Notes on normed algebras

All vector spaces and so forth here will be defined over the complex numbers. If z = x+i y is a complex number, where x, y are real numbers, then the complex conjugate of z is denoted z and defined to be x− i y. The complex conjugate of a sum or product of complex numbers is equal to the corresponding sum or product of complex conjugates. The modulus of a complex number z is the nonnegative real number |z| such that |z| is equal to the product of z and its complex conjugate. Thus the modulus of a product of complex numbers is equal to the product of their moduli, and one can show that the modulus of a sum of two complex numbers is less than or equal to the sum of the moduli of the complex numbers. By a finite-dimensional algebra we mean a finite dimensional complex vector space A equipped with a binary operation which satisfies the usual associativity and distributivity properties, and which has a nonzero multiplicative identity element e. In other words, e x = x e = x for all x ∈ A. Thus A should have positive dimension in particular. Notice that the multiplicative identity element e is unique. As a basic class of examples, let V be a finite-dimensional complex vector space with positive dimension, and consider L(V ), the space of linear mappings from V to itself. This is a vector space whose dimension is equal to the square of the dimension of V . It also becomes an algebra with respect to the usual composition of linear transformations, with the identity transformation I on V , which sends every vector in V to itself, as the multiplicative identity element. If A is any finite-dimensional algebra, then we can identify A with a subalgebra of L(A), the algebra of linear transformations on A considered simply as a vector space. Namely, each element a of A can be identified with the linear transformation x 7→ a x on A.