The concept of a laminated near-ring was introduced in [2]. We recall briefly what it is. Let N be a near-ring and let a∈N. Define a new multiplication on N by x * y = xay for all x,y∈N. With this new multiplication and the same addition as before we have another near-ring which we denote by Na. The near-ring Na is referred to as a laminated near-ring, the original near-ring N is the base near-ring and a is the laminator or laminating element.

Abstract. Let G be a locally compact abelian group and A" be a Banach space. Let Ll(G, X) be the Banach space of X-valued Bochner integrable functions on G. We prove that the space of bounded linear translation invariant operators of L'(G, X) can be identified with L(X, M(G, X)), the space of bounded linear operators from X into M(G, X) where M(G, X) is the space of Jf-valued regular, Borel measures of bounded variation on G. Furthermore, if A is a commutative semisimple Banach algebra with identity of norm 1 then L\G, A) is a Banach algebra and we prove that the space of multipliers of L\G, A) is isometrically isomorphic to M{G, A). It also follows that if dimension of A is greater than one then there exist translationinvariant operators of Ll(G, A) which are not multipliers of L1(G, A).

Let I = (0, ∞) with the usual topology. For x, y ∈ I, we define xy = max(x, y). Then I becomes a locally compact commutative topological semigroup. The Banach space L 1 (I) of all Lebesgue integrable functions on I becomes a commutative semisimple Banach algebra with order convolution as multiplication. A bounded linear operatorThe space of multipliers of L 1 (I) was determined by Johnson and Lahr. Let X be a Banach space and L 1 (I, X) be the Banach space of all X-valued Bochner integrable functions on I. We show that L 1 (I, X) becomes an L 1 (I)-Banach module. Suppose X and Y are Banach spaces. A bounded linear operator T from L 1 (I, X) to L 1 (I, Y ) is called a multiplier if T (f ⋆ g) = f ⋆ T g for all f ∈ L 1 (I) and g ∈ L 1 (I, X). In this paper, we characterize the multipliers from L 1 (I, X) to L 1 (I, Y ).

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

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