Anthony To-Ming Lau

The second duals of Beurling algebras

Introduction Definitions and preliminary results Repeated limit conditions Examples Introverted subspaces Banach algebras of operators Beurling algebras The second dual of

Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

\ell^1(G,\omega)

Topological centers of certain dual algebras

Algebras on discrete, Abelian groups Beurling algebras on

Uniformly continuous functionals on the Fourier algebra of any locally compact group

\mathbb{F}_2

Semigroup of nonexpansive mappings on a Hilbert space

Topological centres of duals of introverted subspaces The second dual of

Ideals with bounded approximate identities in Fourier algebras

L^1(G,\omega)

On -amenability of Banach algebras

Derivations into second duals Open questions Bibliography Index Index of symbols.

Inner amenable locally compact groups

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map ( a , b ) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G ) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC ( G ) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f e C(G ) such that the map a → l a f of G into C(G ) is continuous when C(G ) has a norm topology, where ( l a f )( x ) = f (ax) (a, x e G ). Then LUC ( G ) is a closed subalgebra of C(G ) invariant under translations. Furthermore, if m e LUC ( G )*, f e LUC ( G ), then the function is also in LUC ( G ). Hence we may define a product for n, m e LUC( G )*. LUC ( G )* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l ∞ ( G )* = l 1 ( G )** to LUC ( G )*. We refer the reader to [ 1 ] and [ 10 ] for more details.

The second conjugate algebra of the Fourier algebra of a locally compact group

Let A be a Banach algebra with a bounded approximate identity. Let Z1 and Z2 be, respectively, the topological centers of the algebras A** and (A*A)*. In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras L1 (G) and A(G), we study the sets Z1, Z2, the relations between them and with several other subspaces of A** or A*.

Invariant means and fixed point properties for non-expansive representations of topological semigroups

Let G be any locally compact group. Let VN(G) be the von Neumann algebra generated by the left regular representation of G. We study in this paper the closed subspace UBC(G) of VN(G) consisting of the uniformly continuous functionals as defined by E. Granirer. When G is abelian, UBC(G) isprecisely the bounded uniformly continuous functions on the dual group G. We prove among other things that if G is amenable, then the Banach algebra UBC(G)* (with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore2 UBC(G)* is commutative if and only if G is discrete. We characterize W(G), the weakly almost periodic functionals, as the largest subspace X of VN(G) for which the Arens product makes sense on X* and X* is commutative. We also show that if G is amenable, then for certain subspaces Y of VN(G) which are invariant under the action of the Fourier algebra A(G), the algebra of bounded linear operators on Y commuting with the action of A(G) is isometric and algebra isomorphic to X* for some X C UBC(G).