Closed Lie Ideals Research Articles

Results on Lie ideals of prime rings with homoderivations

Let R be a prime ring of characteristic not 2 and U be a noncentral square closed Lie ideal of R. An additive mapping H on R is called a homoderivation if H(xy) = H(x)H(y)+H (x) y+xH(y) for all x, y ∈ R. In this paper we investigate homoderivations satisfying certain differential identities on square closed Lie ideals of prime rings.

Square closed Lie ideals and multiplicative (Generalized) (α, β) reverse derivation of prime rings

Let R be a 2-torsion free prime ring, U non-zero square closed Lie ideal of R, α be anti-automorphism of R and β be automorphism of R. A mapping F: R → R is called a multiplicative (generalized) (α, β) reverse derivation if F(xy) = F(y)α (x) + β (y)d(x) for all x, y ∈R where d: R→ R is any map (not necessarily additive). The purpose of this paper is to introduce the concept of multiplicative left reverse α-centralizer and investigate some identities of a 2-torsion free prime ring admitting a multiplicative (generalized) (α, β) reverse derivation.

On closed Lie ideals and center of generalized group algebras

For any locally compact group G and any Banach algebra A, a characterization of the closed Lie ideals of the generalized group algebra L1(G,A) is obtained in terms of left and right actions by G and A. In addition, when A is unital and G is an [SIN] group, we show that the center of L1(G,A) is precisely the collection of all center valued functions which are constant on the conjugacy classes of G. As an application, we establish that Z(L1(G)⊗γA)=Z(L1(G))⊗γZ(A), for a class of groups and Banach algebras. And, prior to these, for any finite group G, the Lie ideals of the group algebra C[G] are identified in terms of some canonical spaces determined by the irreducible characters of G.

CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

AbstractFor a locally compact Hausdorff spaceXand aC*-algebraAwith only finitely many closed ideals, we discuss a characterization of closed ideals ofC0(X,A) in terms of closed ideals ofAand a class of closed subspaces ofX. We further use this result to prove that a closed ideal ofC0(X)⊗minAis a finite sum of product ideals. We also establish that for a unitalC*-algebraA,C0(X,A) has the centre-quotient property if and only ifAhas the centre-quotient property. As an application, we characterize the closed Lie ideals ofC0(X,A) and identify all the closed Lie ideals ofHC0(X)⊗minB(H),Hbeing a separable Hilbert space.

On closed Lie ideals of certain tensor products of C∗‐algebras II

AbstractWe identify all closed Lie ideals of and , where is either the Haagerup tensor product, the Banach space projective tensor product or the operator space projective tensor product, A is any simple ‐algebra, B is any ‐algebra with one of them admitting no tracial states, and H is an infinite dimensional separable Hilbert space. Further, generalizing a result of Marcoux, we also identify all closed Lie ideals of , where A is a simple ‐algebra with at most one tracial state and B is any commutative ‐algebra.

A GENERALIZED STUDY ON CLOSED LIE IDEALS WITH (α,α)-DERIVATIONS

In this paper, we study square closed Lie ideals of semi-prime rings with generalized derivations and investigate commutative properties of square closed Lie ideals under different conditions. Also, we take generalized derivation with determined derivation on prime ring and prove that is commuting on Lie ideal. Finally, we reach the corollaries about commutativity of prime rings by using the theorems we prove.

GENERALIZED REVERSE DERIVATIONS ON CLOSED LIE IDEALS

In this study, we investigate commutavity of prime ring R with generalized reverse derivations F and G. Also, we proved that if L is square closed Lieideal, then L is contained in center Z (R) under given conditions in theorems.

On closed Lie ideals of certain tensor products of ‐algebras

AbstractFor a simple ‐algebra A and any other ‐algebra B, it is proved that every closed ideal of is a product ideal if either A is exact or B is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi‐central approximate identity is described in terms of the commutator of the Banach algebra. If α is either the Haagerup norm, the operator space projective norm or the ‐minimal norm, then this allows us to identify all closed Lie ideals of , where A and B are simple, unital ‐algebras with one of them admitting no tracial functionals, and to deduce that every non‐central closed Lie ideal of contains the product ideal . Closed Lie ideals of are also determined, A being any simple unital ‐algebra with at most one tracial state and X any compact Hausdorff space. And, it is shown that closed Lie ideals of are precisely the product ideals, where A is any unital ‐algebra and α any completely positive uniform tensor norm.

Generalized (Jordan) left derivations on rings associated with an element of rings

In this paper, we introduce a new notion of generalized (Jordan) left derivation on rings as follows: let R be a ring, an additive mapping F : R → R is called a generalized (resp. Jordan) left derivation if there exists an element w ∈ R such that F(xy) = xF(y) + yF(x) + yxw (resp. F(x 2) = 2xF(x) + x 2 w) for all x, y ∈ R. Then, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.

