Operator Jensen＇s Inequality on C＊-algebras.pdf

ActaM athematicaSinica，EnglishSeries Jan．，2014，Vo1．30，No．1，PP．35 50 ActaMathematicaSinica， Published online：December15，2013 EnglishSeries DOI：101007／s10114—013—2065—8 @ Springer-VerlagBerlinHeidelberg＆ Http：／／www．ActaMath．com TheEditoria1OfficcofAMS 2014 OperatorJensen’SInequality on C 一algebras Xin LI W eliW U DepartmentofMathematics，EastChinaNormalUniversity，Shanghai200241，P．R．China E-mail：tsoete@163．corn wwu@math．ecnu．edu．cn Abstract A realcontinuousfunction which iSdefined onan intervalissaid tobe92．convex ifitiS convexonthesetofsel~adjointelements，withspectraintheinterval，inallmatrixalgebrasofthe unitalC 一algebra ．W egiveageneralformationofJensen’Sinequalityfor 一convexfunctions． K eywords Jensen’Sinequality,C algebra， 一convexfunction，Bochnerintegral，unitalQ一【field MR(2010)SubjectClassification 46L05，46L07，47A63 1 lIltroduction ClassicJensen’Sinequalitysaysthatif isaconvexfunctionfrom (a，b)to ，then (∑ )∑ (z)， whenever 。 0fori=1，2，．．．，mwith∑mi：1。=1andXi∈(a，b)ofralli(see[11，12】)． Let(X，E，)beameasurespacewith ()=1．Iff：X _卜÷ isanintegrablefunctionwith f(X) (Ol，)，and西isaconvexfunctionfrom (，)to ，then ( ) c ． ThisisthecontinuousversionofJensen’Sinequality． VariousextensionsofJensen’Sinequalitytomoregeneralcontexthavebeenconsideredofr anumberofdecades(see[1，3，4，6，8—10，13，14】)．Iffisarealcontinuousfunctiononsonic intervalIin andXisaboundedself-adjointoperatoronaHilbertspacewithspectrum inI， wecanusespectraltheorytodefineaboundedself-adjointoperator ．厂()= ／ f(A)dEx(入)， whereExisthespectralmeasureofz．A continuousfunction ：IH issaidtobeoperator convexif O(ax+(1一 ))≤aO(x)+(1一