Notes on normed algebras , 2 Stephen William Semmes
نویسنده
چکیده
LetA be a finite-dimensional commutative algebra over the complex numbers, with identity element e. Thus A is a finite-dimensional complex vector space equipped with an additional binary operation of multiplication which satisfies the usual rules of associativity, commutativity, and distributivity, and e is a nonzero element of A such that e x = x for all x ∈ A. As a basic class of examples, one can take X to be a finite set, and A the algebra of complex-valued functions on X, with respect to pointwise multiplication. As another class of examples, suppose that A is a commutative semigroup with identity. In other words, A is a set equipped with an identity element 0 and a binary operation + which satisfies the usual associativity and commutativity rules, and with x + 0 = x for all x ∈ A. Let A be the vector space of complex-valued functions on A, and define the convolution of two such functions f1, f2 by
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