Covering entropy for types in tracial \(\mathrm{W}^\)-algebras

• Published in: arXiv.org 2023-04
• Main Author: Jekel, David
• Format: Article
• Language: English
• Subjects: Algebra; Entropy
• ISSN: 2331-8422

We study embeddings of tracial \(\mathrm{W}^*\)-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined \(1\)-bounded entropy through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples \((X_1^{(N)},X_2^{(N)},\dots)\) having approximately the same \(*\)-moments as the generators \((X_1,X_2,\dots)\) of a given tracial \(\mathrm{W}^*\)-algebra. We study the analogous covering entropy for microstate spaces defined through formulas that use not only \(*\)-algebra operations and the trace, but also suprema and infima, such as arise in the model theory of tracial \(\mathrm{W}^*\)-algebras initiated by Farah, Hart, and Sherman. By relating the new theory with the original \(1\)-bounded entropy, we show that if \(h(\mathcal{N}:\mathcal{M}) \geq 0\), then there exists an embedding of \(\mathcal{M}\) into a matrix ultraproduct \(\mathcal{Q} = \prod_{n \to \mathcal{U}} M_n(\mathbb{C})\) such that \(h(\mathcal{N}:\mathcal{Q})\) is arbitrarily close to \(h(\mathcal{N}:\mathcal{M})\). We deduce if all embeddings of \(\mathcal{M}\) into \(\mathcal{Q}\) are automorphically equivalent, then \(\mathcal{M}\) is strongly \(1\)-bounded and in fact has \(h(\mathcal{M}) \leq 0\).