Proof of Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the $L^2$ Lipschitz Bound, and Extension to Self-Adjoint Vectors

Each result cited is universally quantified over the data in its own statement. $(H,M,\Omega)$ is a cyclic tracial operator algebra with $M=M''$ by Tracial W*-Probability Spaces §space. Norm bounds for sums, scalar multiples and composites are those of 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 $\lVert T\rVert_{\mathrm{op}}$ is a bound for $T$ 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 conjugation $J$ satisfies $\lVert J\eta\rVert=\lVert\eta\rVert$ for every $\eta\in H$ and $J\Omega=\Omega$, by Conjugation of a Complex Hilbert Space §conjugation and The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation.

Claim 1. The first inequality holds because $\lVert S\rVert_{\mathrm{op}}$ is a bound for $S$. For the second, The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §right-action applied to $T^{*}\in M$ gives $JT^{*}J(S\Omega)=S(T^{*})^{*}\Omega=ST\Omega$, and $\lVert JT^{*}J\rVert_{\mathrm{op}}=\lVert T^{*}\rVert_{\mathrm{op}}=\lVert T\rVert_{\mathrm{op}}$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §conjugation and 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 $\lVert ST\Omega\rVert\le\lVert T\rVert_{\mathrm{op}}\lVert S\Omega\rVert$. For the identity put $u=\Delta\Omega$ and $v=\Delta^{*}\Omega$; since $\Delta\in M$, $\lVert v\rVert=\lVert u\rVert$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §trace. Expanding the inner product, $\lVert u+v\rVert^{2}+\lVert u-v\rVert^{2}=2\lVert u\rVert^{2}+2\lVert v\rVert^{2}$, and $|\tfrac{1}{2i}|=\tfrac12$, so

Claim 2. Since $M=M''$ and $a\in M$, Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §adjoint gives $R_{y}(a)\in M$ and $R_{-y}(a)\in M$, and $R_{y}(a)^{*}=R_{-y}(a)$. The set $M=(M')'$ is closed under sums and complex multiples by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra, so for a self-adjoint $d$-tuple $s$ in $M$ the entries $\tfrac12(R_{1}(s_{j})+R_{1}(s_{j})^{*})$ and $\tfrac{1}{2i}(R_{1}(s_{j})-R_{1}(s_{j})^{*})$ of $\mathbf{R}(s)$ belong to $M$; they are self-adjoint with operator norm at most $1$ by Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §parts. Hence $\mathbf{R}(s)$ is a self-adjoint $2d$-tuple in $M$.

Claim 3. Put $\Delta=R_{y}(a)-R_{y}(b)$. By Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation §identity, $\Delta=R_{y}(a)\,(b-a)\,R_{y}(b)$, where $b-a\in M$ and $R_{y}(a),R_{y}(b)\in M$ by Claim 2. By Claim 1 (first inequality, then second) and $\lVert R_{y}(\cdot)\rVert_{\mathrm{op}}\le|y|^{-1}$ from The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A §invertible,

Claim 4. A sequence $(s_{k})$ of self-adjoint elements of $M$ with $s_{k}\Omega\to\xi$ exists by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint. Fix such a sequence. By Claim 3 with $y=1$, $\lVert R_{1}(s_{k})\Omega-R_{1}(s_{l})\Omega\rVert\le\lVert s_{k}\Omega-s_{l}\Omega\rVert$; since the convergent sequence $(s_{k}\Omega)$ is Cauchy, so is $(R_{1}(s_{k})\Omega)$, and it converges to some $\zeta\in H$ because $H$ is a complex Hilbert space. For any other such sequence $(t_{k})$, $\lVert R_{1}(s_{k})\Omega-R_{1}(t_{k})\Omega\rVert\le\lVert s_{k}\Omega-t_{k}\Omega\rVert\to0$, so $(R_{1}(t_{k})\Omega)$ also converges to $\zeta$.

Let $S\in M$. By The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §right-action, $JSJ\in M'$, so $JSJ$ commutes with $R_{1}(s_{k})\in M$, and

using $\lVert R_{1}(s_{k})\rVert_{\mathrm{op}}\le1$ and $J\Omega=\Omega$. Since $JSJ\in\mathcal{L}(H)$ is continuous, $\lVert JSJ\zeta\rVert\le\lVert S\Omega\rVert$. By Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §right-bounded with $C=1$ there is exactly one $T\in M$ with $T\Omega=\zeta$, and $\lVert T\rVert_{\mathrm{op}}\le1$. This $T$ has the stated property; if $T'\in M$ also has it, then $T'\Omega=\zeta=T\Omega$ by uniqueness of limits (Uniqueness of Limits in a Metric Space), so $T'=T$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §separating. Finally, if $\xi=s\Omega$ with $s\in M$ self-adjoint, the constant sequence $s_{k}=s$ is admissible, so $T\Omega=R_{1}(s)\Omega$ with $R_{1}(s)\in M$ by Claim 2, and $T=R_{1}(s)$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §separating.