Schatten norms on Hilbert C^*-modules via pure states

Abstract

Let (ℰ, ⟨ ·, ·⟩) be a Hilbert C^*-module over a C^*-algebra A. The space of adjointable operators on ℰ is denoted by ℒ(ℰ). The sets of all states and pure states on A are denoted by 𝒮(A) and 𝒫( A), respectively. For τ∈𝒮( A), let us define 𝒩^ℰ_τ:={ x∈ℰ:τ( ⟨ x,x⟩)=0 }. The Hilbert completion of ℰ/𝒩^ℰ_τ is denoted by ℰ_τ. For T∈ℒ(ℰ), the operator T_{ℰ_τ}∈𝔹( ℰ_τ), is defined by T_{ℰ_τ}(x+𝒩^ℰ_τ)=Tx+𝒩^ℰ_τ. In this paper, we show that ℰ_τ=ℰ/𝒩^ℰ_τ when A either is a C^*-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert C^*-modules, denoted by π^ℰ_k(·) for k≥1. We prove that π^ℰ_k(T)≤sup_{τ∈𝒫( A)}|T_{ℰ_τ}|_(k)≤π^{ℰ^♯}_k(T_ℰ^♯) for every T∈ℒ(ℰ), where the space ℰ^♯ is constructed as the extension of ℰ by the embedding of A into its enveloping von Neumann algebra A^**.

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BibTeX RIS

BibTeX

@misc{abedi2026schatten,
  title = {{Schatten norms on Hilbert $C^*$-modules via pure states}},
  author = {Sajjad Abedi and Mohammad Sal Moslehian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13944},
  eprint = {2609.13944},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13944},
}

RIS

TY  - UNPB
TI  - Schatten norms on Hilbert C^*-modules via pure states
AU  - Sajjad Abedi
AU  - Mohammad Sal Moslehian
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13944
UR  - https://arxiv.org/abs/2609.13944
AB  - Let (ℰ, ⟨ ·, ·⟩) be a Hilbert C^*-module over a C^*-algebra A. The space of adjointable operators on ℰ is denoted by ℒ(ℰ). The sets of all states and pure states on A are denoted by 𝒮(A) and 𝒫( A), respectively. For τ∈𝒮( A), let us define 𝒩^ℰ_τ:={ x∈ℰ:τ( ⟨ x,x⟩)=0 }. The Hilbert completion of ℰ/𝒩^ℰ_τ is denoted by ℰ_τ. For T∈ℒ(ℰ), the operator T_ℰ_τ∈𝔹( ℰ_τ), is defined by T_ℰ_τ(x+𝒩^ℰ_τ)=Tx+𝒩^ℰ_τ. In this paper, we show that ℰ_τ=ℰ/𝒩^ℰ_τ when A either is a C^*-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert C^*-modules, denoted by π^ℰ_k(·) for k≥1. We prove that π^ℰ_k(T)≤sup_τ∈𝒫( A)|T_ℰ_τ|_(k)≤π^ℰ^♯_k(T_ℰ^♯) for every T∈ℒ(ℰ), where the space ℰ^♯ is constructed as the extension of ℰ by the embedding of A into its enveloping von Neumann algebra A^**.
ER  -

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