On a class of topological algebras

Discontinuity of the product in multiplier algebras

“…In many interesting situations the product is continuous. This is the case for the example Cb(R) of [4] . It is also so for any multiplier algebra M(E) where E admits factorization, Le for every z E E, there exist x and y in E such that z = x .y.…”

“…Then E is said to be locally A-convex ( [4]) if for every x E E and every A E A there exist M(A, x) > 0, N(A, x) > 0 such that Pa(xy) < M(A,x)px(y) (y E E) .…”

“…Some examples of all these classes of algebras and relationships among them can be seen in ( [4]) and ( [5]). …”

“…The multiplier algebra M(E) of E is the space of linear operators T verifying T(x .y) = xT(y)(x, y E E) . Endowed with the strong topology given by the family of semi-norms (Px)xEE, where p,,(T) = JITxIj(x E E), M(E) is a unital, complete, uniformly locally A-convex algebra which is not always mconvex since it is a generalization (see [1, p. 139]) of the algebra Cb(R), of [4], that is not m-convex .…”

Algèbres $A$-convexes et problème de Michael

“…Dans le cas de l'exemple précédent, la topologie M(x) n'est autre que la topologie de la convergence uniforme sur les compacts de R. Pour voir ceci, on passe aux algebres quotients C[-n, n] de C(R) par F~'(0), zz > 1. On munit chaque C[-n, n] de la ^4-norme quotient Pn et on considère qn , la norme de Cochran [4] associée à Pn . On remarque alors que qn(f) = Qn(f), f e C(R).…”