Proof of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation

Throughout write $\sigma=\sigma_{a}$, and for a tracial state $\lambda$ on $\mathcal{P}_{d}$ write $h_{\lambda}(p,q)=\lambda(p^{*}q)$. Since $\gamma\in\Sigma_{m}$, there is a real $r>0$ with $\gamma\in\Sigma_{m,r}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, so Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation applies to $\gamma$; by claim 1 below it applies to $\mu$ as well. Sums, scalar multiples and composites of bounded linear maps are bounded linear maps, composition distributes over sums and commutes with scalar multiples, and $\lVert RX\rVert_{\mathrm{op}}\le\lVert R\rVert_{\mathrm{op}}\lVert X\rVert_{\mathrm{op}}$, by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §operations. Every bounded linear map $X$ between complex inner product spaces is continuous, since $\lVert X\xi-X\eta\rVert=\lVert X(\xi-\eta)\rVert\le\lVert X\rVert_{\mathrm{op}}\lVert\xi-\eta\rVert$ 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; the conjugations $J_{\lambda}$ are continuous by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation. Every bounded linear map $X$ between the complex Hilbert spaces below has an adjoint $X^{*}$, bounded with $\lVert X^{*}\rVert_{\mathrm{op}}=\lVert X\rVert_{\mathrm{op}}$, by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint; by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint we have $\langle X^{*}y,\xi\rangle=\langle y,X\xi\rangle$, and since $X$ is the adjoint of $X^{*}$ 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, also $\langle\xi,X^{*}y\rangle=\langle X\xi,y\rangle$. Adjoints are unique 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.

Preliminary (P). Let $\lambda$ be a tracial state on $\mathcal{P}_{d}$, let $K$ be a complex Hilbert space and let $F,G:\mathcal{H}_{\lambda}\to K$ be continuous maps with $F\widehat{p}=G\widehat{p}$ for every $p\in\mathcal{P}_{d}$. Then $F=G$. Indeed, let $\beta=\operatorname{Re}h_{\lambda}$. By The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §completion and The Hilbert Completion of a Complex Vector Space with a Positive Semidefinite Hermitian Form §canonical-map, $\mathcal{H}_{\lambda}$ is the set $H_{\beta}$ of the Hilbert completion of $(\mathcal{P}_{d},\beta)$ and $\widehat{p}=J_{\beta}p$; by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §hilbert the norm of $\mathcal{H}_{\lambda}$ is the norm $|\cdot|$ of $H_{\beta}$, so the two carry the same metric (and $J_{\beta}(\mathcal{P}_{d})$ is the dense image of The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense). Let $\xi\in\mathcal{H}_{\lambda}$. By The Hilbert Completion of a Real Vector Space with a Positive Semidefinite Symmetric Bilinear Form §completion, $\xi=[u]$ for a $\beta$-Cauchy sequence $u=(u_{k})_{k\in\mathbb{N}}$, and by The Hilbert Completion is a Real Hilbert Space Containing a Dense Isometric Image, and Bounded Linear Maps Extend to It §dense the sequence $(\widehat{u_{k}})_{k}=(J_{\beta}u_{k})_{k}$ converges to $\xi$. By continuity (Continuous Map Between Metric Spaces), the sequence $(F\widehat{u_{k}})_{k}=(G\widehat{u_{k}})_{k}$ converges both to $F\xi$ and to $G\xi$, so $F\xi=G\xi$ by Uniqueness of Limits in a Metric Space. We use (P) for bounded linear maps and for composites of such maps with conjugations.

1. (The marginal law) Let $C>0$ be real with $\lVert L_{a_{j}}\rVert_{\mathrm{op}}\le C$ for all $j\in[n]$. For $j\in[n]$ and $p\in\mathcal{P}_{m}$, by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication and 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,

Since every $a_{j}$ lies in $\mathcal{P}_{m,\mathrm{sa}}$, The Norm Bound of a Noncommutative Law: Multiplication by a Variable is Bounded, the Bound is Detected by Even Moments, and Laws Pull Back under Self-Adjoint Substitutions §pullback (with $d=m$ and $\lambda=\gamma$) shows that $\mu=\gamma\circ\sigma$ is a tracial state on $\mathcal{P}_{n}$ with norm bound $C$, that is $\mu\in\Sigma_{n,C}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound. Such a $C$ exists, for instance $C=1+\sum_{j=1}^{n}\lVert L_{a_{j}}\rVert_{\mathrm{op}}$; hence $\mu\in\Sigma_{n}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

