Matrix amplification thresholds for real-rank-zero inclusions

Following Bell’s construction of a real-rank-zero inclusion whose 2×2 amplification does not have real rank zero, we obtain arbitrarily long finite initial segments of matrix amplifications having real rank zero. For every integer m ≥ 1, there are a separable unital C*-algebra Bm and a unital embedding ιm: C(D2m+1) → Bm such that ιm(k) has real rank zero exactly for k < 2m. The threshold comes from the least matrix size in which a projection on S2m can represent a nonzero reduced K-theory class. A primitive class occurs in size 2m, and gives a norm-one selfadjoint whose image has distance exactly one from the invertible selfadjoints. We give the higher-dimensional adaptation of Bell’s argument and retain all positive levels in a single separable codomain. GitHub repository: eulogica/matrix-amplification-rr0-inclusions. Use of artificial intelligence. ChatGPT by OpenAI, OpenAI Codex, and Claude by Anthropic were used during the preparation and revision of this manuscript for mathematical exploration, proof development, source retrieval, editorial and structural revision, and to assist in checking proofs, notation, terminology, internal consistency, cross-references, and bibliographic citations.