Locally compact abelian group

In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them.

For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups.

A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff; the topological group is called abelian if the underlying group is abelian.

Examples of locally compact abelian groups include: If

is a locally compact abelian group, a character of

is a continuous group homomorphism from

with values in the circle group

can be made into a locally compact abelian group, called the dual group of

The group operation on the dual group is given by pointwise multiplication of characters, the inverse of a character is its complex conjugate and the topology on the space of characters is that of uniform convergence on compact sets (i.e., the compact-open topology, viewing

as a subset of the space of all continuous functions from

This topology is in general not metrizable.

is a separable locally compact abelian group, then the dual group is metrizable.

More abstractly, these are both examples of representable functors, being represented respectively by

is isomorphic to the circle group

A character on the infinite cyclic group of integers

under addition is determined by its value at the generator 1.

Moreover, this formula defines a character for any choice of

The topology of uniform convergence on compact sets is in this case the topology of pointwise convergence.

This is the topology of the circle group inherited from the complex numbers.

is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology.

The group of real numbers

, is isomorphic to its own dual; the characters on

With these dualities, the version of the Fourier transform to be introduced next coincides with the classical Fourier transform on

(In fact, any finite extension of

Pontryagin duality asserts that the functor induces an equivalence of categories between the opposite of the category of locally compact abelian groups (with continuous morphisms) and itself: Clausen (2017) shows that the category LCA of locally compact abelian groups measures, very roughly speaking, the difference between the integers and the reals.

More precisely, the algebraic K-theory spectrum of the category of locally compact abelian groups and the ones of Z and R lie in a homotopy fiber sequence