $h(\eta,\xi)=\overline{h(\xi,\eta)}$ and $h(\xi,\xi)=\lVert\xi_{n}\rVert^{2}$, a real number $\ge0$. So $h$ satisfies the hypotheses of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §real-structure.

Let $H=H_{h}$ be the complex Hilbert completion of $(\mathcal{V},h)$, write $\langle\cdot,\cdot\rangle$ for its pairing $\langle\cdot,\cdot\rangle_{h}$, and let $J_{h}:\mathcal{V}\to H$ be its canonical map. By The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert, $H$ is a complex Hilbert space whose norm is induced by $\langle\cdot,\cdot\rangle$; by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, $J_{h}$ is complex-linear with $\langle J_{h}\xi,J_{h}\eta\rangle=h(\xi,\eta)$; and by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense, $D=J_{h}(\mathcal{V})$ is dense in $H$. Being the image of a complex vector space under a complex-linear map, $D$ is a linear subspace of $H$.

(2.1) If $\xi,\eta\in\mathcal{V}$ agree from some $m$ on, then $J_{h}\xi=J_{h}\eta$. Indeed $\zeta=\xi+(-1)\eta\in\mathcal{V}$ has $\zeta_{n}=0$ for $n\ge m$; taking an admissible $n\ge m$ for $(\zeta,\zeta)$, $\lVert J_{h}\zeta\rVert^{2}=h(\zeta,\zeta)=\lVert\zeta_{n}\rVert^{2}=0$, so $J_{h}\zeta=0$ and $J_{h}\xi=J_{h}\eta$ by linearity.

Step 3. (The maps $\iota_{k}$.) For $k\in\mathbb{N}$ and $\zeta\in H_{k}$ let $e_{k}(\zeta)\in\mathcal{W}$ have $n$-th term $V_{k,n}\zeta$ for $n\ge k$ and $0$ for $n<k$. It is coherent from $k$ since $V_{k,n+1}\zeta=V_{n}V_{k,n}\zeta$, so $e_{k}:H_{k}\to\mathcal{V}$, and $e_{k}$ is complex-linear. Set $\iota_{k}=J_{h}\circ e_{k}:H_{k}\to H$, which is complex-linear. With the admissible index $n=k$ and $V_{k,k}=I$,

so $\lVert\iota_{k}\zeta\rVert=\lVert\zeta\rVert$; thus $1$ is a bound for $\iota_{k}$ and $\iota_{k}\in\mathcal{L}(H_{k},H)$ (Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded).

(3.1) $\iota_{k+1}V_{k}=\iota_{k}$: for $\zeta\in H_{k}$ and $n\ge k+1$, $e_{k+1}(V_{k}\zeta)_{n}=V_{k+1,n}V_{k}\zeta=V_{k,n}\zeta=e_{k}(\zeta)_{n}$ by Step 1, so apply (2.1).

(3.2) If $\xi\in\mathcal{V}$ is coherent from $k$, then $J_{h}\xi=\iota_{k}(\xi_{k})$, because $\xi_{n}=V_{k,n}\xi_{k}=e_{k}(\xi_{k})_{n}$ for $n\ge k$ (Step 2) and (2.1) applies. Hence $D$ is the union of the sets $\iota_{k}(H_{k})$, $k\in\mathbb{N}$, the inclusion $\supseteq$ holding since $e_{k}(H_{k})\subseteq\mathcal{V}$.

(3.3) Let $\Omega=\iota_{1}(\Omega_{1})$. By induction on $k$, $\Omega=\iota_{k}(\Omega_{k})$ for every $k$, since $\iota_{k+1}(\Omega_{k+1})=\iota_{k+1}(V_{k}\Omega_{k})=\iota_{k}(\Omega_{k})$ by (3.1). Moreover $\lVert\Omega\rVert=\lVert\Omega_{1}\rVert=1$.

Step 4. (The operators $\rho_{k}(S)$.) Fix $k\in\mathbb{N}$ and $S\in M_{k}$. For $\xi\in\mathcal{V}$ let $A_{k,S}\xi\in\mathcal{W}$ have $n$-th term $\pi_{k,n}(S)\xi_{n}$ for $n\ge k$ and $0$ for $n<k$. If $\xi$ is coherent from $m$ and $n$ is at least $k$ and $m$, then by Step 1 and A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §intertwining for $\pi_{n}$ at $\pi_{k,n}(S)\in M_{n}$,

So $A_{k,S}\xi\in\mathcal{V}$ is coherent from the larger of $k,m$, and $A_{k,S}:\mathcal{V}\to\mathcal{V}$ is complex-linear because each $\pi_{k,n}(S)$ is. For $n\ge k$ admissible for $(\xi,\xi)$, hence also for $(A_{k,S}\xi,A_{k,S}\xi)$, Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound, Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded and (1.1) give

Hence the complex-linear map $J_{h}\circ A_{k,S}:\mathcal{V}\to H$ satisfies $\lVert J_{h}A_{k,S}\xi\rVert^{2}=h(A_{k,S}\xi,A_{k,S}\xi)\le\lVert S\rVert_{\mathrm{op}}^{2}h(\xi,\xi)$ by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, and The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §extension-linear, with $K=H$ and $C=\lVert S\rVert_{\mathrm{op}}$, gives exactly one continuous map $\rho_{k}(S):H\to H$ with

and $\rho_{k}(S)\in\mathcal{L}(H)$.

