POSITIVE FUNCTIONALS ON ALGEBRAS

Hsia Tao-shing

(Presented by Academician P. S. Aleksandrov, 11 III 1958)

Let \(A\) be a commutative algebra over the field of real numbers with identity element \(e\). Denote by \(\mathfrak M\) the space of all homomorphisms \(M\) of the algebra \(A\) into the field of real numbers satisfying the condition \(e(M)=1\) (i.e., the maximal ideals). We introduce a topology on the set \(\mathfrak M\) as the weak topology defined by the family of all functions \(x(M)\), \(x\in A\). An element \(x\in A\) is called non-negative if for every homomorphism \(M\in\mathfrak M\) the number \(x(M)\ge 0\). A linear functional \(F\) on the algebra \(A\) is called positive if \(F\ne 0\) and for every non-negative element \(x\in A\) we have \(F(x)\ge 0\).

If \(F\) is positive, then \(F(x^2)\ge 0\) for \(x\in A\), since \(x^2(M)=(x(M))^2\ge 0\) for \(M\in\mathfrak M\). But the converse is false. For example, if \(A\) is the algebra of all polynomials \(p(t,s)\) in two variables with real coefficients, then \(\mathfrak M\) will be the plane of the variables \((t,s)\). In this case, as R. B. Zarkhina showed \((^1)\), on \(A\) there exists a linear functional \(F\), not identically zero, satisfying the condition \(F(x^2)\ge 0\) for every \(x\in A\), but not positive.

In the present note we give a necessary and sufficient condition for the positivity of a functional on an algebra with a countable number of generators.

Theorem. Let \(F\) be a linear functional on an algebra with a countable number of generators. Then the following three conditions are equivalent:

1. The functional \(F\) is positive.

2. On the set \(\mathfrak M\) there exists a positive measure \(\mu(M)\) such that

\[ F(x)=\int_{\mathfrak M} x(M)\,d\mu(M). \]

1. For every real positive polynomial \(p(t_1,\ldots,t_n)\) of degree \(4\) and any \(x_1,\ldots,x_n\in A\) the inequality

\[ F\bigl(p(x_1,\ldots,x_n)\bigr)\ge 0 \]

holds.

We note that in the formulation of condition 3 of this theorem, positive polynomials of degree \(4\) cannot be replaced by positive polynomials of degree \(2\).

Proof. It is enough to prove that condition 3 implies condition 2. Let \(\{x_\alpha\}\) be a linear basis of the algebra \(A\), where \(\alpha\in I\), \(I\) being the set of all integers. Put \(x_0=e\) and \(F(e)=1\). To each \(\alpha>0\), \(\alpha\in I\), assign a one-dimensional space \(E_\alpha\). Let

\[ E=\prod_{\alpha>0} E_\alpha \]

be the product of the spaces \(E_\alpha\). We regard \(u\in E\) as a linear functional on \(A\), putting \(u(x_\alpha)=u_\alpha\), the \(\alpha\)-th coordinate of \(u\), and put \(u(e)=1\). Then \(\mathfrak M\subset E\).

Let \(C_4(E)\) be the space of all continuous functions \(\varphi(u)\) on \(E\) such that \(\varphi(u)\) depends only on a finite number of coordinates \(u_1,u_2,\ldots,u_n\) and there exists a constant \(c\) such that
\[ |\varphi(u)| \leq c\left(\sum_{\alpha=1}^{n} u_\alpha^4+1\right). \]

Let \(P_4(E)\) be the subspace of the space \(C_4(E)\) consisting of all polynomials \(p(u)\in C_4(E)\) in the \(u_\alpha\) of degree \(4\). Define a linear functional \(\widetilde F\) on \(P_4(E)\) by the formula
\[ \widetilde F\bigl(p(u_1,u_2,\ldots,u_n)\bigr) = F\bigl(p(x_1,x_2,\ldots,x_n)\bigr) \tag{1} \]
for \(p(u_1,u_2,\ldots,u_n)\in P_4(E)\). From condition 3 it follows that
\[ \widetilde F(p)\geq 0, \]
when \(p(u_1,u_2,\ldots,u_n)\geq 0\). From the lemma of M. Krein \((^2)\) it follows that \(\widetilde F\) can be extended to \(C_4(E)\) with preservation of positivity, i.e. \(\widetilde F(\varphi(u))\geq 0\) for \(\varphi(u)\geq 0\) and \(\varphi\in C_4(E)\).

