{"title":"Homotopy commutativity in Hermitian symmetric spaces","authors":"D. Kishimoto, Masahiro Takeda, Yichen Tong","doi":"10.1017/S0017089522000118","DOIUrl":"https://doi.org/10.1017/S0017089522000118","url":null,"abstract":"Abstract Ganea proved that the loop space of \u0000$mathbb{C} P^n$\u0000 is homotopy commutative if and only if \u0000$n=3$\u0000 . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but \u0000$mathbb{C} P^3$\u0000 are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds \u0000$G/T$\u0000 for a maximal torus T of a compact, connected Lie group G.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"746 - 752"},"PeriodicalIF":0.5,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44703460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"KMS states on \u0000$C_c^{*}(mathbb{N}^2)$","authors":"Anbu Arjunan, Sruthymurali, S. Sundar","doi":"10.1017/S0017089523000071","DOIUrl":"https://doi.org/10.1017/S0017089523000071","url":null,"abstract":"Abstract Let \u0000$C_c^{*}(mathbb{N}^{2})$\u0000 be the universal \u0000$C^{*}$\u0000 -algebra generated by a semigroup of isometries \u0000${v_{(m,n)},:, m,n in mathbb{N}}$\u0000 whose range projections commute. We analyse the structure of KMS states on \u0000$C_{c}^{*}(mathbb{N}^2)$\u0000 for the time evolution determined by a homomorphism \u0000$c,:,mathbb{Z}^{2} to mathbb{R}$\u0000 . In contrast to the reduced version \u0000$C_{red}^{*}(mathbb{N}^{2})$\u0000 , we show that the set of KMS states on \u0000$C_{c}^{*}(mathbb{N}^{2})$\u0000 has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"501 - 528"},"PeriodicalIF":0.5,"publicationDate":"2022-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44842606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}