AF C~*-代数中的子代数上的保幂等映射和局部导子

证明了从AFC-代数E中的子代数A到任意赋范代数B上的范数连续保幂等映射是Jordan同态,以及从A到任意赋范E-双模M上的局部导子是导子,从而推广了Crist关于局部导子的结果．     

CSL代数上的Lie导子

证明了不相关的有限宽度CSL代数上的每一个Lie导子都是内导子与作用在交换子上为零的中心值线性映射之和.     

COMPLETELY BOUNDED COHOMOLOGY OF NON-SELFADJOINT OPERATOR ALGEBRAS

The authors prove that all n-th completely bounded cohomology groups of a nest algebra T(N) acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T(N)-bimodule containing the nest algebra. They also prove that Hcbn(A, M)(?)Hcbn(A, A) for all n≥1 and a CSL algebra A with an ultraweakly closed A-bimodule M containing A.     

Commutants modulo the compact operators of certain CSL algebras II

The commutant modulo compacts, or essential commutant, of a reflexive algebra with commutative subspace lattice is a C^* algebra which is the sum of the compact operators in L(H) and a C^* subalgebra of the core. We give a characterization of the essential commutant of a separably acting CSL algebra in terms of properties of the spectral measure of an operator in the intersection of the essential commutant and the core. This is used to determine some sufficient conditions on the lattice for when the essential commutant is norm generated by the projections it contains.     

Jordan triple derivations on J-subspace lattice algebras

Let L be a J-subspace lattice on a Banach space X and Alg L the associated J-subspace lattice algebra. Let A be a standard operator subalgebra (i.e., it contains all finite rank operators in AlgL) of AlgL and M■B(X) the Alg L-bimodule. It is shown that every linear Jordan triple derivation from A into M is a derivation, and that every generalized Jordan (triple) derivation from A into M is a generalized derivation.     

All-derivable points of operator algebras

Let A be an operator subalgebra in B(H), where H is a Hilbert space. We say that an element Z∈A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ(ST)=φ(S)T+Sφ(T) for any S,T∈A with ST=Z) is a derivation. In this paper, we show that every invertible operator in the nest algebra is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2×2 upper triangular matrixes is an all-derivable point of the algebra.     

Strongly Flat and PO-Flat S-Posets

Abstract

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves.     

The Euler-Maclaurin sum formula for an inner derivation

An operator form of the Euler-Maclaurin sum formula is obtained, expressing the sum of the Euler-Maclaurin infinite series in an inner derivation as the difference between a summation operator and an inner antiderivation, on a closed subalgebra of a Banach algebra.     

Dilations of operator-valued measures with bounded p-variations and framings on Banach spaces

The dilations for operator-valued measures (OVMs) and bounded linear maps indicate that the dilation theory is in general heavily dependent on the Banach space nature of the dilation spaces. This naturally led to many questions concerning special type of dilations. In particular it is not known whether ultraweakly continuous (normal) maps can be dilated to ultraweakly continuous homomorphisms. We answer this question affirmatively for the case when the domain algebra is an abelian von Neumann algebra. It is well known that completely bounded Hilbert space operator valued measures correspond to the existence of orthogonal projection-valued dilations in the sense of Naimark and Stinespring, and OVMs with bounded total variations are completely bounded but not the vice-versa. With the aim of classifying OVMs from the dilation point of view, we introduce the concept of total p-variations for OVMs. We prove that any completely bounded OVM has finite 2-variation, and any OVM with finite p-variation can be dilated to a (but usually non-Hilbertian) projection-valued measure of the same type. With the help of framing induced OVMs, we prove that conventional minimal dilation space of a non-trivial framing contains $c_0$, then does not have bounded p-variation.     

Lie algebras of infinitesimal norm isometries

Given a norm on a finite dimensional vector space V, we may consider the group of all linear automorphisms which preserve it. The Lie algebra of this group is a Lie subalgebra of the endomorphism algebra of V having two properties: (1) it is the Lie algebra of a compact subgroup, and (2) it is “saturated” in a sence made precise below. We show that any Lie subalgebra satisfying these conditions is the Lie algebra of the group of linear automorphisms preserving some norm. There is an appendix on elementary Lie group theory.     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C^∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C^∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.     

Some subalgebras of indefinite type Kac-Moody Lie algebras

We show that every Kac-Moody Lie algebra of indefinite type contains a subalgebra with a Dynkin diagram having two adjacent vertices whose edge labels multiply to a number greater than or equal to five. Consequently, every Kac-Moody algebra of indefinite type contains a subalgebra of strictly hyperbolic type, and a free Lie algebra of rank two.     

The structure of module derivations of Banach algebras of differentiable functions

This paper studies the structure and continuity of derivations of the Banach algebra Cⁿ(I) of n times continuously differentiable functions on an interval I into Banach Cⁿ(I)-modules. The structure of derivations into finite dimensional modules is completely determined. The question of when an arbitrary derivation splits into the sum of continuous and singular parts is discussed. An example is constructed of a derivation of C¹(I) which is discontinuous on every dense subalgebra.     

Nonlinear *-Lie derivations of standard operator algebras

Let ? be an infinite dimensional complex Hilbert space and 𝒜 be a standard operator algebra on ? which is closed under the adjoint operation. We prove that every nonlinear *-Lie derivation δ of 𝒜 is automatically linear. Moreover, δ is an inner *-derivation.     

导子的范数估计

本文给出了因子 ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上导子的范数估计．利用此结果得到一个距离公式,并证明了因子 ｖｏｎ Ｎｅｕｍａｎｎ代数中的任何套子代数都具有ＡＩＰ（ｒ, ｓ）性质     

On some types of maximal abelian subalgebras

Let H be a Hilbert space and B(H) the algebra of all bounded linear operators on H. It is known that there are two kinds of maximal abelian sub-algebras in B(H), to one of which there exists a unique faithful normal projection of norm one from B(H) and to the other any projection of norm one is singular. Any maximal abelian subalgebra A contains a projection e such that Ae is a maximal abelian subalgebra of B(eH) of the first kind and A(1 − e) is the one of the second kind in B((1 − e)H). This will be generalized to an arbitrary von Neumann algebra together with the existence problem of those kinds of maximal abelian subalgebras.     

Generalized Jordan derivations on triangular matrix algebras

In this note, we prove that every generalized Jordan derivation from the algebra of all upper triangular matrices over a commutative ring with identity into its bimodule is the sum of a generalized derivation and an antiderivation.     

The structure of algebraic jordan algebras without nonzero nilpotent elements

An algebraic, linear Jordan algebra without nonzero nil-potent elements is proved to be a subdirect sum of prime Jordan algebras each of which has finite capacity or contains simple subalgebras of arbitrary capacity. If in addition the base field has nonzero character-istic or the algebra satisfies a polynomial identity, then each of the summands is determined to be simple of finite capacity. Further, it is proved that algebraic, PI Jordan algebras without nonzero nilpotent elements are locally finite in the sense that any finitely generated subalgebra has finite capacity.