SOME ALGEBRAS WITHOUT SUBMULTIPLICATIVE NORMS OR POSITIVE FUNCTIONALS

We prove a conjecture of Yood regarding the nonexistence of submultipl icative norms on the algebra C(T) of all continuous functions on a top ological space T which admits an unbounded continuous function. We als o exhibit a quotient of C(T) which does not admit a nonzero positive l inear functional. Finally, it is shown that the algebra L(X) of all li near operators on an infinite-dimensional vector space X admits no non zero submultiplicative seminorm.