Definition request: Radical of a Banach algebra

Question

I am reading the definition of a semi-simple Banach algebra which is given as

The Banach algebra $A$ having the property that Rad$(A)=\{0\}$ is called a semi -simple Banach algebra.

I can not, however, find a proper definition of what exactly the radical of a Banach algebra $A$ is. Can anybody please clarify what exactly it is by providing me with a nicely written definition?

Answer 1

I believe this term refers to the Jacobson radical. This has several equivalent characterizations, so which one you take as the definition is to some extent a matter of taste.

Incidentally, in abstract algebra, rings with vanishing Jacobson radical are often called semiprimitive instead of semisimple, with semisimple being reserved for a much smaller class of rings (those whose modules are completely reducible).