(4.2) $\rho_{k}(S)\iota_{k}\zeta=\iota_{k}(S\zeta)$ for $\zeta\in H_{k}$: by (1.1), $A_{k,S}e_{k}(\zeta)$ has $n$-th term $\pi_{k,n}(S)V_{k,n}\zeta=V_{k,n}S\zeta$ for $n\ge k$ and $0$ for $n<k$, so it equals $e_{k}(S\zeta)$; apply $J_{h}$ and (4.1). In particular $\rho_{k}(S)\Omega=\iota_{k}(S\Omega_{k})$ by (3.3).

Step 5. (Algebraic properties.) Fix $k$, and let $S,T\in M_{k}$, $c\in\mathbb{C}$ and $\xi\in\mathcal{V}$. Throughout, two elements of $\mathcal{L}(H)$ are shown equal by checking that they agree at every $J_{h}\xi$, $\xi\in\mathcal{V}$; this suffices by Bounded Linear and Conjugate-Linear Maps on a Dense Subspace of a Complex Hilbert Space Extend Uniquely §equality applied to the dense linear subspace $D$ of Step 2. Sums, scalar multiples and composites of elements of $\mathcal{L}(H)$ lie in $\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations.

(a) Compatibility. For $n\ge k+1$, $(A_{k+1,\pi_{k}(S)}\xi)_{n}=\pi_{k+1,n}(\pi_{k}(S))\xi_{n}=\pi_{k,n}(S)\xi_{n}=(A_{k,S}\xi)_{n}$ by Step 1, so these sequences agree from $k+1$ on, and (4.1) with (2.1) gives $\rho_{k+1}(\pi_{k}(S))J_{h}\xi=\rho_{k}(S)J_{h}\xi$. Hence $\rho_{k+1}(\pi_{k}(S))=\rho_{k}(S)$.

(b) Unit. $\pi_{k,n}(I)=I$ by Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding, so $A_{k,I}\xi$ agrees with $\xi$ from $k$ on, and $\rho_{k}(I)J_{h}\xi=J_{h}\xi$ by (4.1) and (2.1). Hence $\rho_{k}(I)=I$.

(c) Linearity and multiplicativity. The identities of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding for $\pi_{k,n}$ give, term by term (the terms with $n<k$ being $0$), $A_{k,S+T}\xi=A_{k,S}\xi+A_{k,T}\xi$, $A_{k,cS}\xi=c\,A_{k,S}\xi$ and $A_{k,ST}\xi=A_{k,S}(A_{k,T}\xi)$. Applying the complex-linear map $J_{h}$ and (4.1) (the last one twice),

Hence $\rho_{k}(S+T)=\rho_{k}(S)+\rho_{k}(T)$, $\rho_{k}(cS)=c\,\rho_{k}(S)$ and $\rho_{k}(ST)=\rho_{k}(S)\rho_{k}(T)$.

(d) Adjoints. $\rho_{k}(S)$ has an adjoint $\rho_{k}(S)^{*}\in\mathcal{L}(H)$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint. Let $\xi,\eta\in\mathcal{V}$ and choose $n\ge k$ admissible for both pairs $(\eta,A_{k,S^{*}}\xi)$ and $(A_{k,S}\eta,\xi)$. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, $\pi_{k,n}(S)$ is the adjoint of $\pi_{k,n}(S)^{*}$ and $\rho_{k}(S)$ that of $\rho_{k}(S)^{*}$; with (4.1), The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, $\pi_{k,n}(S^{*})=\pi_{k,n}(S)^{*}$ and Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint this gives

By Bounded Linear and Conjugate-Linear Maps on a Dense Subspace of a Complex Hilbert Space Extend Uniquely §equality with $E=D$, $\rho_{k}(S^{*})=\rho_{k}(S)^{*}$.

(e) Change of level. For $l\ge k$, $\rho_{l}(\pi_{k,l}(S))=\rho_{k}(S)$, by induction on $l\ge k$: for $l=k$, $\pi_{k,k}(S)=S$; and $\rho_{l+1}(\pi_{k,l+1}(S))=\rho_{l+1}(\pi_{l}(\pi_{k,l}(S)))=\rho_{l}(\pi_{k,l}(S))$ by Step 1 and (a).

Step 6. (A cyclic tracial operator algebra.) Let $\mathcal{A}=\{\rho_{k}(S):\ k\in\mathbb{N},\ S\in M_{k}\}\subseteq\mathcal{L}(H)$. By (e), any two elements of $\mathcal{A}$ can be written as $\rho_{l}(S)$, $\rho_{l}(T)$ with a common $l$ and $S,T\in M_{l}$ (take $l$ the larger level and replace the lower-level operator $S_{0}\in M_{k}$ by $\pi_{k,l}(S_{0})$).

