Showing papers in "Tohoku Mathematical Journal in 1960"

On differentiable manifolds with certain structures which are closely related to almost contact structure i

Abstract: then M is said to be a differentiable manifold with almost complex structure. (Tensor fields of the form given above may exist only for some manifolds with even dimension.) We shall call φ the fundamental collineation of the almost complex structure. The set of differentiable manifolds with almost complex structure is wider than the set of complex manifolds. Every differentiable manifold with almost complex structure φ admits a poistive definite Riemannian metric g such that

On a conjecture of kaplansky

Abstract: PROOF. Let A be a C*-algebra, ' a derivation of A. It is enough to show that the derivation is continuous on the self-adjoint portion As of A. Therefore if it is not continuous, by the closed graph theorem there is a sequence \\xn\\ (xn 4= 0) in As such that xn -»0 and xn -> a + ίέ(φθ), where a and b are self-adjoint. First, suppose that a =f= 0 and there exists a positive number λ(> 0) in the spectrum of a (otherwise consider { —xn}). It is enough to assume that λ = 1. Then there is a positive element Λ(||/*|| = 1) of A such that hah S — h 2 . Put yn = xn + 3 \\\\xn\\\\ 7, then yn -> 0, yn = xn and (hyji)' = tiyji + hyn h + hynh' hence (hynh)' —> h(a + ib)h. Therefore

An enumeration of the primitive recursive functions without repetition

Abstract: In a theorem and its corollary [1] Friedberg gave an enumeration of all the recursively enumerable sets without repetition and an enumeration of all the partial recursive functions without repetition. This note is to prove a similar theorem for the primitive recursive functions. The proof is only a classical one. We shall show that the theorem is intuitionistically unprovable in the sense of Kleene [2]. For similar reason the theorem by Friedberg is also intuitionistically unprovable, which is not stated in his paper.