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Linear maps preserving Kasparov cycles and the characterization of induced automorphisms

  • Kamran Sharifi
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Abstract

Let E be a Hilbert C*-module and ℒ(E) the C*-algebra of all bounded adjointable operators on E. An operator T ∈ ℒ(E) is a Kasparov cycle if T*T − 1 and TT*−1 are compact operators on E. If ℒ(E)→ ℒ(E) is a prime C*-algebra, and φ:ℒ(E)→ ℒ(E) is a linear map which is unital and surjective up to compact operators, and preserves Kasparov cycles in both directions, then the induced map ψ:ℒ(E)/𝒦(E) → ℒ(E)/𝒦(E) is either a *-automorphism or a *-anti-automorphism. Our work extends the main result of [J. Math. Anal. Appl. 354 (2009), 625-629] to the set of Kasparov cycles, showing that the assumption ``ℒ(E)/𝒦(E) has real rank zero'' is redundant in our results.

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Cite

BibTeXRIS
BibTeX
@misc{sharifi2026linear,
  title = {{Linear maps preserving Kasparov cycles and the characterization of induced automorphisms}},
  author = {Kamran Sharifi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18619},
  eprint = {2609.18619},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18619},
}
RIS
TY  - UNPB
TI  - Linear maps preserving Kasparov cycles and the characterization of induced automorphisms
AU  - Kamran Sharifi
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18619
UR  - https://arxiv.org/abs/2609.18619
AB  - Let E be a Hilbert C*-module and ℒ(E) the C*-algebra of all bounded adjointable operators on E. An operator T ∈ ℒ(E) is a Kasparov cycle if T^*T − 1 and TT^*−1 are compact operators on E. If ℒ(E)→ ℒ(E) is a prime C*-algebra, and φ:ℒ(E)→ ℒ(E) is a linear map which is unital and surjective up to compact operators, and preserves Kasparov cycles in both directions, then the induced map ψ:ℒ(E)/𝒦(E) → ℒ(E)/𝒦(E) is either a *-automorphism or a *-anti-automorphism. Our work extends the main result of [J. Math. Anal. Appl. 354 (2009), 625-629] to the set of Kasparov cycles, showing that the assumption ``ℒ(E)/𝒦(E) has real rank zero'' is redundant in our results.
ER  -