In 1888 D. HILBERT showed the existence of a real polynomial in two variables of sixth degree which is non-negative on the real plane without being a finite sum of squares of polynomials ([2]; see e.g. [l]). The existence of such polynomials has important consequences in the theory of unbounded operator algehras (see for example [4]) and for the n-dimensional problem of moments because i t implies two different notions of positivity of linear functionals. A linear functional B on the free commutative polynom algebra can be represented by a positive BOREL measure if and only if it is strongly positive ( [ 5 ] ) , i.e. P ( p ) z O for all polynomials p which are non-negative for real variables.
Up to the present time the author has not found an explicitly given polynomial of this kind in the appropriate literature. Furthermore, for the construction of HILBERT'S example one must have certain results from the theory of algebraic curves (for instance the theorem of BEZOUT and CRAMER'S paradoxon). To prove the existence of a positive, not strongly positive linear functional one has to prove the stronger assertion that this polynomial is not in the closure of the cone of all finite sums of squares of polynomials in the strongest locally convex topology. It was noted by GELPAND and WILENKIN (see the remark in [l], p. 218) that this can he done.
The aim of the present paper is to give an explicit example of such a polynomial and to prove these properties by an elementary computation without applying the theory of algebraic curves. Using this polynomial we construct a positive, but not strongly positive linear functional on the free polynom algebra of two hermitian generators. Notice that each positive polynomial can be represented as a finite sum of squares of rational functions according to another theorem of
HILBERT ([3]).
By e ( x , , . . . , 2,) we denote the *-algebra of all polynomials p ( z , , . . . , zn) with complex coefficients in n commuting hermitian indeterminants xl, . . . , xn.
Let P(e) be the cone of all finite sums of squares of polynomials, i.e. P ( e ) -{ C p i ( z l , * * * 9 x,)+ pi(~,v * * 9 ~n ) , pii~e(z1, * * * , xn)).
finite 25 Math. Nachr. Bd. 88