{"paper":{"title":"Joint spectrum and infinite dihedral group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Rongwei Yang, Rostilav Grigorchuk","submitted_at":"2016-05-05T10:43:09Z","abstract_excerpt":"For a tuple $A=(A_1,\\ A_2,\\ ...,\\ A_n)$ of elements in a unital Banach algebra ${\\mathcal B}$, its {\\em projective joint spectrum} $P(A)$ is the collection of $z\\in {\\mathbb C}^n$ such that the multiparameter pencil $A(z)=z_1A_1+z_2A_2+\\cdots +z_nA_n$ is not invertible. If ${\\mathcal B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_1,\\ A_2,\\ ...,\\ A_n$ with respect to a representation $\\rho$, then $P(A)$ is an invariant of (weak) equivalence for $\\rho$. This paper computes the joint spectrum of $(1,\\ a,\\ t)$ for the infinite dihedral group $D_{\\infty}=<a,\\ t\\ |\\ a^2=t^2="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01547","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}