Proof of The Conditional Expectation of a Trace-Preserving Embedding: Values in the Subalgebra, Bimodule Property, Trace, Positivity, Contraction and the Jones Projection

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

By Tracial W*-Probability Spaces §space, $(H_{0},M_{0},\Omega_{0})$ and $(H,M,\Omega)$ are cyclic tracial operator algebras; let $J_{0}$ and $J$ be their conjugations. By Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding, $\pi$ satisfies 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 and the uniqueness in A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry, $V$ is the operator of that clause. Hence $V\in\mathcal{L}(H_{0},H)$, $VS\Omega_{0}=\pi(S)\Omega$ for $S\in M_{0}$, $V^{*}V=I_{H_{0}}$, $\lVert V\xi\rVert=\lVert\xi\rVert$ and $V\Omega_{0}=\Omega$; and by A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §intertwining, for every $S\in M_{0}$,

By Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §expectation, $E(b)=V^{*}bV$ for $b\in M$. Adjoints exist and lie in $\mathcal{L}$ 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, obey $\langle v,T^{*}w\rangle=\langle Tv,w\rangle$ by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint and conjugate symmetry, and are computed 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; sums, scalar multiples and composites of bounded maps are bounded, and composition distributes over sums and commutes with scalars, 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. Compositions of maps (linear or not) are associative.

Claim 1 (Values). Let $b\in M$. Then $E(b)=V^{*}bV\in\mathcal{L}(H_{0})$. Let $S\in M_{0}$; then $J_{0}SJ_{0}\in\mathcal{L}(H_{0})$ and $J\pi(S)J\in\mathcal{L}(H)$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §conjugation. Using $VJ_{0}=JV$ twice and $\pi(S)V=VS$,

Since $J\pi(S)J\in JMJ=M'$ by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §commutant applied to $(H,M,\Omega)$, and $b\in M$, we have $b\,(J\pi(S)J)=(J\pi(S)J)\,b$ by The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant. Using $V^{*}J=J_{0}V^{*}$ twice and $V^{*}\pi(S)=SV^{*}$,

Therefore

As this holds for every $S\in M_{0}$, $E(b)\in(J_{0}M_{0}J_{0})'$, which equals $M_{0}$ by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §commutant applied to $(H_{0},M_{0},\Omega_{0})$.

For $b,c\in M$ and $\lambda\in\mathbb{C}$, $E(b+c)=V^{*}(b+c)V=V^{*}bV+V^{*}cV$ and $E(\lambda b)=V^{*}(\lambda b)V=\lambda\,V^{*}bV$, so $E$ is linear. By the adjoint calculus, $E(b)^{*}=(V^{*}bV)^{*}=V^{*}b^{*}(V^{*})^{*}=V^{*}b^{*}V=E(b^{*})$. Since $\lVert V\xi\rVert=\lVert\xi\rVert$, the number $1$ is a bound for $V$, so $\lVert V\rVert_{\mathrm{op}}\le1$ 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, and $\lVert V^{*}\rVert_{\mathrm{op}}=\lVert V\rVert_{\mathrm{op}}\le1$ 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. Hence

Claim 2 (Bimodule property). For $b\in M$ and $S,T\in M_{0}$ we have $\pi(S)b\pi(T)\in M$ by Cyclic Tracial Operator Algebras and Their Traces §star-algebra, and by the intertwining relations

Also $E(I)=V^{*}IV=V^{*}V=I$. With $b=I$ and $T=I$ (so $\pi(T)=I$) this gives $E(\pi(S))=S\,E(I)=S$.

Claim 3 (Trace). Let $b\in M$. By Claim 1, $E(b)\in M_{0}$, and by Tracial W*-Probability Spaces §trace and Cyclic Tracial Operator Algebras and Their Traces §trace,

For $S\in M_{0}$, Claim 2 with $T=I$ gives $S\,E(b)=E(\pi(S)b)$, and $\pi(S)b\in M$; so by the first identity $\tau_{0}(S\,E(b))=\tau_{0}(E(\pi(S)b))=\tau(\pi(S)b)$.

Claim 4 (Positivity). Let $b\in M$ and put $T=bV\in\mathcal{L}(H_{0},H)$. By the adjoint calculus $T^{*}=V^{*}b^{*}$, so $E(b^{*}b)=V^{*}b^{*}bV=T^{*}T$, which is self-adjoint and positive semi-definite 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. As $E(b^{*}b)\in\mathcal{L}(H_{0})$, this is $E(b^{*}b)\ge0$ in the sense of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps.

Claim 5 (Jones projection). The operator $e=VV^{*}$ belongs to $\mathcal{L}(H)$ and is linear. By the adjoint calculus $e^{*}=(V^{*})^{*}V^{*}=VV^{*}=e$, so $e$ is an adjoint of itself and hence self-adjoint 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. Moreover $ee=V(V^{*}V)V^{*}=VI_{H_{0}}V^{*}=e$. Thus $e$ is an orthogonal projection by Orthogonal Projection. For $S\in M_{0}$,

For $b\in M$, $E(b)\Omega_{0}=V^{*}bV\Omega_{0}=V^{*}b\Omega$; and since $E(b)\in M_{0}$ by Claim 1, the defining property of $V$ gives