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3.4 Function spaces

A connection 1-form Aia can be reconstructed uniquely if all its holonomies are known [172]. It is thus sufficient to parameterize the configuration space by matrix elements of he for all edges in space. This defines an algebra of functions on the infinite-dimensional space of connections š’œ, which are multiplied as ā„‚-valued functions. Moreover, there is a duality operation by complex conjugation, and if the structure group G is compact a supremum norm exists, since matrix elements of holonomies are then bounded. Thus, matrix elements form an Abelian C ∗-algebra with unit as a sub-algebra of all continuous functions on š’œ.

Any Abelian C∗-algebra with unit can be represented as the algebra of all continuous functions on a compact space ¯ š’œ. The intuitive idea is that the original space š’œ, 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 ∗ C-algebra. We thus obtain a compactification ¯ š’œ, the space of generalized connections [24], which densely contains the space š’œ.

There is a natural diffeomorphism invariant measure dμAL on š’œ¯, the Ashtekar–Lewandowski measure [20], which defines the Hilbert space ā„‹ = L2 (š’œ¯, dμ ) AL of square integrable functions on the space of generalized connections. A dense subset Cyl of functions is given by cylindrical functions f (he1,...,hen ), which depend on the connection through a finite but arbitrary number of holonomies. These cylindrical functions are associated with graphs g formed by the edges e1, …, en. For functions cylindrical with respect to two identical graphs the inner product can be written as

∫ ∫ ∏n āŸØf |gāŸ© = d μAL(A )f(A )∗g(A ) = dμH (hi)f(h1,...,hn)∗g(h1,...,hn ), (10 ) ¯š’œ SU(2)n i=1
with the Haar measure d μH on G. The importance of generalized connections can be seen from the fact that the space š’œ of smooth connections is a subset of measure zero in š’œ¯ [224].

With the dense subset Cyl of ā„‹ we obtain the Gel’fand triple

∗ Cyl ⊂ ā„‹ ⊂ Cyl (11 )
with the dual Cyl ∗ of linear functionals from Cyl to the set of complex numbers. Elements of Cyl ∗ are distributions, and there is no inner product on the full space. However, one can define inner products on certain subspaces defined by the physical context. Often, those subspaces appear when constraints with continuous spectra are solved following the Dirac procedure. Other examples include the definition of semiclassical or, as we will use it in Section 7, symmetric states.
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