By a normed algebra we mean a real or complex (possibly nonassociative) algebra A endowed with a norm ||·|| satisfying ||xy|| ≤ ||x||.||y|| for all x,y ∈ A. A complete normed associative algebra will be called a Banach algebra. A normed algebra is called norm-unital if it has a unit 1 such that ||1|| = 1. Unitary elements of a normunital normed associative algebra A are defined as those invertible elements u of A satisfying ||u|| = ||u−1|| = 1. By a unitary normed associative algebra we mean a norm-unital associative normed algebra A such that the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. Relevant examples of unitary Banach algebras are all unital C^∗-algebras and the discrete group algebras l1(G) for every group G. We are interested in the development of a general theory of unitary Banach algebras. For a real (respectively, complex) norm unital Banach algebra A, consider Property (S) which follows:

(S) There exists a linear (respectively, conjugate-linear) algebra involution on A mapping each unitary element to its inverse.

It is known that Property (S) is fulfilled in the case that A is a unital C^∗-algebra, a discrete group algebra, or a finite-dimensional unitary Banach algebra. However, in general, unitary Banach algebras need not satisfy Property (S). We show that unitary semisimple commutative complex Banach algebras satisfy Property (S), and that, endowed with the involution given by such a property, they become hermitian ∗-algebras. From this theorem we get new characterization of unital C^∗-algebras. Later, we study Property (S) in the noncommutative case. To this end, we introduce “good” groups and prove that, if A is a real or complex unitary semisimple Banach algebra such that the group UA is good, then A satisfies Property (S). It seems to be an open problem whether or not every group is good. We give some equivalent reformulations in terms of unitary normed algebras.

Historically, unitary Banach algebras have been considered only from an associative point of view, and the main topic of interest has been characterize C^∗-algebras among them. In this talk, we leave the associative scope in order to deal by the first time with nonassociative unitary normed algebras.