On the set of orthogonal positive sums in a C^*-algebra

Abstract

For a C^*-algebra A let ℐ(A) be the set of all elements a+b with a,b∈ A₊, ab=0 and |a|=|b|=1. We study the algebraic, metric and topological properties of ℐ(A). We describe ℐ(A⊕ B) and show that ℐ(A/J)⊆π(ℐ(A)) for the quotient map π:A→ A/J. For finite-dimensional A we prove ℐ(A)=Sph⁺_A(E(A)), where E(A) is the set of minimal projections of A and Sph⁺_A(E) is the set of positive norm-one elements at distance one from every element of E; we also show that the analogous statement with the minimal projections of the bidual fails in general. When ℐ(A)≠∅ the distance from $1$ to ℐ(A) is $0$ or $1$, according to whether A has a nontrivial projection. Moreover, ℐ(A) is norm compact if and only if A is finite-dimensional, closedness of ℐ(A) passes to hereditary subalgebras, ℐ(A) is norm closed for every von Neumann algebra, and the extreme points of ℐ(A) are exactly its projections. For a compact, locally path connected space X the set ℐ(C(X)) is closed, whereas ℐ(ℓ_c(A)) fails to be closed when A is unital without nontrivial projections.

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BibTeX RIS

BibTeX

@misc{taghavi2026orthogonal,
  title = {{On the set of orthogonal positive sums in a $C^*$-algebra}},
  author = {Ali Taghavi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15167},
  eprint = {2610.15167},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15167},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - On the set of orthogonal positive sums in a C^*-algebra
AU  - Ali Taghavi
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15167
UR  - https://arxiv.org/abs/2610.15167
AB  - For a C^*-algebra A let ℐ(A) be the set of all elements a+b with a,b∈ A_+, ab=0 and |a|=|b|=1. We study the algebraic, metric and topological properties of ℐ(A). We describe ℐ(A⊕ B) and show that ℐ(A/J)⊆π(ℐ(A)) for the quotient map π:A→ A/J. For finite-dimensional A we prove ℐ(A)=Sph^+_A(E(A)), where E(A) is the set of minimal projections of A and Sph^+_A(E) is the set of positive norm-one elements at distance one from every element of E; we also show that the analogous statement with the minimal projections of the bidual fails in general. When ℐ(A)≠∅ the distance from $1$ to ℐ(A) is $0$ or $1$, according to whether A has a nontrivial projection. Moreover, ℐ(A) is norm compact if and only if A is finite-dimensional, closedness of ℐ(A) passes to hereditary subalgebras, ℐ(A) is norm closed for every von Neumann algebra, and the extreme points of ℐ(A) are exactly its projections. For a compact, locally path connected space X the set ℐ(C(X)) is closed, whereas ℐ(ℓ_c(A)) fails to be closed when A is unital without nontrivial projections.
ER  -

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