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 π<inf>y,λ</inf> ∈ 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 L<sup>1</sup>(G), associated to some unitary representation of G. We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in LUC(A) are all coordinate functions of some norm continuous representation of A on a dual Banach space.
|Journal||Transactions of the American Mathematical Society|
Filali, M., Neufang, M, & Monfared, M.S. (M. Sangani). (2015). Representations of Banach algebras subordinate to topologically introverted spaces. Transactions of the American Mathematical Society, 367(11), 8033–8050.