Absolute Valued Algebras.

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) = ÇKx)y = x(ky) 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 a^O, 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.e., a norm satisfying, in addition to the usual requirements, the condition \xy\ =\x\ -\y\ for any x, y m 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 2«-i xlAQis a Hubert space over R with respect to the norm |x| = (2Z"-i xn)ll2< and with the usual addition and scalar multiplication : {x„} + {yn} = {xn +yn}, X {xn ) = {Xx„}. Let