If G is a locally compact abelian group, there is another locally
compact abelian group, the Pontryagin dual group [^(G)], which
characterises G in the sense that [^([^(G)])] is
homeomorphically isomorphic with G. In studying the algebraic and
topological properties of G, it is frequently necessary to study its
group algebra L1(G), the algebra of Haar integrable functions on
G. In the case that G has discrete topology this algebra contains
the group ring C[G]; exactly as the algebra of Laurent
series with summable coefficients contains the algebra of Laurent
polynomials C[Z].
If G is not abelian, there is often no object which directly plays
as rich a role as does [^(G)] from the abelain case. However,
there is a construction of an abelian Banach algebra by Eymard, the
Fourier algebra A(G), which in the abelian case is exactly
L1([^(G)]). Categorically, A(G) plays the role of
"[^(G)]". Although A(G) is ostensibly commutative, spatially
it is not commutative. The theory of operator spaces is thus required
to understand the actual structure of A(G), in its capacity as a
dual object of G.
I will explain some background on this subject, and highlight results
of mine and of many leading researchers, which show that the operator
space structure is indispensible in our modern understanding of
A(G). In particular I want to note recent achievements in
understanding cohomological properties, ideals, multipliers and
homomomorphisms.
