# Loop Quantization and Symmetry: Configuration Spaces

### Abstract

Given two sets **S**_{1}, **S**_{2} and unital *C*^{*}-algebras
\({\mathfrak{A}_1}\)
,
\({\mathfrak{A}_2}\)
of functions thereon, we show that a map
\({\sigma : {\bf S}_1 \longrightarrow {\bf S}_2}\)
can be lifted to a continuous map
\({\bar\sigma : {\rm spec}\, \mathfrak{A}_1 \longrightarrow {\rm spec}\, \mathfrak{A}_2}\)
iff
\({\sigma^\ast \mathfrak{A}_2 := \{\sigma^\ast f\, |\, f \in \mathfrak{A}_2\} \subseteq \mathfrak{A}_1}\)
. Moreover,
\({\bar \sigma}\)
is unique if existing, and injective iff
\({\sigma^\ast \mathfrak{A}_2}\)
is dense. Then, we apply these results to loop quantum gravity and loop quantum cosmology. For all usual technical conventions, we decide whether the cosmological quantum configuration space is embedded into the gravitational one; indeed, both are spectra of some *C*^{*}-algebras, say,
\({\mathfrak{A}_{\rm cosm}}\)
and
\({\mathfrak{A}_{\rm grav}}\)
, respectively. Typically, there is no embedding, but one can always get an embedding by the defining
\({\mathfrak{A}_{\rm cosm} := C^\ast(\sigma^\ast \mathfrak{A}_{\rm grav})}\)
, where
\({\sigma}\)
denotes the embedding between the classical configuration spaces. Finally, we explicitly determine
\({C^\ast(\sigma^\ast \mathfrak{A}_{\rm grav})}\)
in the homogeneous isotropic case for
\({\mathfrak{A}_{\rm grav}}\)
generated by the matrix functions of parallel transports along analytic paths. The cosmological quantum configuration space so equals the disjoint union of
\({\mathbb{R}}\)
and the Bohr compactification of

-Abstract Truncated-

Publisher URL: https://link.springer.com/article/10.1007/s00220-017-3030-7

DOI: 10.1007/s00220-017-3030-7

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