Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions

TL;DR

The paper investigates whether hereditarily essential $C^*$-inclusions admit a unique pseudo-expectation. It proves positive results for inclusions arising from discrete and twisted crossed products $\mathcal{A}\subseteq\mathcal{A}\rtimes_{\alpha,r}^{\sigma}G$, using stabilization and dynamical properties to obtain a unique, faithful pseudo-expectation; it also provides a counterexample showing non-uniqueness in general via $C_r^*(G)\subseteq C(X)\rtimes_{\alpha,r}G$, highlighting the role of regularity. The work thus clarifies the landscape: regular, hereditarily essential inclusions may exhibit uniqueness, while non-regular irreducible inclusions can fail to do so, motivating a reformulation in terms of equivalent conditions (hereditarily essential, aperiodic, and unique pseudo-expectation) for regular inclusions. The results have implications for understanding when injective envelopes and crossed-product structures yield rigid extension maps, with potential impacts on the study of dynamical $C^*$-algebras and their ideal structure. Open questions remain about the full equivalence of the three reformulated conditions in the regular setting and the exact boundaries of the positive results beyond crossed-product frameworks.

Abstract

The $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is said to be hereditarily essential if for every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$ and every non-zero ideal $\{0\} \neq \mathcal{J} \unlhd \mathcal{C}$, we have that $\mathcal{J} \cap \mathcal{A} \neq \{0\}$. That is, $\mathcal{A}$ detects ideals in every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$. By a result of Pitts and Zarikian, a unital $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is hereditarily essential if and only if every pseudo-expectation $θ:\mathcal{B} \to I(\mathcal{A})$ for $\mathcal{A} \subseteq \mathcal{B}$ is faithful. A decade-old open question asks whether hereditarily essential $C^*$-inclusions must have unique pseudo-expectations? In this note, we answer the question affirmatively for some important classes of $C^*$-inclusions, in particular those of the form $\mathcal{A} \subseteq \mathcal{A} \rtimes_{α,r}^σG$, for a twisted $C^*$-dynamical system $(\mathcal{A},G,α,σ)$. On the other hand, we settle the general question negatively by exhibiting $C^*$-irreducible inclusions of the form $C_r^*(G) \subseteq C(X) \rtimes_{α,r} G$ with multiple conditional expectations. Our results leave open the possibility that the question might have a positive answer for regular hereditarily essential $C^*$-inclusions.

Theorems & Definitions (21)

• Theorem 1.3.1: PittsZarikian2015
• proof
• Theorem : main positive result
• Theorem : main negative result
• Lemma 2.1.1
• proof
• Theorem 2.1.2
• proof
• Example 2.1.3
• proof