Noncomplex methods in real Banach algebras

AbstractThe principal result is a new proof, independent of complex function theory, that the spectrum of any element of a real Banach algebra is nonempty (Theorem III). The methods used include elementary properties of plane convex sets, and the existence in a commutative algebra, of a continuous square root on any convex set of invertible elements which contains a square —this square root being uniquely determined by its value at one point. The proof (Theorem II) of the existence of a continuous square root (or of a continuous nth root for n ⩾ 2) is derived from a more general result (Theorem I) and depends heavily on commutativity, invertibility, and convexity. Theorem I deals with a function whose values lie in a locally convex linear topological space, and gives conditions which guarantee when several local continuous right inverses of this function may be combined to form a continuous right inverse with “large” domain. The “large” domain may be any convex subset of the range of the function, provided certain local conditions (Definition 3) are satisfied by the local inverses