Proof of Trace-Preserving Embeddings Preserve Operator Norms and Compose

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

Write $\tau_{j}=\tau_{M_{j}}$ for $j=0,1,2$ for the traces. By Tracial W*-Probability Spaces §space, each $(H_{j},M_{j},\Omega_{j})$ is a cyclic tracial operator algebra with $M_{j}=M_{j}''$, so by Cyclic Tracial Operator Algebras and Their Traces §star-algebra each $M_{j}\subseteq\mathcal{L}(H_{j})$ contains $I$ and is closed under sums, complex scalar multiples, products and adjoints. By Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding, $\pi$ and $\pi'$ satisfy the six identities displayed there, which are exactly the hypotheses of A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry; and by Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry, $V_{\pi}$ and $V_{\pi'}$ are the operators $V$ of A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry for $\pi$ and for $\pi'$.

Step 1. (Claim 1, lower bound.) Let $S\in M_{0}$. By A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §injective applied to $\pi$, $\lVert S\rVert_{\mathrm{op}}\le\lVert\pi(S)\rVert_{\mathrm{op}}$.

Step 2. (A positive element of $M_{0}$.) Let $c=\lVert S\rVert_{\mathrm{op}}$. By Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §operator-norm, $c$ is the greatest lower bound of the set of bounds for $S$, each of which is $\ge0$ by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded; so $0$ is a lower bound of that set and $c\ge0$. Set

By Cyclic Tracial Operator Algebras and Their Traces §star-algebra, $S^{*}$, $S^{*}S$ and hence $T$ belong to $M_{0}\subseteq\mathcal{L}(H_{0})$. 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, $S^{*}S$ is self-adjoint, so it is an adjoint of itself, and $T$ has the adjoint $c^{2}I+(-1)S^{*}S=T$ (the scalars $c^{2}$ and $-1$ are real); by the same claim $T$ is self-adjoint. For $\xi\in H_{0}$ the same claim gives

By 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, $c$ is a bound for $S$, so $\lVert S\xi\rVert\le c\lVert\xi\rVert$ by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded; both sides are $\ge0$, so squaring shows that $\langle\xi,T\xi\rangle$ is a real number $\ge0$. Thus $T$ is positive semi-definite in the sense of Positive Semi-Definite Operator, and $T\ge0$ in the sense of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps.

Step 3. (The square root lies in $M_{0}$.) By Square Roots of Positive Bounded Operators on a Complex Hilbert Space, Commuting with Everything that Commutes with the Operator §square-root applied to $T$ there is an operator, which we call $R\in\mathcal{L}(H_{0})$, with $R\ge0$ and $RR=T$; by Square Roots of Positive Bounded Operators on a Complex Hilbert Space, Commuting with Everything that Commutes with the Operator §commutation, $R$ commutes with every $B\in\mathcal{L}(H_{0})$ that commutes with $T$. Let $B\in M_{0}'$, the commutant of The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant. Since $S,S^{*}\in M_{0}$, we have $BS=SB$ and $BS^{*}=S^{*}B$; as composition distributes over sums and commutes with scalar multiples (Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations),

Hence $RB=BR$. As $B\in M_{0}'$ was arbitrary, $R\in(M_{0}')'=M_{0}''$ by The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant, and $M_{0}''=M_{0}$ by Tracial W*-Probability Spaces §space. So $R\in M_{0}$.

Step 4. (Claim 1, upper bound.) By Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps, $R\ge0$ includes that $R$ is self-adjoint, so $R$ is an adjoint of $R$ 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, and $R^{*}=R$ 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-unique. The identities of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding give $\pi(R)^{*}=\pi(R^{*})=\pi(R)$ and

Here $\pi(S),\pi(R)\in M_{1}\subseteq\mathcal{L}(H_{1})$ have adjoints 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. For $\xi\in H_{1}$, 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 gives

Hence $\lVert\pi(S)\xi\rVert\le c\lVert\xi\rVert$ for every $\xi\in H_{1}$, so the real number $c\ge0$ is a bound for $\pi(S)$ (Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded), and $\lVert\pi(S)\rVert_{\mathrm{op}}\le c=\lVert S\rVert_{\mathrm{op}}$ by 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. With Step 1 this proves claim 1.

Step 5. (Claim 2, the embedding identities.) Let $\sigma=\pi'\circ\pi:M_{0}\to M_{2}$, which is defined because $\pi$ takes values in $M_{1}$. For $S,T\in M_{0}$ and $c\in\mathbb{C}$, applying the identities of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding first for $\pi$ and then for $\pi'$ at the elements $\pi(S),\pi(T)\in M_{1}$:

So $\sigma$ is a trace-preserving embedding of $(H_{0},M_{0},\Omega_{0})$ into $(H_{2},M_{2},\Omega_{2})$.

Step 6. (Claim 2, the implementing isometry.) 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, $V_{\pi'}V_{\pi}\in\mathcal{L}(H_{0},H_{2})$. For $S\in M_{0}$, Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry for $\pi$, and then for $\pi'$ at $\pi(S)\in M_{1}$, gives

By Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry, $V_{\sigma}$ is the unique element of $\mathcal{L}(H_{0},H_{2})$ with $V_{\sigma}S\Omega_{0}=\sigma(S)\Omega_{2}$ for every $S\in M_{0}$, the uniqueness being that of A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry applied to $\sigma$. Hence $V_{\pi'\circ\pi}=V_{\pi'}V_{\pi}$, which completes claim 2.