Locally convex *-algebras having stronger unbounded C^*-norms
by
Atsushi Inoue
Fukuoka University, Fukuoka, Japan

A mapping p of a *-subalgebra D(p) of a *-algebra A into R⁺ = [0, \infty) is said to be an unbounded C^*-(semi)norm on A if it is a C^*-(semi)norm on D(p). Unbounded C^*-seminorms on *-algebras have appeared in many mathematical and physical subjects (for example, locally convex *-algebras, the moment problem, the quantum field theory etc.). But, it seems that this systematical study has still been insufficient. So, we have tried to study systematically unbounded C^*-seminorms and to apply such studies to those of locally convex *-algebras.

A locally convex *-algebras is a *-algebra which is also a Hausdorff locally convex space such that the multiplication is separately continuous and the involution is continuous. The studies of locally convex *-algebras were begun with those of locally m-convex *-algebras by R. Arens, E. Michael and the others. A locally convex *-algebra A[\tau] is said to be m-convex (resp. C^*-convex) if the topology \tau is determined by a directed family {p_\lambda}_{\lambda in \Lambda} of m^*-seminorms (resp. C^*-semionorms), that is, p_\lambda(xy) <= p_\lambda(x) p_\lambda(y) and p_\lambda(x^*) = p_\lambda(x) for each \lambda in \Lambda and x, y in A (resp. p_\lambda(x^*x) = p_\lambda(x)² for each x in A and \lambda in \Lambda). A complete locally C^*-convex algebra is said to be a pro-C^*-algebra. It is known that every complete locally m-convex *-algebra (resp. pro-C^*-algebra) is a projective limit of Banach *-algebras (resp. C^*-algebras). But, it is difficult to study general locally convex *-algebras which are not m-convex or C^*-convex even if the multiplication is jointly continuous. So, we defined and studied recently the notions of M^*-like (or C^*-like) locally convex *-algebras as follows: Let A[\tau] be a locally convex *-algebra. A directed family \Gamma = { p_\lambda}_{\lambda in \Lambda} of seminorms determining the topology \tau is said to be M^*-like if for any \lambda in \Lambda there exists \lambda' in \Lambda such that p_\lambda(xy) <= p_\lambda'(x) p_\lambda'(y) and p_\lambda(x^*) <= p_\lambda'(x), ^{for all} x, y in A, and further if p_\lambda(x)² <= p_\lambda'(x^*x), x in A, then \Gamma is said to be C^*-like. Then each p_\lambda is not necessarily m-convex (or C^*-convex), but the unbounded m^*-(or C^*-)norm p_\Gamma is defined by

D(p\Gamma) = { x in A; sup \lambda in \Lambda  p\lambda(x) < \infty} and p\Gamma(x) = sup \lambda in \Lambda  p\lambda(x), x in D(p\Gamma).

A locally convex *-algebra A[\tau] is said to be M^*-like (resp. C^*-like) if it is complete and there exists an M^*-like (resp. C^*-like) family \Gamma = {p_\lambda }_{\lambda in \Lambda} of seminorms determining the topology \tau such that D(p_\Gamma) is \tau-dense in A. G. R. Allan and P. G. Dixon defined the notion of GB^*-algebras which is a generalization of C^*-algebra (B^*-algebra): Let A[\tau] be a locally convex *-algebra with identity 1. We denote by B^* the collection of closed, bounded absolutely convex subsets B of A satisfying 1 in B, B^* = B and B² subset B. For every B in B^*, the linear span A[B] of B forms a normed *-algebra equipped with the Minkowski functional ||  ||_B of B. If A[B] is complete for every B in B^*, then AA pseudo-complete locally convex *-algebra A is said to be a GB^*-algebra if B^* has the greatest member B₀ and (1 + x^*x)^-1 in A[B₀] for every x in A. If A is a GB^*-algebra, then A[B₀] is a C^*-algebra and ||  ||_B0 is an unbounded C^*-norm on A. Thus the studies of unbounded C^*-seminorms may be useful for those of locally convex *-algebras.

In this talk we shall investigate the structure of a locally convex *-algebra A[\tau] having an unbounded C^*-norm p satisfying the conditions: (S₁) the topology defined by p is stronger than the topology \tau on D(p) (simply, \tau\prec p); (S₂) \tau and p are compatible in the sense that any Cauchy net {x_\alpha } in D(p)[p] such that x_\alpha\overset\tau --> 0 implies x_\alpha\oversetp --> 0. The unbounded C^*-norms p_\Gamma and ||  ||_B0 stated above have this property. Such an unbounded C^*-norm is said to be stronger. is said to be pseudo-complete. We shall show that a locally convex *-algebra A[\tau] has a stronger unbounded C^*-norms if and only if the completion [`(A)][\tau] of A[\tau] contains continuously a C^*-algebra, and further characterise GB^*-algebras by stronger unbounded C^*-norms.

Date received: April 24, 2000