Conjugations on Hilbert C*-modules
Abstract
This work initiates a systematic development of conjugation theory within the framework of Hilbert C*-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of *-conjugate-automorphisms ♯ on C*-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set 𝕃c(ℰ) of all adjointable conjugate-linear operators on a Hilbert C*-module ℰ possesses the structure of a Hilbert 𝕃(ℰ)-module. We also introduce the key notion of ♯-conjugation on a Hilbert C*-module and fully characterize such modules in terms of real forms and real C*-algebras. Moreover, we investigate the geometric properties of ♯-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert C*-module equipped with a ♯-conjugation C, thereby extending the celebrated Garcia--Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913--3931]. As a consequence, we show that any C-symmetric unitary operator factors as a product of two ♯-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition ⟨ Cx,y⟩ = ⟨ x,Cy⟩* would force the underlying C*-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.
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BibTeX
@misc{faregh2026conjugations,
title = {{Conjugations on Hilbert $C^*$-modules}},
author = {Mahboubeh Faregh and Rasoul Eskandari and Mohammad Sal Moslehian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13931},
eprint = {2609.13931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13931},
}
RIS
TY - UNPB
TI - Conjugations on Hilbert C^*-modules
AU - Mahboubeh Faregh
AU - Rasoul Eskandari
AU - Mohammad Sal Moslehian
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13931
UR - https://arxiv.org/abs/2609.13931
AB - This work initiates a systematic development of conjugation theory within the framework of Hilbert C^*-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of *-conjugate-automorphisms ♯ on C^*-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set 𝕃_c(ℰ) of all adjointable conjugate-linear operators on a Hilbert C^*-module ℰ possesses the structure of a Hilbert 𝕃(ℰ)-module. We also introduce the key notion of ♯-conjugation on a Hilbert C^*-module and fully characterize such modules in terms of real forms and real C^*-algebras. Moreover, we investigate the geometric properties of ♯-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert C^*-module equipped with a ♯-conjugation C, thereby extending the celebrated Garcia--Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913--3931]. As a consequence, we show that any C-symmetric unitary operator factors as a product of two ♯-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition ⟨ Cx,y⟩ = ⟨ x,Cy⟩^* would force the underlying C^*-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.
ER -