Involutions and Trivolutions in Algebras Related to Second Duals of Group Algebras

We define a trivolution on a complex algebra $A$ as a non-zero conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized inverse of itself, that is, $\tau^3=\tau$. We give several characterizations of trivolutions and show with examples that they appear naturally on many Banach algebras, particularly those arising from group algebras. We give several results on the existence or non-existence of involutions on the dual of a topologically introverted space. We investigate conditions under which the dual of a topologically introverted space admits trivolutions

Almost periodic functionals and finite-dimensional representations

Abstract We show that if $A$ is a $\mathrm{C}^*$-algebra and $\lambda \in A^*$ is a nonzero almost periodic functional which is a coordinate functional of a topologically irreducible involutive representation $\pi$, then $\dim\pi \lt \infty$. We introduce the RFD transform $\alpha _A : A \rightarrow U(A)$ of a Banach algebra $A$ and establish its universal property. We show that if $A$ has a bounded two-sided approximate identity, then almost periodic functionals on $A$ which are limits of coordinate functionals of finite-dimensional representations have lifts to almost periodic functionals on $U(A)$. Other connections with almost periodicity and harmonic analysis are also discussed

Representations of Banach algebras subordinate to topologically introverted spaces

Let A be a Banach algebra, X a closed subspace of A∗, Y a dual Banach space with predual Y∗, and π a continuous representation of A on Y. We call π subordinate to X if each coordinate function πy,λ ∈ X, for all y ∈ Y, λ ∈ Y∗. If X is topologically left (right) introverted and Y is reflexive, we show the existence of a natural bijection between continuous representations of A on Y subordinate to X, and normal representations of X∗ on Y. We show that if A has a bounded approximate identity, then every weakly almost periodic functional on A is a coordinate function of a continuous representation of A subordinate to WAP(A). We show that a function f on a locally compact group G is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of L1(G), associated to some unitary representation of G. We generalize the latter