In this paper, we show that Jordan $ au$-centralizers and local $ au$-centralizers are $ au$-centralizers under certain conditions. We also discuss a new type of generalized derivations associated with Hochschild 2-cocycles and introduce a special local generalized derivation associated with Hochschild 2-cocycles. We prove that if $cal{L}$ is a CDCSL and $cal{M}$ is a dual normal unital Banach $algcal{L}$-bimodule, then every local generalized derivation of above type from $algcal{L}$ into $cal{M}$ is a generalized derivation.