简介：Assumethateachcompletelyirrationalnoncommutativetorusisrealizedasaninductivelimitofcirclealgebras,andthatforacompletelyirrationalnoncommutativetorusAwofrankmthereareacompletelyirrationalnoncommutativetorusAρofrankmandapositiveintegerdsuchthattr(Aw)=1/d.tr(Aρ).ItisprovedthatthesetofallC^*-algebrasofsectionsoflocallytrivialC^*-algebrabundlesoverS^2withfibresAωhasagroupsturcture,denotedbyπ1^s(Aut(Aω)),whichisisomorphictoZifEd>1and{0}ifd>1.LetBcdbeacd-homogeneousC^*-algebraoverS^2×T^2ofwhichnonon-trivialmatrixalgebracanbefactoredout.ThesphericalnoncommutativetorusSρ^cdisdefinedbytwistingC^*(T2×Z^m-2)inBcd×C^*(Z^m-3)byatotallyskewmultiplierρonT^2×Z^m-2。ItisshownthatSρ^cd×Mρ∞isisomorphictoC(S^2)×C^*(T^2×Z^m-2,ρ）×Mcd(C)×Mρ∞ifandonlyifthesetofprimefactorsofcdisasubsetofthesetofprimefactorsofp.