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Universal symbols on locally compact abelian groups

Gorin E. A.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

199-204

Let G be a locally compact abelian group, let X be its dual, let M(G) be the convolution algebra of all regular Borel measures of bounded variation on G, and let M (X) = {my | my is an element of M (G)}. A complex-valued function q on X is called symbol, if (the restriction) q|E is an element of M(X)|E for any compact subset E is a subset of X. If A is a Banach algebra then the group of invertible elements of A is denoted by A* and the spectral radius of a is an element of A is denoted by |a| . We consider continuous representations G is an element of g --> Tg is an element of A* such that ||Tg|| = 1 for all g is an element of G. In this case put my(T) = integral of GTg[my]dg). Obviously, we obtain a representation of M (G) (this representation is denoted by the same symbol T). It can easily be checked that hull(Ker(T)) is a subset of X. It leads to the correct definition of q(T) for all symbols q. Symbol q is called universal, if ||q(T)|| = |q(T)| for all T. The aim of the paper is to give a function-theoretic description of universal symbols.