2. Varieties of Banach Algebras: the Birkhoff Theorem. 3. Semivarieties. 4. Varieties of C*-algebras.

Consider linear associative algebras over the complex field. Definition 1 A variety of algebras is a (non-empty) class of algebras closed under subalgebras, quotients, products and isomorphic images. Definition 2 Let p(X1,..., Xn) be a polynomial (unless otherwise specified, this will mean a polynomial in non-commuting indeterminates, without constant term). We say that an algebra A obeys the law p = 0 iff p(x1,..., xn) = 0 (x1,..., xn ∈ A). Example 1 An algebra “obeys the law XY − Y X = 0 ” iff it is commutative. Theorem 1 (Birkhoff’s Theorem) For a class V of algebras the following are equivalent: 1. V is a variety (closed under subalgebras, quotients, products and isomorphic images); 2. For some set P of polynomials V = {A: A obeys the law p = 0 ∀p ∈ P}