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The purpose of this paper is to give a brief general account of the completely reducible finite-dimensional representations of a locally finite group G over a given algebraically closed field K. Theorem 1 shows that all such representations of G can be brought down to the algebraic closure F in K of the prime field of K. This reduces all further considerations in this account to countable groups. Theorem 2 characterizes the existence of a faithful completely reducible representation of G of degree n over K in terms of the existence of such representations for appropriate finite subgroups of G. Throughout the paper, G denotes a locally finite group, K denotes an arbitrary algebraically closed field and F denotes the algebraic closure in K of the prime field of K. F denotes an w-dimensional vector space over K. An F-form of V is an -F-subspace W of V such that W and K are linearly disjoint over K and Vis the X-span of W. (Equivalently, an F-form of V is the i^-span of a basis of V.) Il A is an Falgebra, AK denotes the algebra A®pK.

1. This paper announces results concerning the cohomology groups H*(N, N) where A*" is in a certain class of finite-dimensional nilpotent Lie algebras over a field k and T is an abelian Lie algebra faithfully represented as a maximal diagonalizable algebra of derivations of N; we shall refer to such an iV as a T-algebra. The additional hypotheses to be placed on the pair N, T are inspired by the case when J is a Cartan subalgebra and T+N=B is a Borel subalgebra of a complex semisimple Lie algebra. In that case Kostant has shown [2] that H%N, N)=0 for i^.2 and the authors applied this result in [3] to conclude that H*(B, B) = 0. (A similar argument shows H*(P9 P ) = 0 for P parabolic.) Here we are concerned with the relations between the vanishing ofH%N, N), especially for i=2 , and the structure of the algebras N. Let W denote the set of weights of T in N. If dirn(r)=dim(7\T/A)=m then the subset of W arising from the induced representation of T on N/N has precisely m elements, say {a1? • • • , am}. Every a e Wthen has a unique representation a = 2 ^a* with each c? a nonnegative integer and ct 0. For such an a we call the sum (in Z) 2 i the height of a and denote it by |a|. For a in W, denote by Aa the weight space for a in N. DEFINITION. A T-algebra is called positive if (i) dim(70=dim(A7iV), (ii) N is graded by the heights of the weights, i.e., if N(j) = Q)\al=j Na then [N(j)9N(k)]cN(j+k). REMARK. Condition (ii) is superfluous in characteristic 0. However, in characteristic p>0 it has such consequences as N=0 for r > ( / > l ) d i m ( r ) .

Introduction. Let X be a finite dimensional complex Euclidean space, and let T be a linear operator acting on X. The Jordan decomposition theorem states that T has a unique decomposition T= S + N, where S = f(T)Z dE(z), E is a spectral measure supported by the spectrum a(T) of T, and N is a nilpotent operator commuting with S. Our main result (Theorem 2.1) is a generalization of the Jordan theorem for operators with real spectrum to infinite dimensional reflexive Banach spaces. We consider operators T satisfying the growth condition leitTI = O(jtjk) for some integer k ? 0 and all real t. In ?1, we construct the "Jordan manifold" for T, on which T is shown to have a unique Jordan decomposition, if the spectrum (which is real because of the growth condition) has Lebesgue measure zero (?2). Related results are described in ?2. The theory is illustrated by examples in ?3. This work is clearly related to Dunford's theory of spectral operators. However, the latter is needed as a prerequisite only for Theorem 2.12. The standard reference is [1], [2]. Many thanks are due to Professors H. Furstenberg and C. A. McCarthy for discovering an error in the original version of this paper.