Zero-divisor graphs and zero-divisor real rank of C*-algebras
Abstract
We study two invariants of C*-algebras built from positive zero divisors. First, we associate with a C*-algebra A the graph Γ+(A) whose vertices are the nonzero positive zero divisors of A, two vertices being adjacent when they are orthogonal. We show that Γ+ of the Calkin algebra is connected of diameter $3$, that Γ+(B(H)) is connected of diameter $4$ (for dim H≥3), that Γ+(A) is connected of diameter at most $4$ for every von Neumann factor A≠ℂ,M2(ℂ), and of diameter at most $3$ if A is not of type~I, whereas Γ+(M2(ℂ)) is disconnected. Second, we study the zero-divisor real rank ZDRR(A), obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that ZDRR(A)≤ RR(A). For commutative algebras C(X) we prove ZDRR(C(X))=RR(C(X))=dim X when X is a compact almost-P-space, while ZDRR(C(X))=0 for separable X. Using Levy's connected, totally ordered almost-P-space we construct unital commutative C*-algebras An with ZDRR(An)=n for every n∈ℕ0. On the other hand ZDRR(C(X×[0,1]))=0 for every compact Hausdorff space X, so ZDRR and RR can differ arbitrarily.
Related
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BibTeX
@misc{taghavi2026zerodivisor,
title = {{Zero-divisor graphs and zero-divisor real rank of $C^*$-algebras}},
author = {Ali Taghavi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15160},
eprint = {2610.15160},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15160},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zero-divisor graphs and zero-divisor real rank of C^*-algebras
AU - Ali Taghavi
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15160
UR - https://arxiv.org/abs/2610.15160
AB - We study two invariants of C^*-algebras built from positive zero divisors. First, we associate with a C^*-algebra A the graph Γ_+(A) whose vertices are the nonzero positive zero divisors of A, two vertices being adjacent when they are orthogonal. We show that Γ_+ of the Calkin algebra is connected of diameter $3$, that Γ_+(B(H)) is connected of diameter $4$ (for dim H≥3), that Γ_+(A) is connected of diameter at most $4$ for every von Neumann factor A≠ℂ,M_2(ℂ), and of diameter at most $3$ if A is not of type~I, whereas Γ_+(M_2(ℂ)) is disconnected. Second, we study the zero-divisor real rank ZDRR(A), obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that ZDRR(A)≤ RR(A). For commutative algebras C(X) we prove ZDRR(C(X))=RR(C(X))=dim X when X is a compact almost-P-space, while ZDRR(C(X))=0 for separable X. Using Levy's connected, totally ordered almost-P-space we construct unital commutative C^*-algebras A_n with ZDRR(A_n)=n for every n∈ℕ_0. On the other hand ZDRR(C(X×[0,1]))=0 for every compact Hausdorff space X, so ZDRR and RR can differ arbitrarily.
ER -