  We study the 3-algebraic structure involved in the recently shown M2-branes
worldvolume gauge theories. We first extend an arbitrary finite dimensional
3-algebra into an infinite dimensional 3-algebra by adding a mode number to
each generator. A unique central charge in the algebra of gauge transformations
appears naturally in this extension. We present an infinite dimensional
extended 3-algebra with a general metric and also a different extension with a
Lorentzian metric. We then study ordinary finite dimensional 3-algebras with
different signatures of the metric, focusing on the cases with a negative
eigenvalue and the cases with a zero eigenvalue. In the latter cases we present
a new algebra, whose corresponding theory is a decoupled abelian gauge theory
together with a free theory with global gauge symmetry, and there is no
negative kinetic term from this algebra.
