A Local Kernel Property of Closed Derivations on C(I × I)

Abstract

In this note we show a local behavior of closed derivations on C([O, 1] x [0, 1]), which is essentially different from one-dimensional derivations. Roughly speaking, any closed derivations on C([0, 1] x [0, 1]) has a nonconstant kernel locally. Our concerns are unbounded densely defined derivations on the Banach algebra C(Q) of continuous real-valued functions on a compact Hausdorff space U. A (closed) derivation 8 on C(Q) means a (closed) linear map in C(Q) defined on a dense subalgebra 9(8) which satisfies the derivation property, i.e., 8(fg) = 8(f )g + f6( g) for every f, g in 6(8). Without loss of generality, we shall assume that the unit function 1 belongs to 9(8) (so that 8(1) = 0 follows from the derivation property). The kernel and range of 8 are denoted by ran 8 and ker 8, respectively. A closed subset E in Q is called a self-determining set for 8 if (f)E = ? wheneverf E(8) and f |E = O, where fl E means the restriction of f to E. If E is a self-determining set, the formula 6E(f I E) = (f ) I E defines a derivation 6E with domain { fl E; f E 9(8)}. It is known (see [1, Lemma 4.1 or 2, Lemma 1.1.10]) that if 8 is a closed derivation, for every open subset U in Q the closure U is a self-determining set for 8 and, moreover, Su is closable. For closable derivation Su, the closure will be written by Au. Closed derivations on the unit interval I = [0, 1] have been studied by various authors and recently the structure has been made clear (see references). On the contrary, even the derivations on C(I x I) are almost left untouched. The following theorem gives a property of derivations on C(I x I) which clearly holds for partial derivatives, but which does not hold for derivations on C(I). THEOREM. For any closed derivation 8 on C(I x I) with ran 8 = C(I x I) and any open subset V in I x I, there is a nonempty connected open subset U contained in V such that ker Au contains some nonconstant functions in 9(8u). PROOF. Without loss of generality we may assume that V is connected. We consider two cases. Case 1. There exists at least one nonempty open connected subset W of V such that 6is not closed (but closable). In this case, it is easily shown that ker8w Received by the editors October 5, 1984 and, in revised form, January 14, 1985. 1980 Mathematics Subject Classification. Primary 46L05.