Representations of bornological algebras

Bounded representations of bornological algebras are considered. The left and right bornological radicals in bornological algebras are introduced. It is shown that the left (right) bornological radical of a bornological algebra $A$ is equal to the intersection of all bornologically closed maximal regular left (respectively, right) ideals of $A$ and these both radicals of $A$ and the Jacobson radical of $A$ coincide when $A$ is an advertive and simplicial bornological algebra (in particular, a bornological $Q$-algebra).