We present an operator-algebraic approach to the quantization and reduction
of lattice field theories. Our approach uses groupoid C*-algebras to describe
the observables and exploits Rieffel induction to implement the quantum gauge
symmetries. We introduce direct systems of Hilbert spaces and direct systems of
(observable) C*-algebras, and, dually, corresponding inverse systems of
configuration spaces and (pair) groupoids. The continuum and thermodynamic
limit of the theory can then be described by taking the corresponding limits,
thereby keeping the duality between the Hilbert space and observable C*-algebra
on the one hand, and the configuration space and the pair groupoid on the
other. Since all constructions are equivariant with respect to the gauge group,
the reduction procedure applies in the limit as well.Comment: 23 pages, 6 figure