Frames in $C^*$-algebras and the stabilization theorem

Kasparov's theorem states that every countably generated Hilbert $C^*$-module over a $C^*$-algebra $A$ stabilizes, i.e., it can be represented as a direct summand of the direct sum of countably many copies of $A$. This property fails for uncountably generated modules and uncountable direct sums. In fact, this property is almost equivalent to the existence of the so-called standard frame. In this talk, we discuss the existence of such frames and some other related problems in the case of a commutative $C^*$-algebra considered as a module over itself.