We prove automatic continuity theorems for "decomposable" or "local" lineartransformations between certain natural subspaces of operator algebras. Thetransformations involved are not algebra homomorphisms but often are modulehomomorphisms. We show that all left (respectively quasi-) centralizers of thePedersen ideal of a C*-algebra A are locally bounded if and only if A has noinfinite dimensional elementary direct summand. It has previously been shown byLazar and Taylor and Phillips that double centralizers of Pedersen's ideal arealways locally bounded.
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