Operator Valued Positive Definite Functions On Unimodular Locally Compact Groups

Positive definite function on a unimodular locally compact group G is important notion of abstract harmonic analysis.It has been widely used in the group representation and functinal analysis.In 1948,Godement first showed continuous square integrable positive definite function f on a unimodular locally compact group G has square in-tegrable positive definite square root g,that is f= g*g,and among others,the inner product(g,g)0.In 2011,H.Y.HE[3]introduced a definition of locally integrable Matrix-valued pos-itive definitive function on a unimodular locally compact group G,and generalized two import,ant results of Godemcnt on square integrable positive definitive functions.Ho showed that a imatrix-valued cointirnuous square integrable positive definitive function can always be written as the convolution of an matrix-valued square integrable posi-tive definitive function with itself.Meanwhile,given two square integrable matrix-valued positive definitive functions f and g,the inner<f,g)=?GTrf(x)g(x)td?(x)? 0,in addition,<f,g)= Oif and only if f*9 0.These conclusions have very important application in the representation theory.In this paper,in the above context,we introduced a definition of locally inte-grable trace class operator-valued positive definitive function on a unimodular locally compact group G.We denote the set of the functions by P(G,T(H)),where,the set of trace 'class operators denoted by T(H).Meanwhile,we introduced two notions of trace class operator-valued bounded continuous functions in the ?.?Hs(H)topology and Hilbert-Schmidt operator-valued square integrable functions on a locally compact group G,where,the set of Hilbert-Schmidt operators denoted by HS(H).We denote the set of above two functions by Cb(G,T(H))and L2(G,HS(H)respectively.Then,we define P2(G)as the se.t of both T(H)-valued bounded continuous positive definite functions in the ?.?HS(H)topology and HS(H)-valued square integrable functions on G,meanwhile,P2(G)is defined as the set of T(H)-valued positive definite functions and HS(H)-valued square integrable functions.Furtherly,we generalized three results of H.Y.HE,that is,given two elements ?,?in P2(G)on a unimodular locally compact group G,there exists an element ? in P2(G)such that? = ?*?.Meanwhile,given two elements ?,? inP2(G),the inner...