Absolute-valued algebras

Abstract

An algebra A over the real field R is a vector space over R which is closed with respect to a product xy which is linear in both x and y, and which satisfies the condition X(xy) = (Xx)y = x(Xy) for any X in R and x, y in A. The product is not necessarily associative. An element e of the algebra A is called a unit element if ex = xe =x for any x in A. Given any subset B of A, dim B will denote the linear dimension of B; i.e., the power of a maximal set of linearly independent elements of B. Further, [B] will denote the linear set spanned by the elements of B. For each x in A, we shall denote by A (x) the subalgebra generated by x. The algebra A is called algebraic if A (x) is finite dimensional for every x in A. The algebra A is said to be a division algebra if for every a, b in A, with a5O, the equations ax=b and ya=b are solvable in A. An algebra over R is called absolute-valued if it is a normed space under a multiplicative norm I j; i.e., a norm satisfying, in addition to the usual requirements, the condition I xyI = I x| I | yj for any x, y in A. It is obvious that an absolute-valued algebra contains no divisors of zero. A. A. Albert has shown [2, p. 768] that: (*) An absolute-valued algebraic algebra with a unit element is isomorphic to either the real field R, the complex field C, the quaternion algebra Q, or the Cayley-Dickson algebra D. F. B. Wright has proved [6, p. 332] the same theorem for absolute-valued division algebras with a unit element. In the present note we extend this result to an arbitrary absolute-valued algebra with a unit element. First, we shall give a simple example of an infinite dimensional algebra which is absolute-valued. The existence of such an algebra shows that the assumption of the presence of a unit element is essential. Let Ao be the space of all sequences x = xn } of real numbers with convergent series x Xn. Ao is a Hilbert space over R with respect to the norm I xI= = X)1/2, and with the usual addition and scalar multiplication: { Xn} + { yn } = { Xn +Yn } X { Xn } = {XxXn }. Let d>