The "linear" counterpart of the problem of analytic group extensions of the Poincare group is presented in terms of the considerably simpler (but less general) analysis of Lie algebra extensions of the Poincare algebra P. After easily proving with this technique that every C kernel (P, () has an extension and that every such extension is inessential, the problem of analyzing the central extensions of P is carried out with the well-expected result that every such extension is trivial. But contrary to some claims, we exhibit an example which explicitly shows an essential noncentral extension of P.
We show that, if A is a finite-dimensional *-simple associative algebra with involution (over the field K of real or complex numbers) whose hermitian part H(A, * > is of degree > 3 over its center, if B is a unital algebra with involution over 06, and if (I.11 is an algebra norm on H(A @ B, * 1, then there exists an algebra norm on A @ B whose restriction to H(A @ B, *> is equivalent to 11. 11. Applying zel'manovian techniques, we prove that the same is true if the finite dimensionality of A is relaxed to the mere existence of a unit for A, but the unital algebra B is assumed to be associative. We also obtain results of a similar nature showing that, for suitable choices of algebras A and B over K, the continuity of the natural product of the algebra A @ B for a given norm can be derived from the continuity of the symmetrized product.
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