Certain Additive Maps on (m+n+1)-Power Closed Lie Ideals

Let R be a prime ring and m, n be fixed non-negative integers such that m+n ≠ 0. Suppose L is an (m+n+1)-power closed Lie ideal, and this means um+n+1∈ L for all u ∈ L. If char R=0 or a prime p > 2(m+n), we characterize the additive maps d: L → R satisfying d(um+n+1)=(m+n+1)umd(u)un (resp., d(um+n+1)=umd(u)un) for all u ∈ L.

Topological Radicals and Frattini Theory of Banach Lie Algebras

In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codimensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via subspace-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of closed Lie subalgebras of finite codimension (closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) produce different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained.

Certain additive maps on m-power closed Lie ideals

Let R be a prime ring with extended centroid C and m a fixed positive integer > 1. A Lie ideal L of R is called m-power closed if \({u^m \in L}\) for all \({u \in L}\) . We prove that if char R = 0 or a prime p > m, then every non-central, m-power closed Lie ideal L of R contains a nonzero ideal of R except when dimCRC = 4, m is odd, and \({u^{m-1} \in C}\) for all \({u \in L}\) . Moreover, the additive maps d : L → R satisfying d(um) = mum-1d(u) (resp. d(um) = um-1d(u)) for all \({u \in L}\) are completely characterized if char R = 0 or a prime p > 2(m − 1).

On Lie submodules and tensor algebras

Let be a bimodule over an algebra B, and let be the algebra of operators on generated by all operators x ↦ ax − xa, where a ∈ B. We show that in many (but not all) cases, consists of all elementary operators x ↦ Σa i xb i whose coefficients satisfy the conditions Σ i a i b i = Σ i b i a i = 0. Analogs of these results are proved for Banach bimodules over Banach algebras. Using them, we obtain the description of the structure of closed Lie ideals for a class of Banach algebras and prove some density theorems for Lie algebras of operators on Hilbert spaces.

Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L 0 that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L 0 has finite codimension in L and there are Lie subalgebras L 0 = L 0 ⊂ L 1 ⊂ ⋯ ⊂ L p = L such that L i + 1 = L i + [ L i , L i + 1 ] for all i; (2) L 0 is a Lie ideal of L and dim ( L 0 ) = ∞ . These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.

CONJUGATION-INVARIANT SUBSPACES AND LIE IDEALS IN NON-SELFADJOINT OPERATOR ALGEBRAS

It is shown that a weakly closed subspace S of a nest algebra A is closed under conjugation by invertible elements in A, that is, a−1Sa=S if and only if S is a Lie ideal. A similar result holds for not-necessarily-closed subspaces of algebras of infinite multiplicity. An explicit characterisation of weakly closed Lie ideals in a nest algebra is given.

Lie ideals in triangular operator algebras

We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if 2 is a closed Lie ideal of the triangular operator algebra A, then there exist a closed associative ideal IC and a closed subalgebra Or of the diagonal A ni A* so that K C 2 C IC + Oc. INTRODUCTION Let A be an associative complex algebra. Under the Lie multiplication [x, y] xy yx, A becomes a Lie algebra. A Lie ideal in A is a linear manifold Z in A for which [a, k] c ? for every a E A and k c 2. In many instances, there is a close connection between the Lie ideal structure and the associative ideal structure of A. This connection has been investigated for prime rings in [6], in [3] for B($5) the set of bounded operators on a Hilbert space Si, and in [10] for certain von Neumann algebras. (See also [9, 4, 11].) In this paper we pursue this line of investigation for two classes of triangular operator algebras, namely nest algebras anld triangular UHF algebras. The authors would like to thank Frank A. Zorzitto for many helpful conversations. 1. WEAKLY CLOSED LIE IDEALS IN NEST ALGEBRAS Recall that a nest JK on a Hilbert space Si is a chain of closed subspaces of Si which is closed under the operations of arbitrary intersections and closed linear spans, and which includes {O} and S5. The nest algebra T(KJ) is the algebra of all operators on Se leaving every member of JK invariant. This is always closed in the weak operator topology. The diagonal D(KJ) of a nest algebra T(JK) is the von Neumann subalgebra 7(K) n T(K)*. If E, F E KV with E < F, then F E is called an interval of the nest. The nonzero minimal intervals are called atoms. A nest is atomic if the atoms of the nest span 55. We refer the reader to [1] for more information on nest algebras. Our main result, Theorem 12, shows that for every weakly closed Lie ideal Z in 7(JK), there exist a corresponding weakly closed associative ideal IC and a von Neumann subalgebra SK of O(A/) such that

On the closed Lie ideals of certainC *-algebras

In this article we determine the closed Lie ideals of a uniformly hyperfiniteC*-algebra, and of the tensor product of such an algebra withC(X), the space of continuous functions on a compact, Hausdorff space. This is done by localizing the Lie ideals in algebras of the form\(\mathbb{A} = \mathbb{M}_n \otimes \mathbb{B}\), where\(\mathbb{B}\) is an algebra over a field of characteristic not equal to 2.

Closed Lie Ideals in Operator Algebras

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.