WEAK INVERTIBILITY AND STRONG SPECTRUM

A notion of weak invertibility in a unital associative algebra A and a corresponding notion of strong spectrum of an element of A is defined . It is shown that many relationships between the Jacobson radical, th e group of invertibles and the spectrum have analogues relating the st rong radical, the set of weakly invertible elements and the strong spe ctrum. The nonunital case is also discussed. A characterization is giv en of all (submultiplicative) norms on A in which every modular maxima l ideal M subset-or-equal-to A is closed.