Projections on weak*-closed subspace of dual Banach algebras

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on M _a (S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.     

Strongly and weakly almost periodic linear maps on semigroup algebras

Let S be a foundation locally compact topological semigroup. Two new topologies τ _c and τ _w are introduced on M _a (S)^*. We introduce τ _c and τ _w almost periodic functionals in M _a (S)^*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M _a (S)^*, it is proved that T _f ∈ℬ(M _a (S),M _a (S)^*) is strong almost periodic if and only if f is τ _c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M _a (S)) has the semiright invariant isometry property, it is shown that the set of τ _w -almost periodic functionals has a topological left invariant mean.     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

Completely almost periodic functionals

Using the notion of complete compactness introduced by H.  Saar, we define completely almost periodic functionals on completely contractive Banach algebras. We show that, if (M, Γ) is a Hopf–von Neumann algebra with M injective, then the space of completely almost periodic functionals on M _* is a C*-subalgebra of M.     

On the Space of Weakly Almost Periodic Functionals and Its Amenability

Let S be a locally compact semitopological semigroup with measure algebra M(S), M₀(S) the set of all probability measures in M(S) and WF(S) the space of weakly almost periodic functionals on M(S)^*. Assuming that M₀(S) has the semiright invariant isometry property, it is shown that WF(S) has a topological left invariant mean (TLIM) whenever the center of M₀(S) is nonempty; in particular if either the center of S is nonempty or S has a left identity, then WF(S) has a TLIM. Finally if, for each M₀(S), the mapping v v * of M₀(S) into itself is surjective and the center of M₀(S) is nonempty, then WF(S) has a TLIM. We also generalize some results from discrete case to topological one.AMS Subject Classification (1991): 43A07     

On a Banach Algebra Generated in its Multiplier Space and Applications to Locally Compact Groups

In this paper, we present a general version of the algebra A_M(G) which was introduced by B. Forrest. Indeed, for a faithful commutative Banach algebra A, we embed it in ?(A), the multiplier algebra of A, and obtain Banach algebra A_M. Then, we study the spaceability of A_M? A and A_M (G) ? ?A(G). These results give some characterizations of compactness and discreteness of locally compact groups. Also, we show that A_M(G) is an ideal in its second dual if and only if G is discrete. Finally, we study the BSE-property of A_M(G).

     

The Amenability of Measure Algebras

In this paper we shall prove that the measure algebra M(G) ofa locally compact group G is amenable as a Banach algebra ifand only if G is discrete and amenable as a group. Our contributionis to resolve a conjecture by proving that M(G) is not amenablein the case where the group G is not discrete. Indeed, we shallprove a much stronger result: the measure algebra of a non-discrete,locally compact group has a non-zero, continuous point derivationat a certain character on the algebra.     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a *-representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π:A(G)→B(H) is similar to a *-representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a *-representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].     

M0(G)-boundaries are M(G)-boundaries

Let M(G) denote the convolution algebra of finite regular complex-valued Borel measures on a locally compact abelian group G, and let M₀(G) be the ideal consisting of those measures whose Fourier-Stieltjes transforms vanish at infinity. Then there is a natural inclusion of the maximal ideal space Δ₀ of M₀(G) in the maximal ideal space of M(G). The main result states that any subset of Δ₀ which is a boundary for M₀(G) is a boundary for M(G). An immediate corollary is that the ?ilov boundary of M₀(G) is dense in the ?ilov boundary of M(G).     

Hyper-Tauberian algebras and weak amenability of Figà-Talamanca-Herz algebras

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A_p(G) of a locally compact group G for p∈(1,∞). We show that A_p(G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A_p(G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A_p(G) is operator weakly amenable.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

Every semitopological semigroup compactification of the group H + [0,1] is trivial

Let G=H ₊ [0, 1] be the topological group of all orientation-preserving selfhomeomorphisms of the closed interval [0,1] endowed with the usual compact open topology. We show that every weakly almost periodic function on G is constant. Consequently, G does not admit nontrivial (weakly) continuous representations by linear isometries in reflexive Banach spaces. September 6, 1999     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.     

Witt groups and the elementary type conjecture

Let A be an algebra with involution * over a field F of characteristic zero and Id(A, *) the ideal of the free algebra with involution of *-identities of A. By means of the representation theory of the hyperoctahedral group Z ₂wrS _n we give a characterization of Id(A, *) in case the sequence of its *-codimensions is polynomially bounded. We also exhibit an algebra G ₂ with the following distinguished property: the sequence of *-codimensions of Id(G ₂, *) is not polynomially bounded but the *-codimensions of any T-ideal U properly containing Id(G ₂, *) are polynomially bounded.     

A remark on discontinuous translation invariant functionals onL p (G) for certain compact groupsG

We show some criterion for the existence of discontinuous translation invariant linear functionals onC (G) andL _p (G) for a compact groupG.     

Multiplication operators on sobolev disk algebra

In this paper,we study the algebra consisting of analytic functions in the Sobolev space W~(2,2) (D) (D is the unit disk),called the Sobolev disk algebra,explore the properties of the multiplication operators M_f on it and give the characterization of the corn- mutant algebra A′(M_f) of M_f.We show that A′(M_f) is commutative if and only if M_f~* is a Cowen-Douglas operator of index 1.     

On One Class of Matrix Topological *-Algebras

We study a matrix algebra M _n(U), where U is a commutative topological nuclear entire (bounded, analytic) *-algebra. We prove that M _n(U) is also a topological nuclear entire (bounded, analytic) *-algebra.     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.