The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra
Abstract
We show how a simple metric condition in terms of the C*-norm of an arbitrary C*-algebra A can be employed to determine those norm-one positive elements which are projections. Concretely, if we write SA+ for the intersection of the positive cone and the unit sphere of A, we establish that an element a ∈ SA+ is a projection if, and only if, {a} = {x∈ SA+: |x−z| = 1 for all z∈ SA+ with |a−z|=1 }. This proves a conjecture which has been previously confirmed for von Neumann algebras and unital C*-algebras. In the setting of not necessarily unital JB*-algebras, the Shirshov--Cohn theorem is invoked to obtain a similar metric characterisation of projections. In the wider setting of JB*-triples, we show how a metric characterisation in terms of inner ideals can be established to determine non-zero tripotents.
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BibTeX
@misc{peralta2026norm,
title = {{The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra}},
author = {Antonio M. Peralta and Pedro Saavedra},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35290},
eprint = {2610.35290},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35290},
}
RIS
TY - UNPB
TI - The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra
AU - Antonio M. Peralta
AU - Pedro Saavedra
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35290
UR - https://arxiv.org/abs/2610.35290
AB - We show how a simple metric condition in terms of the C^*-norm of an arbitrary C^*-algebra A can be employed to determine those norm-one positive elements which are projections. Concretely, if we write S_A^+ for the intersection of the positive cone and the unit sphere of A, we establish that an element a ∈ S_A^+ is a projection if, and only if, {a} = {x∈ S_A^+: |x−z| = 1 for all z∈ S_A^+ with |a−z|=1 }. This proves a conjecture which has been previously confirmed for von Neumann algebras and unital C^*-algebras. In the setting of not necessarily unital JB^*-algebras, the Shirshov--Cohn theorem is invoked to obtain a similar metric characterisation of projections. In the wider setting of JB^*-triples, we show how a metric characterisation in terms of inner ideals can be established to determine non-zero tripotents.
ER -