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1.
Let G be a locally compact group, and ZL1(G) be the centre of its group algebra. We show that when G is compact ZL1(G) is not amenable when G is either non-abelian and connected, or is a product of infinitely many finite non-abelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G).  相似文献   

2.
It is shown that for amenable groups, all finite-dimensional extensions of Ap(G) algebras split strongly. Furthermore, each extension of Ap(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GLn(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of Ap(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of Ap(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of Bp(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between Ap(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.  相似文献   

3.
We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of ?1-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.  相似文献   

4.
We study power boundedness in the Fourier and Fourier-Stieltjes algebras, A(G) and B(G), of a locally compact group G as well as in some other commutative Banach algebras. The main results concern the question of when all elements with spectral radius at most one in any of these algebras are power bounded, the characterization of power bounded elements in A(G) and B(G) and also the structure of the Gelfand transform of a single power bounded element.  相似文献   

5.
Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with , we use the operator space structure on to equip the Figà-Talamanca-Herz algebra Ap(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p?q?2 or 2?q?p and amenable G, the canonical inclusion Aq(G)⊂Ap(G) is completely bounded (with cb-norm at most , where is Grothendieck's constant). As an application, we show that G is amenable if and only if Ap(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan.  相似文献   

6.
We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.  相似文献   

7.
In this paper we study the ideal amenability of Banach algebras. LetA be a Banach algebra and letI be a closed two-sided ideal inA, A isI-weakly amenable ifH 1(A,I *) = {0}. Further,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenability of Banach algebras on locally compact groups.  相似文献   

8.
We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.  相似文献   

9.
Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Generalizing the notion of spectral sets in Δ(A), the considerably larger class of weak spectral sets was introduced and studied in [C.R. Warner, Weak spectral synthesis, Proc. Amer. Math. Soc. 99 (1987) 244-248]. We prove injection theorems for weak spectral sets and weak Ditkin sets and a Ditkin-Shilov type theorem, which applies to projective tensor products. In addition, we show that weak spectral synthesis holds for the Fourier algebra A(G) of a locally compact group G if and only if G is discrete.  相似文献   

10.
The space L p (G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces L p (G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra L p (G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.  相似文献   

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