Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product

1. (Data)¶ The conventions of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation are in force. $m_{1},m_{2},n\in\mathbb{N}$; the letter $\varepsilon$ ranges over $\{1,2\}$, and $\bar{\varepsilon}=3-\varepsilon$ is the other index. For each $\varepsilon$, $\gamma_{\varepsilon}\in\Sigma_{m_{\varepsilon}}$ and $a^{\varepsilon}$ is an $n$-tuple in $\mathcal{P}_{m_{\varepsilon},\mathrm{sa}}$, and the two laws have the common marginal

which belongs to $\Sigma_{n}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law.

2. (Algebras)¶ $N=\mathcal{M}_{\mu}$ and $A_{\varepsilon}=\mathcal{M}_{\gamma_{\varepsilon}}$ are the tracial algebras of $\mu$ and $\gamma_{\varepsilon}$, with traces $\tau_{\mu}$ and $\tau_{\gamma_{\varepsilon}}$; $\mathcal{H}_{\mu}$ and $\mathcal{H}_{\gamma_{\varepsilon}}$ are the complex GNS spaces on which they act, and $I$ is the identity map of whichever of these spaces is in play.

3. (Embeddings and expectations)¶ $\pi_{\varepsilon}:N\to A_{\varepsilon}$ and $E_{\varepsilon}:A_{\varepsilon}\to N$ are the embedding and the conditional expectation of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation for $\gamma=\gamma_{\varepsilon}$ and $a=a^{\varepsilon}$.

4. (Background)¶ The following results are in force and may be used without restating them: Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation, 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, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation, The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It, Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators, Square Roots of Positive Bounded Operators on a Complex Hilbert Space, Commuting with Everything that Commutes with the Operator and Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus.