Any Abelian -algebra with unit can be represented as the algebra of all continuous functions on a compact space . The intuitive idea is that the original space , which has many more continuous functions, is enlarged by adding new points to it. This increases the number of continuity conditions and thus shrinks the set of continuous functions. This is done until only matrix elements of holonomies survive when continuity is imposed, and it follows from general results that the enlarged space must be compact for an Abelian unital -algebra. We thus obtain a compactification , the space of generalized connections , which densely contains the space .
There is a natural diffeomorphism invariant measure on , the Ashtekar–Lewandowski measure , which defines the Hilbert space of square integrable functions on the space of generalized connections. A dense subset of functions is given by cylindrical functions , which depend on the connection through a finite but arbitrary number of holonomies. These cylindrical functions are associated with graphs formed by the edges , …, . For functions cylindrical with respect to two identical graphs the inner product can be written as.
With the dense subset of we obtain the Gel’fand triple
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