2. (The isometry) Define $T_{0}:\mathcal{P}_{n}\to\mathcal{H}_{\gamma}$ by $T_{0}p=\widehat{\sigma(p)}^{\,\gamma}$. It is complex-linear, because $\sigma$ is linear by Substitution of Noncommutative Polynomials into the Variables §substitution and the canonical map is complex-linear by The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry. For $p,q\in\mathcal{P}_{n}$, by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint (the $a_{j}$ being self-adjoint) and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism,

in particular $\lVert T_{0}p\rVert^{2}=h_{\mu}(p,p)$. By The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns and The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes, $\mathcal{H}_{\mu}$ is the complex Hilbert completion of $(\mathcal{P}_{n},h_{\mu})$ and $\widehat{p}^{\,\mu}$ is the image of $p$ under its canonical map. Hence 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 $C=1$, gives a continuous $V:\mathcal{H}_{\mu}\to\mathcal{H}_{\gamma}$ with $V\widehat{p}^{\,\mu}=\widehat{\sigma(p)}^{\,\gamma}$ for every $p$, which belongs to $\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ with bound $1$ and satisfies $\langle V\xi,V\eta\rangle=\langle\xi,\eta\rangle$ for all $\xi,\eta\in\mathcal{H}_{\mu}$. Any $V'\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ with the same values on classes is continuous, hence $V'=V$ by (P); so $V$ is the only such map. 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, $\lVert V\rVert_{\mathrm{op}}\le1$, hence also $\lVert V^{*}\rVert_{\mathrm{op}}\le1$.

For $\xi\in\mathcal{H}_{\mu}$ let $w=V^{*}V\xi-\xi$. Then $\langle w,w\rangle=\langle w,V^{*}V\xi\rangle-\langle w,\xi\rangle=\langle Vw,V\xi\rangle-\langle w,\xi\rangle=0$, so $w=0$ by claim 4 of Elementary Properties of a Complex Inner Product; thus $V^{*}V=I$. By The Complex GNS Space of a Tracial State on Noncommutative Polynomials §vacuum and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $V\Omega_{\mu}=V\widehat{1}=\widehat{\sigma(1)}=\widehat{1}=\Omega_{\gamma}$.

For $p,q\in\mathcal{P}_{n}$, by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism,

All four composites are bounded linear maps, so (P) gives $VL_{p}=L_{\sigma(p)}V$ and $VR_{p}=R_{\sigma(p)}V$. For the adjoint forms, apply the first identity to $p^{*}$ and use $\sigma(p^{*})=\sigma(p)^{*}$: $VL_{p^{*}}=L_{\sigma(p)^{*}}V$. By Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint, $L_{p}^{*}=L_{p^{*}}$ and $L_{\sigma(p)}^{*}=L_{\sigma(p)^{*}}$, so 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 $L_{p}$ is the adjoint of $L_{p^{*}}$ and $L_{\sigma(p)}$ is the adjoint of $L_{\sigma(p)^{*}}$; by the same claim (the adjoint of a composite $RX$ is $X^{*}R^{*}$), $L_{p}V^{*}$ is the adjoint of $VL_{p^{*}}$ and $V^{*}L_{\sigma(p)}$ is the adjoint of $L_{\sigma(p)^{*}}V$. These two maps being equal, uniqueness of adjoints gives $V^{*}L_{\sigma(p)}=L_{p}V^{*}$. The same argument with $R$ in place of $L$ gives $V^{*}R_{\sigma(p)}=R_{p}V^{*}$.

Finally, for $p\in\mathcal{P}_{n}$, by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint, $VJ_{\mu}\widehat{p}=V\widehat{p^{*}}=\widehat{\sigma(p)^{*}}=J_{\gamma}\widehat{\sigma(p)}=J_{\gamma}V\widehat{p}$; both $VJ_{\mu}$ and $J_{\gamma}V$ are continuous, so $VJ_{\mu}=J_{\gamma}V$ by (P).

3. (The embedding) Fix $T\in\mathcal{M}_{\mu}$, put $\zeta=VT\Omega_{\mu}\in\mathcal{H}_{\gamma}$, write $\Omega=\Omega_{\mu}$, and fix $q\in\mathcal{P}_{m}$. By Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation applied to $q^{*}$, and $(q^{*})^{*}=q$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, we have $J_{\gamma}L_{q^{*}}J_{\gamma}=R_{q}$, hence $J_{\gamma}R_{q}=J_{\gamma}J_{\gamma}L_{q^{*}}J_{\gamma}=L_{q^{*}}J_{\gamma}$; as $J_{\gamma}$ preserves norms by the same claim, $\lVert R_{q}\zeta\rVert=\lVert L_{q^{*}}J_{\gamma}\zeta\rVert$. By claim 2 and The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §vacuum, $J_{\gamma}\zeta=VJ_{\mu}T\Omega=VT^{*}\Omega$. Let $F=L_{q^{*}}V\in\mathcal{L}(\mathcal{H}_{\mu},\mathcal{H}_{\gamma})$ and $D=F^{*}F\in\mathcal{L}(\mathcal{H}_{\mu})$; thus $\lVert R_{q}\zeta\rVert=\lVert FT^{*}\Omega\rVert$.

