Banach Algebras With an Adjoint Operation

This paper is concerned with a certain class of Banach algebras introduced by Gelfand and Neumark [4]. The terminology,, "Banach algebra" or simply "B-algebra," refers here to what was originally called a "normed ring" by Gelfand [3]. Thus a B-algebra is simply a vector algebra (not necessarily commutative or finite dimensional) in which the underlying vector space is a complex Banach space. The norm in the Banach space is related to multiplication by the condition jj xy IJ I JI x I II y I1, which holds for all elements x, y. We will assume in addition that the algebra contain an identity element e and that 1e 11 = 1. A B-algebra is called a "B*-algebra" if to each element x there corresponds a unique element x*, called the "adjoint" of x, with the following properties: (i) (x*)* = x. (ii) (xy)* = y*x*. (iii) If X, 1t are complex numbers, then (Ax + Ay)* = Ax* + fy*. (iv) 11 x*x 11 = 11 x I1'. An important type of B*-algebra is an algebra of bounded operators on Hilbert space. In this case the norm of an element is the bound of the operator and the adjoint of an element is the adjoint operator defined in the usual way. Gelfand and Neumark [4] have shown that a B*-algebra, which satisfies the additional condition that elements of the form x*x + e possess inverses, is isomorphic2 to a B*-algebra of bounded operators on Hilbert space (not necessarily separable). To the best of our knowledge it is an open question as to whether or not every B*-algebra can be so represented.3 We are primarily concerned here with abstract B*-algebras. However, even in the special case of an algebra of operators on Hilbert space, we believe our main results to be new and the methods to be interesting and useful since they are independent of the Hilbert space. Our general purpose is to study the structure of a B*-algebra in terms of its projections.4 Such a study of course demands the existence of many pro-