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Suppose S is a Banach space and K is the near-ring of all zero preserving Lipschitz transformations from S to S. It is shown that all exponentials of members of K have certain differentiability properties. This leads to the fact that no neighborhood of the identity transformation is filled with exponentials of members of K. It is known ([1], 13] or [2]) that if K is a Banach algebra with identity I and T is an element of K such that IT II < 1, then T = Exp A for some A e K. Here Exp A = lim J( + (I/n)A)n. From this fact it follows easily that the identity component of K is precisely the set of all finite products of exponentials of members of K. One purpose of this note is to show that in a near-ring of zero preserving Lipschitz transformations on a Banach space, it no longer holds that all elements in some neighborhood of I are exponentials. It was shown in [2] that for such near-rings the finite products of exponentials are dense in the identity component. The results of this note indicate that [2] cannot be improved in a certain direction. Denote by S a real Banach space and by K a near-ring of zero preserving Lipschitz transformations from S to S. Precisely, K is a collection of transformations from S to S such that: (1) if each of T and V e K and c is a number, then T+ V, TV and cT e K; (2) if T e K, there is a smallest number Tj (called the norm of T) such that ITx Tyl < I ||Tj x y for all x, y e S; (3) TO = 0; (4) I, the identity transformation on S, is in K; (5) K is complete in the sense that if T is a transformation from S to S such that (2) holds and IT F0= is a sequence of elements of K which converges to T uniformly on each bounded subset of S, then T e K. If A E K then Exp A denotes the element L of K such that Lx = Received by the editors December 31, 1973. AMS (MOS) subject classifications (1970). Primary 471199.

Differentiability of the exponential of a member of a near-ring