MATHEMATICS

K. DE LEEUW, H. MIRKIL (K. DE LEEUW, H. MIRKIL)

ANALYSIS AND SYNTHESIS OF RINGS OF TYPE \(C\)

(Presented by Academician P. S. Aleksandrov on 4 X 1962)

Let \(X\) be a compact group, and let \(R\) be a complete normed ring of continuous real-valued functions on \(X\) (with pointwise multiplication). The ring \(R\) is assumed to be “homogeneous,” i.e., together with each function \(f\) it contains all its translates \(T^x f\), and the mapping \(x \to T^x\) is a continuous representation of the group \(X\) on \(R\). It is also assumed that the ring \(R\) is “regular,” i.e., for any \(x, y \in X\), \(x \ne y\), there exists a function \(f \in R\) that is equal to zero in a neighborhood of \(x\) and equal to 1 at the point \(y\).

If \(\|f\|\) is the norm given on \(R\), then the growth norm at the point \(x\) will be denoted by \(\|f\|^x = \inf\{\|g\| : g = f \text{ in a neighborhood of } x\}\). Among the closed primary ideals contained in the maximal ideal \(M^x = \{f : f(x)=0\}\), the ideal \(I^x = \{f : \|f\|^x=0\}\) is the smallest; it is called the minimal ideal at the point \(x\).

One says that \(R\) is a ring of type \(C\) if the norm \(\|f\|\) is equivalent to the norm \(\sup_x \|f\|^x\). An equivalent formulation is the following.

In a regular homogeneous ring \(R\) put \(L^x = R/I^x\). The ring \(L^x\) is local, with the unique maximal ideal \(N^x = M^x/I^x\), and all the rings \(L^x\) are canonically isomorphic to the ring \(L^0\) (this isomorphism is effected by means of translation). Let \(C(X,L^0)\) be the ring of continuous \(L^0\)-valued functions on \(X\); for each function \(f \in R\) define the function \(f^b \in C(X,L^0)\) by setting \(f^b(x)=f/I^x=T^x f/I^0\) (where, for example, by \(f/I^x\) is denoted the image of \(f\) in \(R/I^x\)). Then \(R\) is a ring of type \(C\) if and only if the mapping \(f \to f^b\) takes \(R\) onto a closed subring of the ring \(C(X,L^0)\).

Rings of type \(C\) were first considered by G. E. Shilov. He established the following basic result on analysis of type \(C\) (see \((^1)\); Shilov’s proof is given for the case of a commutative group \(X\), but his method carries over without difficulty to the case of an arbitrary compact group \(X\)).

Theorem. Let \(R\) be a regular ring of type \(C\), and let \(P\) be the ring of trigonometric polynomials (i.e., such functions \(p\) on \(X\) that only a finite number of translates \(T^x p\) are linearly independent). Then \(R\) contains \(P\) and is isomorphic to the closure in \(C(X,L^0)\) of the ring \(P^b\).

The synthesis of type \(C\) is the process inverse to the one described above. The starting point in it is the specification of a local ring \(L\) (normed, complete) with a unique maximal ideal \(N\) and of such an isomorphism \(b\) of the ring \(P\) into \(C(X,L)\) that \(f^b(x)=(T^x f)^b(0)\). Denoting by \(S\) the closure of \(P^b\) in \(C(X,L)\), suppose that every element of the ring \(L\) occurs as the value \(F(0)\) at zero of some function \(F \in S\). Denote by \(\natural\) the natural homomorphism of the ring \(C(X,L)\) onto \(C(X)\), defined by factorization in the ring \(L\) of the values of functions modulo the ideal \(N\); suppose that \(f^{b\natural}=f\) for all polynomials \(f \in P\). The ring \(S\) of \(L\)-valued functions and the corresponding ring \(S^\natural\) of real-valued functions thus turn out to be synthesized from the ring \(L\) and the isomor-

homomorphism \(b\). But in order to understand this construction properly, we must answer several important questions.

Question 1. Is the mapping \(S \to S^{\natural}\) one-to-one?

Question 2. Is the ring \(S^{\natural}\) regular?

Question 3. If the ring \(S^{\natural}\) is regular, is it a ring of type \(C\)?

G. E. Shilov answered questions 1 and 3 in the affirmative. To question 2 he answered in the negative and gave certain sufficient conditions for regularity.

In the present note we wish to point out one more important question.

Question 4. Let \(S^{\natural}\) be a ring of type \(C\), and let \(I^0\) be the minimal ideal at zero; are the local rings \(S^{\natural}/I^0\) and \(L\) isomorphic?

The answer is negative. Indeed, there always exists a natural homomorphism \(S^{\natural}/I^0 \to L\), but, as we shall see, this homomorphism may be proper. Thus we shall show, in particular, that one and the same ring \(S^{\natural}\) can be synthesized from two different local rings \(L_1\) and \(L_2\).

The example giving an answer to question 4 is based on the properties of elliptic differential operators. Let \(X\) be an \(n\)-dimensional torus, and consider only rings of strict type \(C\), i.e., such that \(R/I^0\) is finite-dimensional. A ring \(R\) is of strict type \(C\) if and only if it contains the ring \(R_\infty\) of all infinitely differentiable functions. In such rings it is convenient to regard the ring \(R_\infty\) as playing the same role as the ring \(P\) in the general case. It is not hard to show that the answers to questions 1, 3, 4 do not change if \(P\) is replaced by \(R_\infty\). Question 2 now, of course, has an affirmative answer.

