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.     

Group algebras in which complements are summands

It is shown that (1) every almost self-injective group algebra is self-injective and (2) if the group algebra KG is continuous, then G is a locally finite group. Furthermore, it follows that the following assertions are equivalent: a CS group algebra KG is continuous; KG is principally self-injective; the group G is locally finite. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 3–11, 2005.     

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L^1 (G) is its group algebra and L^∞(G) is the dual of L^1 (G).Lau has studied the bounded linear operators T:L^∞(G)→L^∞(G) which commute with convolutions and translations. For a subspace H of L^∞(G), we know that M(L^∞(G),H), the Banach algebra of all bounded linear operators on L^∞(G) into H which commute with convolutions, has been studied by Pyre and Lau. In this paper, we generalize these problems to L(K)^*, the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L^1(G) but also most of the semigroup algebras.Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however,we succeed in getting some interesting results.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

The center of the 3-dimensional and 4-dimensional Sklyanin algebras

LetA=A(E, ) denote either the 3-dimensional or 4-dimensional Sklyanin algebra associated to an elliptic curveE and a point E. Assume that the base field is algebraically closed, and that its characteristic does not divide the dimension ofA. It is known thatA is a finite module over its center if and only if is of finite order. Generators and defining relations for the centerZ(A) are given. IfS=Proj(Z(A)) andA is the sheaf ofO _S -algebras defined byA(S _(f))=A[f ^–1]₀ then the centerL ofA is described. For example, for the 3-dimensional Sklyanin algebra we obtain a new proof of M. Artin's result thatSpec L². However, for the 4-dimensional Sklyanin algebra there is not such a simple result: althoughSpec L is rational and normal, it is singular. We describe its singular locus, which is also the non-Azumaya locus ofA.     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

Spectral Synthesis and Topologies on Ideal Spaces for Banach*-Algebras

This paper continues the study of spectral synthesis and the topologies τ_∞ and τ_r on the ideal space of a Banach algebra, concentrating on the class of Banach ^*-algebras, and in particular on L¹-group algebras. It is shown that if a group G is a finite extension of an abelian group then τ_r is Hausdorff on the ideal space of L¹(G) if and only if L¹(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]⁻-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L¹(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τ_r is not Hausdorff on the ideal lattice of the Fourier algebra A(G).     

Automatic continuity over Moore groups

LetG be a Moore group, letB be a Banach algebra, and let :L ¹(G)B be a homomorphism. We show that is continuous if and only if its restriction to the center ofL ¹(G) is continuous. As a consequence, we obtain that (i) every homomorphism fromL ¹(G) orC ^*(G) onto a dense subalgebra of a semisimple Banach algebra, and (ii) every epimorphism fromC ^*(G) onto a Banach algebra is automatically continuous.     

Polynomial Identities of RA and RA2 Loop Algebras

Let F be an algebraically closed field of characteristic zero and L an RA loop. We prove that the loop algebra FL is in the variety generated by the split Cayley–Dickson algebra Z _F over F. For RA2 loops of type M(Dih(A), ^?1,g ₀), we prove that the loop algebra is in the variety generated by the algebra  ₃ which is a noncommutative simple component of the loop algebra of a certain RA2 loop of order 16. The same does not hold for the RA2 loops of type M(G, ^?1,g ₀), where G is a non-Abelian group of exponent 4 having exactly 2 squares.     

Lie Nilpotent Group Algebras and Upper Lie Codimension Subgroups

In this article we introduce the series of the upper Lie codimension subgroups of a group algebra KG of a group G over a field K. By means of this series we give a contribution to the conjecture cl _L (KG) = cl ^L (KG) when G belongs to particular classes of finite p-groups.     

The ordinary quivers of Hochschild extension algebras for self-injective Nakayama algebras

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule Hom_K(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ?-graded second Hochschild homology group HH₂(A) in the sense of Sköldberg.     

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].     

Automorphisms of Relatively Free Nilpotent Lie Algebras

Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N _Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL _n (?) and N _Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL _n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.     

Graded self-injective algebras “are” trivial extensions

For a positively graded artin algebra A=_n0A_n we introduce its Beilinson algebra b(A). We prove that if A is well-graded self-injective, then the category of graded A-modules is equivalent to the category of graded modules over the trivial extension algebra T(b(A)). Consequently, there is a full exact embedding from the bounded derived category of b(A) into the stable category of graded modules over A; it is an equivalence if and only if the 0-th component algebra A₀ has finite global dimension.     

Complements to the Almost Complete Tilting A (m)-Modules

Let A be a finite dimensional hereditary algebra over an algebraically closed field and A ^(m) be the mth replicated algebra of A. We prove that if T is a faithful almost complete tilting A ^(m)-module with pd _{A ^(m)} T ≤ m, then T has exactly m + 1 indecomposable nonisomorphic complements with projective dimensions at most m. Moreover, we give an explicit distribution of the complements to T.     

Simplicity of Skew Group Rings of Abelian Groups

Necessary and sufficient conditions for simplicity of a general skew group ring A ?_σ G are not known. In this article, we show that a skew group ring A ?_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ?_σ G is simple.     

The Semisimplicity of L^1 （K,w） of a Weighted Commutative Hypergroup K

In the present paper, it is shown that, for a locally compact commutative hypergroup K with a Borel measurable weight function w, the Banach algebra L^1 （K, w） is semisimple if and only if L^1 （K） is semisimple. Indeed, we have improved compact groups to the general setting of locally a well-krown result of Bhatt and Dedania from locally compact hypergroups.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

Ideal structure of Beurling algebras on [FC]− groups

It is proved that for an [FC]^? group G, the Beurling algebra L_ω¹(G) is 1-regular if and only if ω is non-quasianalytic. As an application the Wiener property is deduced. Further for a σ-compact [FD]^? group G, points in Prim₁L_ω¹(G) are shown to be spectral provided that ω satisfies Shilov's conditions.