An F-algebra is a Banach algebra A whose dual is a von Neumann algebra in which the identity of \(A^*\) is a complex homomorphism on A. (There are many such algebras including \(L^ 1(G)\) and the Fourier algebra of a locally compact group G, and the convolution measure algebras of Taylor.) The simplest results occur when A has an approximate identity bounded by 1 and in this review we shall restrict ourselves to this case although the paper contains much more general material. The set \(UC_ r(A)\) of right uniformly continuous functionals on A is defined to be the set of right translates of elements of \(A^*\) by elements of A; it is a Banach subspace of \(A^*\) which is also left introverted, that is, invariant under the natural left action of \(A^{**}\) on \(A^*\). Here in addition, \(UC_ r(A)\) is weak* dense in \(A^*\), is invariant under the involution of \(A^*\), and contains the weakly almost periodic functionals on A.
Now let X be a subspace of \(A^*\) which is invariant under both the left and right actions of A and which is left introverted. If also X contains a weak* dense \(C^*\) subalgebra of \(A^*\), then the algebra of right multipliers of A is isometrically anti-isomorphic with the largest subalgebra of \(X^*\) which contains A as a right ideal. Again, if \(Y=X.A\) (the space of right translates of elements of X by elements of A), then the algebra of operators on X which commute with the right action of A on X is isomorphic with \(Y^*\); in particular, when \(X=A^*\), \(Y^*=UC_ r(A)^*.\)
The paper closes with some results on invariant means, in particular that \(A^*\) has a topological right invariant mean if and only if \(UC_ r(A)\) has a topological right invariant mean.