Let \(C'_4(E)\) be the subspace of the space \(C_4(E)\) consisting of all such functions \(\varphi(u)\in C_4(E)\) that
\[ \lim_{|u_\nu|\to\infty} |\varphi(u)|\Big/\sum_{\nu=1}^{n} u_\nu^4=0 \]
for some \(n\). By Riesz’s theorem there exist positive measures
\(\mu(u_{\alpha_1},u_{\alpha_2},\ldots,u_{\alpha_n})\) such that
\[ \widetilde F(\varphi)=\int \varphi(u)\,d\mu(u_{\alpha_1},u_{\alpha_2},\ldots,u_{\alpha_n}), \]
when \(\varphi(u)\in C'_4(E)\) and \(\varphi(u)\) depends only on
\(u_{\alpha_1},\ldots,u_{\alpha_n}\). Note that
\[ \int d\mu(u_{\alpha_1},\ldots,u_{\alpha_n})=F(e)=1. \]
By A. N. Kolmogorov’s theorem \((^2)\), on \(E\) there exists a positive measure \(\mu(u)\) such that
\[ \widetilde F(\varphi)=\int_E \varphi(u)\,d\mu(u) \]
for \(\varphi\in C'_4(E)\). Therefore, if \(\varphi\in C_4(E)\) and \(\varphi(u)\geq 0\), then
\[ \widetilde F(\varphi)\geq \int_E \varphi(u)\,d\mu(u), \tag{2} \]
since \(\varphi(u)\) can be approximated by a monotone sequence of positive functions from \(C'_4(E)\).

For any \(\alpha,\alpha'\in I\) there exist real numbers \(a_{\alpha\alpha'}^{(k)}\) such that
\[ x_\alpha x_{\alpha'}=\sum_k a_{\alpha\alpha'}^{(k)}x_k, \]
where \(a_{\alpha\alpha'}^{(k)}=0\) for all but a finite number of \(k\).

Let \(E_{\alpha\alpha'}\) be the subspace of the space \(E\) consisting of those elements \(u\in E\) which satisfy the condition
\[ L_{\alpha,\alpha'}(u)\equiv u(x_\alpha)u(x_{\alpha'}) -\sum_k a_{\alpha\alpha'}^{(k)}u(x_k) \equiv u_\alpha u_{\alpha'}-\sum_k a_{\alpha\alpha'}^{(k)}u_k. \]
Then
\[ \mathfrak M=\bigcap_{\alpha,\alpha'\in I} E_{\alpha\alpha'}. \]
But from (2) it follows that
\[ 0= F\left(\left(x_\alpha x_{\alpha'}-\sum_k a_{\alpha\alpha'}^{(k)}x_k\right)^2\right) = \widetilde F\bigl(L_{\alpha\alpha'}(u)^2\bigr) \geq \int_E L_{\alpha\alpha'}(u)^2\,d\mu(u). \]

Therefore

\[ \int_{E-E_{\alpha\alpha'}} d\mu(u)=0 \]

and, consequently,

\[ \int_{E-\mathfrak M} d\mu(u)=0. \]

Thus it has been proved that

\[ \widetilde F(\varphi(u))=\int_{\mathfrak M}\varphi(M)\,d\mu(M) \tag{3} \]

for \(\varphi(u)\in C'_4(E)\). If \(x\in A\), then there exist \(a_\alpha\) such that \(x=\sum_{\alpha=1}^{n} a_\alpha x_\alpha\).

From (1) and (3) we obtain that

\[ F(x)=F\left(\sum a_\alpha x_\alpha\right) =\widetilde F\left(\sum a_\alpha u_\alpha\right) =\int_{\mathfrak M}\left(\sum a_\alpha u_\alpha\right)d\mu(u) = \]

\[ =\int_{\mathfrak M}\sum a_\alpha u(x_\alpha)\,d\mu(u) =\int_{\mathfrak M}u(x)\,d\mu(u). \]

Therefore \(F(x)\) satisfies condition (2).

In conclusion, I express my sincere gratitude to I. M. Gel'fand for valuable assistance.

Moscow State University
named after M. V. Lomonosov

Received
4 III 1958

REFERENCES

1. R. B. Zarkhina, On the two-dimensional moment problem, Dissertation, Moscow State University, 1953.
2. N. Akhiezer, M. Krein, On certain questions in the theory of moments, 1938.
3. A. N. Kolmogorov, Basic concepts of probability theory, Moscow–Leningrad, 1936.