Some Automatic Continuity Theorems for Operator Algebras and Centralizers of Pedersen's Ideal

We prove automatic continuity theorems for "decomposable" or "local" linear transformations between certain natural subspaces of operator algebras. The transformations involved are not algebra homomorphisms but often are module homomorphisms. We show that all left (respectively quasi-) centralizers of the Pedersen ideal of a C*-algebra A are locally bounded if and only if A has no infinite dimensional elementary direct summand. It has previously been shown by Lazar and Taylor and Phillips that double centralizers of Pedersen's ideal are always locally bounded.