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

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $(H,M,\Omega)$ be a tracial W*-probability space with conjugation $J$ and fixed vectors $H_{\mathrm{sa}}$ as in Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces. Resolvents $R_{y}(a)$ and resolvent transforms $\mathbf{R}(s)$ of self-adjoint tuples $s$ in $M$ are those of Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple.

1. (Products)¶ For all $S,T\in M$, $\lVert ST\Omega\rVert\le\lVert S\rVert_{\mathrm{op}}\lVert T\Omega\rVert$ and $\lVert ST\Omega\rVert\le\lVert S\Omega\rVert\,\lVert T\rVert_{\mathrm{op}}$. For every $\Delta\in M$,

2. (Membership)¶ Let $a\in M$ be self-adjoint and let $y\ne0$ be real. Then $R_{y}(a)\in M$ and $R_{y}(a)^{*}\in M$. Consequently, for every $d\in\mathbb{N}$ and every self-adjoint $d$-tuple $s$ in $M$, $\mathbf{R}(s)$ is a self-adjoint $2d$-tuple in $M$ whose entries have operator norm at most $1$.

3. (Lipschitz bound)¶ Let $a,b\in M$ be self-adjoint and let $y\ne0$ be real. Then

4. (Extension to self-adjoint vectors)¶ Let $\xi\in H_{\mathrm{sa}}$. There is a sequence $(s_{k})_{k\in\mathbb{N}}$ of self-adjoint elements of $M$ such that $(s_{k}\Omega)$ converges to $\xi$ in $H$, by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint. There is exactly one $T\in M$ such that $(R_{1}(s_{k})\Omega)_{k\in\mathbb{N}}$ converges to $T\Omega$ for every such sequence $(s_{k})$; it satisfies $\lVert T\rVert_{\mathrm{op}}\le1$, and $T=R_{1}(s)$ when $\xi=s\Omega$ for a self-adjoint $s\in M$.