HIGHER POINT DERIVATIONS ON COMMUTATIVE BANACH ALGEBRAS, III fey

i=o hold for each choice of / and g in A and k in { 1 , . . . , q}. A point derivation of infinite order is an infinite sequence {dk} of linear functional such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each fcSl, and degenerate if d1 = 0. Some of the terminology given in the previous paragraph was introduced in our earlier paper (3), where we began a study of the continuity properties of point derivations on commutative Banach algebras. In particular, we asked whether, for a given algebra A, there is a function q •-» p(q) on the set N of natural numbers such that, whenever du ..., dp(<,) is a point derivation on A, the "initial segment" du ..., dq of order q is necessarily continuous. By algebraic methods, we obtained partial results for a number of algebras (see Theorem 2.3 and Examples 2.5 to 2.8 of (3)), and found for the algebra C of n -times continuously differentiate functions on a compact interval a complete description of the point derivations, including a determination of the function p(cj). In a second paper (4), we continued the study by giving a construction which showed that, in general, the function p(q) need not exist: a consequence of the existence of p(q) is that every point derivation of infinite order is continuous, and we constructed algebras of various types with totally discontinuous point derivations of infinite order.