For $p\in\mathcal{P}_{n}$, claim 2 and Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §commute give $FR_{p}=L_{q^{*}}R_{\sigma(p)}V=R_{\sigma(p)}L_{q^{*}}V=R_{\sigma(p)}F$. Applying this to $p^{*}$ and taking adjoints exactly as in claim 2 ($R_{p}$ is the adjoint of $R_{p^{*}}$ and $R_{\sigma(p)}$ that of $R_{\sigma(p^{*})}=R_{\sigma(p)^{*}}$) yields $R_{p}F^{*}=F^{*}R_{\sigma(p)}$. Hence $DR_{p}=F^{*}R_{\sigma(p)}F=R_{p}F^{*}F=R_{p}D$, so $D\in\mathcal{M}_{\mu}$ by The Tracial Algebra of a Noncommutative Law and Its Trace §algebra, and $D\ge0$ 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.

Let $c=\lVert T\rVert_{\mathrm{op}}^{2}$ and $A=cI-T^{*}T$. By The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra, $T^{*}$, $DT^{*}$, $T^{*}T$, $T^{*}TD$ and $A$ lie in $\mathcal{M}_{\mu}$, and $A\ge0$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §positivity. Using 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 for the first equality, the adjoint relation for the second, The Tracial Algebra of a Noncommutative Law and Its Trace §trace for the third, and the trace property and linearity of The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §trace for the rest,

By The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §positivity, $\tau_{\mu}(AD)$ is real and nonnegative. Moreover $\tau_{\mu}(D)=\langle\Omega,F^{*}F\Omega\rangle=\lVert F\Omega\rVert^{2}$ 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 by claim 2, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation, $F\Omega=L_{q^{*}}\Omega_{\gamma}=\widehat{q^{*}}=J_{\gamma}\widehat{q}$, so $\tau_{\mu}(D)=\lVert\widehat{q}\rVert^{2}$. Therefore $\lVert R_{q}\zeta\rVert^{2}\le\lVert T\rVert_{\mathrm{op}}^{2}\lVert\widehat{q}\rVert^{2}$, and taking nonnegative square roots, $\lVert R_{q}\zeta\rVert\le\lVert T\rVert_{\mathrm{op}}\lVert\widehat{q}\rVert$ for every $q\in\mathcal{P}_{m}$. By The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §bounded-vectors for $\gamma$ with $C=\lVert T\rVert_{\mathrm{op}}$, there is exactly one $\pi(T)\in\mathcal{M}_{\gamma}$ with $\pi(T)\Omega_{\gamma}=VT\Omega_{\mu}$, and $\lVert\pi(T)\rVert_{\mathrm{op}}\le\lVert T\rVert_{\mathrm{op}}$.

For $p\in\mathcal{P}_{n}$, by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §vacuum (for $\gamma$ and for $\mu$) and claim 2,

Both $\pi(T)V$ and $VT$ are bounded linear, so $\pi(T)V=VT$ by (P).

