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B(H) is not a twisted groupoid C*-algebra
Alcides Buss, Luiz Felipe Garcia, Tom\'as Pacheco
https://arxiv.org/abs/2603.21946 https://arxiv.org/pdf/2603.21946 https://arxiv.org/html/2603.21946
arXiv:2603.21946v1 Announce Type: new
Abstract: We show that $B(H)$ for an infinite dimensional Hilbert space $H$ cannot be realized as the reduced twisted $C^*$-algebra of any locally compact Hausdorff \'etale groupoid.
The proof is based on the canonical conditional expectation $$C_r^*(G,\Sigma)\to C_0(G^{(0)})$$ and a structural analysis of the resulting diagonal subalgebra inside $B(H)$. We show that this diagonal must be an atomic abelian von Neumann algebra, and then exclude both possibilities for its spectrum.
If the unit space is finite, one obtains a tracial state on $C_r^*(G,\Sigma)$, which is impossible for $B(H)$. If it is infinite, the groupoid structure forces a block-sparsity phenomenon for compactly supported sections, which is incompatible with $B(H)$.
This provides the first examples of $C^*$-algebras that cannot be realized as reduced twisted \'etale groupoid $C^*$-algebras.
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The Self-Replication Phase Diagram: Mapping Where Life Becomes Possible in Cellular Automata Rule Space
Don Yin
https://arxiv.org/abs/2603.25239 https://arxiv.org/pdf/2603.25239 https://arxiv.org/html/2603.25239
arXiv:2603.25239v1 Announce Type: new
Abstract: What substrate features allow life? We exhaustively classify all 262,144 outer-totalistic binary cellular automata rules with Moore neighbourhood for self-replication and produce phase diagrams in the $(\lambda, F)$ plane, where $\lambda$ is Langton's rule density and $F$ is a background-stability parameter. Of these rules, 20,152 (7.69%) support pattern proliferation, concentrated at low rule density ($\lambda \approx 0.15$--$0.25$) and low-to-moderate background stability ($F \approx 0.2$--$0.3$), in the weakly supercritical regime (Derrida coefficient $\mu = 1.81$ for replicators vs. $1.39$ for non-replicators). Self-replicating rules are more approximately mass-conserving (mass-balance 0.21 vs. 0.34), and this generalises to $k{=}3$ Moore rules. A three-tier detection hierarchy (pattern proliferation, extended-length confirmation, and causal perturbation) yields an estimated 1.56% causal self-replication rate. Self-replication rate increases monotonically with neighbourhood size under equalised detection: von Neumann 4.79%, Moore 7.69%, extended Moore 16.69%. These results identify background stability and approximate mass conservation as the primary axes of the self-replication phase boundary.
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