Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915107803496448 |
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| author | Zarikian, Vrej |
| author_facet | Zarikian, Vrej |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19484 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions Zarikian, Vrej Operator Algebras 46L07, 46L55 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. |
| title | Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions |
| topic | Operator Algebras 46L07, 46L55 |
| url | https://arxiv.org/abs/2406.19484 |