Let us first consider analysis of type \(C\). The mapping \(f \to f^{\natural}(x)\) of the ring \(R\) onto \(L^x\) is determined by its restriction to the dense subring \(R_\infty\). Since \(L^x\) is finite-dimensional, the image of \(R_\infty\) is all of \(L^x\). It can be shown that the \(R_\infty\)-ideal

\[ I_\infty^x=\left\{ f\in R_\infty:\ \text{all }\frac{\partial^{\alpha_1+\cdots+\alpha_n}}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}} f(x)=0\right\} \]

is contained in the \(R\)-ideal \(I^x\); consequently, there exists a natural homomorphism \(L_\infty^x=R_\infty/I_\infty^x\) onto \(L^x\). But \(L_\infty^x\) is isomorphic to the ring of all formal power series, and the homomorphism \(R_\infty \to L_\infty^x\) assigns to a function \(f\) its Taylor expansion

\[ \sum \frac{f^{(\alpha)}(x)}{\alpha!} t^\alpha \]

(here, for \(n\)-fold series, we use the abbreviated notation). The homomorphism \(L_\infty^x \to L^x\) is reduction of a series modulo some ideal \(I_\infty^x\) of finite codimension. Thus, if \(I_\infty^x\) is of finite codimension, one can find \(m\) differential operators with constant coefficients \(D_1,\ldots,D_m\) such that the image \(f^{\natural}(x)\) of a function \(f\) in \(L^x\) is equal to zero if and only if \(D_1 f(x)=\cdots=D_m f(x)=0\). The norm of type \(C\) will then be equivalent to the norm

\[ \sup\{|D_1 f|+\cdots+|D_m f|\}. \]

Conversely, for synthesis of type \(C\) we choose an arbitrary ideal of finite codimension in the abstract ring \(L_\infty\) of formal power series and complete \(R_\infty\) in the corresponding norm. By Shilov’s main theorem, if first, from the ring \(R\), by analysis of type \(C\), we construct the ideal \(I_\infty\), and then, by synthesis of type \(C\), obtain a new ring \(R'\), then \(R=R'\). But if we begin with the ideal \(I_\infty\), construct by synthesis the ring \(R\), and then by analysis obtain a new ideal \(I'_\infty\), we know only that \(I'_\infty \subseteq I_\infty\). Question 4 is equivalent to the question whether the ideals \(I'_\infty\) and \(I_\infty\) coincide, or, in other words, whether every ideal \(I_\infty\) in the ring of formal power series can be

can be obtained by analysis of type \(C\) from a certain corresponding ring \(R\) of type \(C\). Next we give a counterexample. On the two-dimensional torus consider the norm

\[ \sup\left\{|f|+\sum_{i=1}^{3}\left|\frac{\partial^i f}{\partial x^i}\right| +\sum_{i=1}^{3}\left|\frac{\partial^i f}{\partial y^i}\right| +\left|\frac{\partial^4 f}{\partial x^4}+\frac{\partial^4 f}{\partial y^4}\right|\right\}, \]

corresponding to the ideal generated by the polynomials \(xy,\ x^4-y^4\) and by all monomials of degree five. Carrying out first the synthesis and then the analysis, we obtain the norm

\[ \sup\left\{|f|+\sum_{1\leq i+j\leq 3}\left|\frac{\partial^{i+j}f}{\partial x^i\partial y^j}\right| +\left|\frac{\partial^4 f}{\partial x^4}+\frac{\partial^4 f}{\partial y^4}\right|\right\}, \]

corresponding to the ideal in the ring of formal power series generated by the polynomials \(x^4-y^4,\ xy^3,\ x^2y^2,\ x^3y\) and by all monomials of degree five. To prove the equivalence of these formally different norms, we turn to the following theorem.

Theorem. Let \(p\) be a homogeneous polynomial in \(n\) variables, equal to zero only at the origin. Let

\[ p\left(\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n}\right) = p\left(\frac{\partial}{\partial x}\right) \]

be the elliptic operator defined by the polynomial \(p\). Then for any differential operator with constant coefficients \(q\left(\frac{\partial}{\partial x}\right)\) of order strictly less than the order of the polynomial \(p\), there exists an \(\varepsilon>0\) such that for all \(f\in R_\infty\) the inequality

\[ \varepsilon \sup\left|q\left(\frac{\partial}{\partial x}\right)f\right| \leq \sup\left\{|f|+\left|p\left(\frac{\partial}{\partial x}\right)f\right|\right\}. \]

holds.

The equivalence of the norms already allows us to answer question 4. But in order to verify the requirement that the second norm really arises as a result of analysis of type \(C\), we must show that there is no third equivalent norm formally stronger than the second. For this we use the following theorem.

Theorem. Let \(p\left(\frac{\partial}{\partial x}\right)\) and \(q\left(\frac{\partial}{\partial x}\right)\) be differential operators with constant coefficients of orders \(p\) and \(q\), respectively, \(p\leq q\), and let the polynomial \(q\) not be a multiple of the polynomial \(p\). Then for any \(\varepsilon>0\) there exists a function \(f\in R_\infty\) such that

\[ \varepsilon \sup\left|q\left(\frac{\partial}{\partial x}\right)f\right| > \sup\left\{|f|+\left|p\left(\frac{\partial}{\partial x}\right)f\right|\right\}. \]

Stanford University
Dartmouth College
Hanover, New Hampshire, USA

Received
2 X 1962

References

1. G. E. Shilov, Uspekhi Mat. Nauk, 6, No. 1 (41), 91 (1951).