4. (Homomorphism) By the uniqueness in claim 3, for $X\in\mathcal{M}_{\mu}$ and $Y\in\mathcal{M}_{\gamma}$ with $Y\Omega_{\gamma}=VX\Omega_{\mu}$ we have $\pi(X)=Y$. All operators below lie in $\mathcal{M}_{\mu}$ or $\mathcal{M}_{\gamma}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra. Let $S,T\in\mathcal{M}_{\mu}$, $c\in\mathbb{C}$ and $p\in\mathcal{P}_{n}$. Then $(\pi(S)+c\pi(T))\Omega_{\gamma}=VS\Omega_{\mu}+cVT\Omega_{\mu}=V(S+cT)\Omega_{\mu}$, so $\pi(S+cT)=\pi(S)+c\pi(T)$. Next, $I\Omega_{\gamma}=\Omega_{\gamma}=VI\Omega_{\mu}$ by claim 2, so $\pi(I)=I$. By claim 3, $\pi(S)\pi(T)\Omega_{\gamma}=\pi(S)VT\Omega_{\mu}=VST\Omega_{\mu}$, so $\pi(ST)=\pi(S)\pi(T)$. By The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §vacuum (for $\gamma$ and for $\mu$) and claim 2, $\pi(T)^{*}\Omega_{\gamma}=J_{\gamma}\pi(T)\Omega_{\gamma}=J_{\gamma}VT\Omega_{\mu}=VJ_{\mu}T\Omega_{\mu}=VT^{*}\Omega_{\mu}$, so $\pi(T^{*})=\pi(T)^{*}$. By The Tracial Algebra of a Noncommutative Law and Its Trace §trace and claim 2, $\tau_{\gamma}(\pi(T))=\langle\Omega_{\gamma},\pi(T)\Omega_{\gamma}\rangle=\langle V\Omega_{\mu},VT\Omega_{\mu}\rangle=\langle\Omega_{\mu},T\Omega_{\mu}\rangle=\tau_{\mu}(T)$. By Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and claim 2, $L_{\sigma(p)}\Omega_{\gamma}=\widehat{\sigma(p)}=V\widehat{p}=VL_{p}\Omega_{\mu}$, and $L_{\sigma(p)}\in\mathcal{M}_{\gamma}$ by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §commute, so $\pi(L_{p})=L_{\sigma(p)}$. Finally, if $\pi(S)=\pi(T)$, then $VS\Omega_{\mu}=VT\Omega_{\mu}$, so $S\Omega_{\mu}=V^{*}VS\Omega_{\mu}=V^{*}VT\Omega_{\mu}=T\Omega_{\mu}$; then $(S-T)\Omega_{\mu}=0$ with $S-T\in\mathcal{M}_{\mu}$, and $S=T$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §vacuum. So $\pi$ is injective.

5. (Conditional expectation) Let $b\in\mathcal{M}_{\gamma}$; then $E(b)=V^{*}bV\in\mathcal{L}(\mathcal{H}_{\mu})$. For $p\in\mathcal{P}_{n}$, by claim 2 and The Tracial Algebra of a Noncommutative Law and Its Trace §algebra (for $\gamma$),

so $E(b)\in\mathcal{M}_{\mu}$. $E$ is linear since composition distributes over sums and commutes with scalar multiples. 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, the adjoint of $V^{*}(bV)$ is $(bV)^{*}(V^{*})^{*}=V^{*}b^{*}V$, $V$ being the adjoint of $V^{*}$; so $E(b)^{*}=E(b^{*})$. Also $\lVert E(b)\rVert_{\mathrm{op}}\le\lVert V^{*}\rVert_{\mathrm{op}}\lVert b\rVert_{\mathrm{op}}\lVert V\rVert_{\mathrm{op}}\le\lVert b\rVert_{\mathrm{op}}$ by claim 2.

Let $S,T\in\mathcal{M}_{\mu}$. By claims 3 and 4, $\pi(S)^{*}V=\pi(S^{*})V=VS^{*}$; since $\pi(S)$ is the adjoint of $\pi(S)^{*}$ and $S$ that of $S^{*}$, taking adjoints with 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 uniqueness gives $V^{*}\pi(S)=SV^{*}$. With $\pi(T)V=VT$ this yields $E(\pi(S)\,b\,\pi(T))=V^{*}\pi(S)\,b\,\pi(T)V=SV^{*}bVT=S\,E(b)\,T$, and $E(\pi(T))=V^{*}\pi(T)V=V^{*}VT=T$. For $\zeta,\zeta'\in\mathcal{H}_{\mu}$, $\langle\zeta,E(b)\zeta'\rangle=\langle\zeta,V^{*}(bV\zeta')\rangle=\langle V\zeta,bV\zeta'\rangle$; with $\zeta=\zeta'=\Omega_{\mu}$, $V\Omega_{\mu}=\Omega_{\gamma}$ and The Tracial Algebra of a Noncommutative Law and Its Trace §trace this gives $\tau_{\mu}(E(b))=\langle\Omega_{\gamma},b\Omega_{\gamma}\rangle=\tau_{\gamma}(b)$.

In particular $E(I)=V^{*}V=I$. For $b\in\mathcal{M}_{\gamma}$, $b^{*}b\in\mathcal{M}_{\gamma}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra, and 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 the adjoint of $bV$ is $V^{*}b^{*}$, so $E(b^{*}b)=(bV)^{*}(bV)$, which is self-adjoint and positive semi-definite by the same claim; that is, $E(b^{*}b)\ge0$.