Symmetry and-commutativity of topological^*-algebras

Abstract

An advertibly complete locallym-convex (lmc)^*-algebraE is symmetric if and only if each normed (inverse limit) factorE/N _α,α ∈ A, ofE is symmetric in the respective Banach factorE _α ,α ∈ A, ofE. Every locally C^*-algebra is symmetric. If

denotes the continuous positive functionals on an lmc^*-algebraE and

withL _f ={x ∈ E: f(x ^* x) =0}, thenE is, by definition,

-commutative if

for anyx, y ∈ E.

-commutativity and commutativity coincide in lmcC ^*-algebras, so that an lmc^*-algebra with a bounded approximate identity is

-commutative if and only if its enveloping algebra is commutative. Several standard results for commutative lmc^*-algebras are also obtained in the

-commutative case, as for instance, the nonemptiness of the Gel'fand space of a suitable

-commutative lmc^*-algebra, the automatic continuity of positive functionals when the algebras involved factor, as well as that the spectral radius is a continuous submultiplicative semi-norm, when the algebras considered are moreover symmetric. An application of the latter result yields a spectral characterization of

-commutativity.