  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.
