AN ALGEBRAIC CHARACTERIZATION OF COMPLETE INNER PRODUCT SPACES

We present a characterization of complete inner product spaces using en involution on the set of all bounded linear operators on a Banach space. As a metric conditions we impose a multiplicative property of the norm for hermitain operators. In the second part we present a simpler proof (we believe) of the Kakutani and Mackney theorem on the characterizations of 
complete inner product spaces. Our proof was suggested by an ingenious proof of a similar result obtained by N. Prijatelj.

INTRODUCTION.Let X be a complex Banach space and L (X) be the set of all bounded linear operators on X.There are conditions about [(X) (or parts of i(X)) which force X to be an inner product space,i.e,these conditions imply the exist- ence of an inner product <, > on X such that for all x in , IIll -< , > where li I I denotes the norm of the Banach space .
One of these sets of conditions was formulated by S.Kakutani and G.W. Mackey  (1944,1946) and reads as follows:suDpose that on i(X) we have a mapping * whose values are in i(X) such that the following relations hold: (T+S)* =T*+ S* 2.
T*)*= T, 3  Further we present a new proof of the Kakutani-Mackey theorem which we believe to be simpler than the original one.Our proof is inspired from the proof in the Prijatelj's paper.
I. AN ALGEBRAIC CHARACTERIZATION OF COMPLETE INNER PRODUCT SPACES.
Let X be a complex Banach space and i(X) be the set of all bounded linear operators on X.
Theorem 1.1.Suppose that on [(X) there exists a mapping '' with values in i(X)such that the following conditions are satisfied: I. (T+S) = T*+ S , 2. (T) = T, 3. (TS)= ST , p2 4 if P is a hermitian rojection (i e =P =P) then Then on X there exists an inner product space <,> with the property that llxll ---< x,x >o Proof.Let us consider a,b be two arbitrary linearly independent elements in which generate a (closed)subspace of X,which we denote by X 2. We consider an element f in X which has the followin pronerties: Using f we deine the ollowing operator by the formula: Pl x= f (x) / llall.a.We choose g in with the properties: 2. IIiI :.
We consider the following operator on X: This is a hermitain projection and we have the relations: PIP2=P2 p1=O which implies that the operator P defined by the formula P =PI +P2 is a hermitian projection on X.From our condition we get that the norm of P is 1.Then the Kakutani-Bohnenblust theorem implies that on the space X there exists an inner product satisfying the theorem.
Remark 1.2 Our conditions in the theorem 1.1 are much like to those of Kakutani-Mackey and Prijatelj; the conditions I-3 are identical while the condition 4 requires in fact that the Prijatelj's condition must be satisfied for hermitian operators which are projections.

A PROOF OF THE KAKUTANI-MACKEY'S THEOREM.
In what follows we present a proof of the Kakutani-Mackey's theorem using some ideas from the proof of N.Prijatelj's elegant paper.In order to make the Note selfcontained we give here more details.
Let us consider an element a # o of X and f in X with the following properties 2. f(a)=a We consider on i(X) the following functional g defined by the formula: and it is clear that this is a positive functional (i.e.g(I)=1= ).
Using this functional we define the following bilinear fo on (we remark that the positivity of g implies that g(T)=(g(T)), <T,S> =g (S" T) which has all the properties of an inner product except the fact that <T,T> =O does not necessarily implies that T=O.Suppose now that the algebra has the following additional property: for any u,v in ,}u =v there exists a unitary operator U (i.e.U'U =UU'=I) such that Uu=v.

V. I. ISTRATESCU
We consider now an operator defined on i(X)with values in which is defined as follows: W(T)=Ta.
Then it is clear that IIWII =llall and if N(W) is the null space of W then N(W) is a left ideal in i(XI.Another left ideal in [(X) is defined as follows: R, R s [(X),<R,R> =o =N.
Then clear we have the inclusion N(W) cN and we show that we have in fact an equality.Now since we suppose that i(X) has the additional property with respect to unitary operators,we consider an arbitrary element x of X,x # o .Then we find an unitary operator U with the property that Since llxll u is in [(X) we get that w(llxll u) -x.Now in order to prove the above equality we must prove the inclusion in the converse sense and for this it suffices to Drove that CN (W) c C N Suppose that T is in CN(W) and then W(T) Ta =x o.We find a unitary operator U in [(X) with the property that Further,since the properties 2 and 3 holds taking into account that the identity operator is hermitian and that the operator P1x=f(x)el is hermitian and a projection as well as the fact that for any operator T,TT is hermitian,we get that the coefficients satisfy the following (where the bar denotes the conjugate).Now we use the fact that for unitary operators U the relations must agrre again and thus the followinq relations must be hlod: a1111 +r a2121 =I =a 1111 +r-1 a12a12 a1211 +r a2221 ==r a1121 +a12 22' Now if x is an arbitrary element of X 2 (of norm I) we must show that for some unitary operator U,U e =x.From the fact that the norm of x is we conclude that a=a a2=(1-a 2) I/2oThUs the matrix of the unitary operator is A e (I A 2) I/2 i (e+s) e (I A 2)I/2 i(8-t) i(+t-s) e A e / Thus the additional property holds for our algebra and this completes the proof of the theorem.
(TS)*=S* T* 4. T*T =O then T =O.N.Pri3atelj (1964) has considered the followinq set of conditions: on [() there exists a map * whose values are in [() and such that the followinq relations hold: TS) = ST ,In what follows we present a new set of conditions on [(XI which imply the existence of an inner product on X <, > satisfying the relation and which simplifies to some extent the conditions imposed by N.Prijatelj. It is not difficult to see that this operator is a hermitian proje- ction.Now since b=P1b+ (I-PI) b we get that (l-P1)b #o (since a,b are supposed linearly independent).
ilxll U) a =o.This implies that (T-llxll U) N(W) = N Now if T,S are in [(X) and T-S is in N then g (TT) =g (SS) Further this means that Ig(TT)-g (ss) I=Ig(T(T-S))+g((T-S) S) =<Ig(T'(S-S))l+Ig((T-S)'S) and applying the Cauchy-Buniakowsky's inequality (for positive functionals) we get that g (T (T-S)) =o g((T-S) *s)=o.Since (T-IIxll u) is in N we have that V. I. ISTRATESCU We must prove that there exists a unitary operator U with the property that Ue1=x.Now ,every operator on X 2 renresented by a 2x2 matrix / a11 a]2\ \ a21 a22] and the adjoint by the matrix Since the property holds the elements a.. are linear combi- nations of the conjugate ,i.e., aij { b kj" are neal numbers (parameters)Thus the matrix representing a unitary operator has the form: