关于C~*-代数中Bohr不等式的一个注记（英文）

A note on the Bohr inequalityin C*-algebras

利用C*-代数到B(H)中的等距*-表示,研究C*-代数中的Bohr不等式,得到了4个推广的Bohr不等式成立的一些充分必要条件1.主要结论如下:设p,q∈R^+,且满足1/p+1/q=1,则A,B∈S(S为有单位元的C*-代数),|A-B|~2+|(1-p)A-B|~2≤p|A|~2+q|B|~2成立当且仅当p≤2;设α,β,u,v∈R,p,q∈R^+,则|αA+βB|~2+|uA+vB|~2≤p|A|~2+q|B|~2成立当且仅当p≥α~2+u^2,q≥β~2+v^2且(p-(α~2+u^2))(q-(β~2+v^2))≥(αβ+uv)~2;设a,b∈R^+,c∈C,则A,B∈S,a|A|~2+b|B|~2+cA*B+cB*A≥0成立当且仅当ab≥|c|2;设α,β∈R,x,y是正数,则A,B∈S,|αA+βB|~2≤x|A|~2+y|B|~2成立,当且仅当x≥α~2,y≥β~2且(x-α~2)(y-β)≥α~2β~2.

By using an isometric*-representation from a C*-algebra into B(H),where H is a Hilbert space,the generalized Bohr inequalities in a C*-algebra were discussed.Some necessary and sufficient conditions for four generalized Bohr inequalities are obtained.The main results are as follows:Let p,q∈R+and 1/p+1/q=1,then for all A,B∈S(Sis a unital C*-algebras),|A-B|~2+|(1-p)A-B|~2≤p|A|~2+q|B|~2iff p≤2.Letα,β,u,v∈R+and p,q∈C+,then for all A,B∈S,|αA+2βB|~2+|uA+vB|≤p|A|~2+q|B|~2 iff p≥α~2+u^2,q≥β~2+v^2 and(p-(α~2+u^2))(q-(β~2+v^2))≥(αβ+uv)2.Let a,b∈R+and c∈C+,then for all A,B∈S,a|A|~2+b|B|~2+cA*B+cB*A≥0iff ab≥|c|~2.Letα,+β∈Rand x,ybe positive numbers,then for all A,B∈S,|αA+βB|~2≤x|A|~2+y|B|~2 iff x≥α~2,y≥ β~2and(x-α~2)(y-β~2)≥α~2β~2.