Algebra. $I=\rho_{1}(I)\in\mathcal{A}$ by (b); and for $S,T\in M_{l}$ and $c\in\mathbb{C}$, (c) and (d) give $\rho_{l}(S)+\rho_{l}(T)=\rho_{l}(S+T)$, $c\,\rho_{l}(S)=\rho_{l}(cS)$, $\rho_{l}(S)\rho_{l}(T)=\rho_{l}(ST)$ and $\rho_{l}(S)^{*}=\rho_{l}(S^{*})$, which lie in $\mathcal{A}$ because $M_{l}$ is closed under these operations. So Cyclic Tracial Operator Algebras and Their Traces §star-algebra holds.

Cyclicity. $\lVert\Omega\rVert=1$ by (3.3). Let $x\in H$ and let $\varepsilon>0$ be real. Since $D$ is dense in $H$, Characterization of the Closure in a Metric Space by Open Balls (claim 1 implies claim 3) and (3.2) give $k$ and $\zeta\in H_{k}$ with $\lVert x-\iota_{k}\zeta\rVert<\varepsilon/2$; since $M_{k}\Omega_{k}$ is dense in $H_{k}$, the same claims give $S\in M_{k}$ with $\lVert\zeta-S\Omega_{k}\rVert<\varepsilon/2$. By (4.2), $\rho_{k}(S)\Omega=\iota_{k}(S\Omega_{k})$, and $\iota_{k}$ is linear and preserves norms (Step 3), so $\lVert\iota_{k}\zeta-\rho_{k}(S)\Omega\rVert=\lVert\zeta-S\Omega_{k}\rVert<\varepsilon/2$, and the triangle inequality gives $\lVert x-\rho_{k}(S)\Omega\rVert<\varepsilon$. By claim 3 implies claim 1 of Characterization of the Closure in a Metric Space by Open Balls, $x$ lies in the closure of $\mathcal{A}\Omega$; so $\mathcal{A}\Omega$ is dense in $H$ (Dense Subset of a Topological Space), and Cyclic Tracial Operator Algebras and Their Traces §cyclic holds.

Traciality. Let $A=\rho_{l}(S)$ and $B=\rho_{l}(T)$ with $S,T\in M_{l}$. By (c), (4.2) and Step 3,

and likewise $\langle\Omega,BA\Omega\rangle=\langle\Omega_{l},TS\Omega_{l}\rangle$. These agree by Cyclic Tracial Operator Algebras and Their Traces §tracial for $M_{l}$, so Cyclic Tracial Operator Algebras and Their Traces §tracial holds for $\mathcal{A}$.

Thus $(H,\mathcal{A},\Omega)$ is a cyclic tracial operator algebra in the sense of Cyclic Tracial Operator Algebras and Their Traces §triple.

Step 7. (The W-closure.)* Let $J$ be the conjugation of $(H,\mathcal{A},\Omega)$ given by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation, and let $M=(J\mathcal{A}J)'$, the commutant of The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant. By The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant §w-star, $(H,M,\Omega)$ is a tracial W*-probability space whose trace $\tau_{M}$ agrees with $\tau_{\mathcal{A}}$ on $\mathcal{A}$; by The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant §contains, $\mathcal{A}\subseteq M$; and by The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant §double-commutant, $M=\mathcal{A}''$, which is claim 3.

Step 8. (The embeddings $\rho_{k}$ and claims 1 and 2.) Fix $k$. The map $\rho_{k}:M_{k}\to M$ takes values in $\mathcal{A}\subseteq M$ and satisfies the first five identities of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding by Step 5 (b), (c), (d). For $S\in M_{k}$, by Step 7, Cyclic Tracial Operator Algebras and Their Traces §trace, (4.2), Step 3 and Tracial W*-Probability Spaces §trace,

So $\rho_{k}$ is a trace-preserving embedding of $(H_{k},M_{k},\Omega_{k})$ into $(H,M,\Omega)$. Since $\iota_{k}\in\mathcal{L}(H_{k},H)$ and $\iota_{k}S\Omega_{k}=\rho_{k}(S)\Omega$ for every $S\in M_{k}$ by (4.2), the uniqueness in Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry (from A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry) gives $V_{\rho_{k}}=\iota_{k}$.

Claim 1: $\rho_{k+1}(\pi_{k}(S))=\rho_{k}(S)$ is Step 5 (a), and $V_{\rho_{k+1}}V_{\pi_{k}}=\iota_{k+1}V_{k}=\iota_{k}=V_{\rho_{k}}$ by (3.1). Claim 2: the set $\{\rho_{k}(S)\Omega:\ k\in\mathbb{N},\ S\in M_{k}\}$ is $\mathcal{A}\Omega$, which is dense in $H$ by Step 6. Claim 3 was shown in Step 7.