Commutative Twisted Group Algebras

A twisted group algebra <!-- MATH ${L^1}(A,G;T,\alpha )$ --> is commutative iff A and G are, T is trivial and is symmetric: <!-- MATH $\alpha (\gamma ,g) = \alpha (g,\gamma )$ --> . The maximal ideal space <!-- MATH ${L^1}(A,G;\alpha )\hat \emptyset$ --> of a commutative twisted group algebra is a principal <!-- MATH $G\hat \emptyset$ --> bundle over <!-- MATH $A\hat \emptyset$ --> . A class of principal <!-- MATH $G\hat \emptyset$ --> bundles over second countable locally compact M is defined which is in 1-1 correspondence with the (isomorphism classes of) <!-- MATH ${C_\infty }(M)$ --> -valued commutative twisted group algebras on G. If G is finite only locally trivial bundles can be such duals, but in general the duals need not be locally trivial.