{"id":3319,"date":"2023-11-12T20:59:11","date_gmt":"2023-11-13T04:59:11","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=3319"},"modified":"2023-11-12T20:59:11","modified_gmt":"2023-11-13T04:59:11","slug":"continued-fractions-directed-graphs-and-defining-spectral-triples-on-effros-shen-af-algebras-samantha-brooker-arizona-state-university","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/continued-fractions-directed-graphs-and-defining-spectral-triples-on-effros-shen-af-algebras-samantha-brooker-arizona-state-university\/","title":{"rendered":"Continued fractions, directed graphs, and defining spectral triples on Effros-Shen AF algebras (Samantha Brooker, Arizona State University)"},"content":{"rendered":"<p>The Effros-Shen algebra corresponding to an irrational number $\\theta$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $\\theta$ encodes the dimensions of the summands, and how the matrix algebras at the nth level fit into the summands at the (n+1)th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the C*-algebra of a category of paths \u2013 a generalization of a directed graph \u2013 determined by the continued fraction expansion of \\theta. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. Drawing on a construction by Christensen and Ivan, we use this inductive limit structure to define a spectral triple, trading the advantages of working with finite-dimensional approximants for the techniques provided by the category of paths, pursuant to studying the algebras as quantum compact metric spaces. I will discuss categories of paths and their precursors, graph C*-algebras, the example of Mitscher and Spielberg, and a bit about the spectral triple construction. This is joint work with Konrad Aguilar and Jack Spielberg.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Effros-Shen algebra corresponding to an irrational number $\\theta$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $\\theta$ encodes [&hellip;]<\/p>\n","protected":false},"author":195,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[14],"class_list":["post-3319","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-analysis-seminar","cat_analysis-seminar"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - 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