Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation

1. (Conventions)¶ The conventions of Square-Integrable Noncommutative Laws: Standing Notation are in force. The letter $M$ names the set of operators of a tracial W*-probability space, while $M(\lambda)$ remains the second moment of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws; the letter $V$, with a subscript, denotes an implementing isometry, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces.

2. (Spaces)¶ $(H,M,\Omega)$ and $(K,N,\Psi)$ denote tracial W*-probability spaces, with traces $\tau_{M}$ and $\tau_{N}$. $J$ is the conjugation of $(H,M,\Omega)$, $M'$ is the commutant of $M$, and $H_{\mathrm{sa}}=H^{J}$ is the set of fixed vectors of $J$.

3. (Embeddings)¶ For a trace-preserving embedding $\pi$, $V_{\pi}$ is its implementing isometry and $E_{\pi}$ its conditional expectation.

4. (Tuples)¶ For $d\in\mathbb{N}$, self-adjoint $d$-tuples $s$ in $M$, their vacuum tuples $s\Omega$ and their laws $\lambda_{s}$ are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws; $L^{2}$ $d$-tuples $X,Y$, their differences $X-Y$ and $L^{2}$ norm $\lVert X\rVert_{2}$, the operations $TX$, $(X,Y)$, $(X,Y,Z)$ and $V_{\pi}X$, and the laws $\mathrm{law}(X)$, $\mathrm{law}(X,Y)$ and $\mathrm{law}(X,Y,Z)$ are those of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws.

5. (Background)¶ The following results are in force and may be used without restating them: Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation, The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra, The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant, Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors, The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications, A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry, The Conditional Expectation of a Trace-Preserving Embedding: Values in the Subalgebra, Bimodule Property, Trace, Positivity, Contraction and the Jones Projection, Trace-Preserving Embeddings Preserve Operator Norms and Compose, Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings, Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation, Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings, Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws and Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs.