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| CATEGORIES ABOUT BOHR COMPACTIFICATION | |
| topological groups | |
| harmonic analysis | |
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DEFINITIONS AND BASIC PROPERTIES Given a Topological Group ''G'', the Bohr compactification of ''G'' is a compact ''Hausdorff'' topological group '''Bohr'''(''G'') and a continuous homomorphism :b: ''G'' → '''Bohr'''(''G'') which is Universal with respect to homomorphisms into compact Hausdorff groups; this means that if ''K'' is another compact Hausdorff topological group and f is a continuous homomorphism, then there is a unique continuous homomorphism :Bohr(''f''): Bohr(''G'') → ''K'' such that ''f'' = Bohr(''f'') '''b'''. Theorem. The Bohr compactification exists and is unique up to isomorphism. This is a direct application of the Tychonoff Theorem . We will denote the Bohr compactification of ''G'' by Bohr(''G'') and the canonical map by : The correspondence ''G'' → Bohr(''G'') defines a covariant functor on the category of topological groups and continuous homomorphisms. The Bohr compactification is intimately connected to the finite-dimensional Unitary Representation theory of a topological group. The Kernel of b consists exactly of those elements of ''G'' which cannot be separated from the identity of ''G'' by finite-dimensional ''unitary'' representations. The Bohr compactification also reduces many problems in the theory of Almost Periodic Function s on topological groups to that of functions on compact groups. A bounded continuous complex-valued function ''f'' on a topological group ''G'' is uniformly almost periodic iff the set of right translates ''g''''f'' where : is relatively compact in the uniform topology as ''g'' varies through ''G''. Theorem. A bounded continuous complex-valued function ''f'' on ''G'' is uniformly almost periodic iff there is a continuous function ''f''1 on '''Bohr'''(''G'') (which is uniquely determined) such that : |
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