The Uniqueness of Norm Problem in Banach Algebras

Abstract

A Banach algebra, or simply B-algebra, is a normed ring in the sense of I. Gelfand [5]. In other words, it is a linear algebra (not necessarily commutative) in which the underlying linear vector space is a B-space. The norm is also assumed to satisfy the multiplicative condition, 11 xy 11 _ 11 x 11 11 y II. The existence of an identity element is not assumed and, unless stated otherwise, the B-algebras considered are real (i.e., the field of scalars is the real numbers). Note that the complex case is included in the real. A B-algebra is said to possess a unique norm in case any two norms x and x II,, under which it is a B-algebra, are necessarily equivalent; that is, 1n 11 o if, and only if, I x, II, -0. It was shown by Gelfand [5, Satz 17] that the norm in a semi-simple [9] complex commutative B-algebra with identity is unique. It has also been shown by Eidelheit [4, Theorem 1] that the B-algebra of all bounded operators on a (real) B-space has a unique norm. The B-algebra is semi-simple in this case as well. The question naturally arises as to whether or not every semi-simple B-algebra has a unique norm. To the best of our knowledge, this is still an open question. The object of this paper is to present a few results relating to this problem. In ?1 some preliminary definitions and results for general B-algebras are given. In ?2, we consider first a linear space X which is a B-space under each of two norms II x I and II x II,. An element s is said to separate the norms provided there exists {xe such that s Xn I> 0 and ILx ,, In -I 0. The set S of all separating elements is a linear subspace of X closed with respect to both norms. The condition S = (0) is necessary and sufficient for equivalence of the norms. Let Z be a linear subspace of X closed with respect to both norms and define be = X, [x] = x + Z, 11 [x] 11 = glbt,6x || x + t ||, || [x] 1l, = glbtfl -| X + t 1ii. Then a necessary condition for the equivalence of the norms 11 [x] ||, || [x] II, in ? is that S a Z and a sufficient condition is that S = Z. In specializing to the case of an algebra A, which is a B-algebra under two norms, the set S of separating elements becomes a 2-sided ideal in 93, closed with respect to both norms and each of whose elements is a topological divisor of zero. This implies that a simple B-algebra with identity element has a unique norm. More generally, any B-algebra which is strongly semi-simple, in the sense defined by I. E. Segal [21, p. 74], also has a unique norm. It is also shown here that the uniqueness of norm problem for semi-simple B-algebras can be reduced to the primitive [10, 11] case. In ?3 a complex, irreducible B-algebra of linear operators on a complex linear