Algebra of quotients with bounded evaluation of a normed prime algebra

Following the construction of the bounded algebra of quotients for ultraprime algebras by M. Mathieu, we introduce a kind of bounded algebra of quotients for normed prime algebras involving the multiplication algebra and that will be called the algebra of quotients with bounded evaluation. In the class of totally prime associative algebras our construction turns out to be the appropriate concept of algebra of quotients. We compute the algebra of quotients with bounded evaluation of some norm ideals on a complex Hilbert space, namely: The ideal of all compact operators and the Schatten p-ideals for 1 less than or equal to p < infinity.