Additive maps on standard operator algebras preserving parts of the spectrum

Let and be standard operator algebras on infinite dimensional complex Banach spaces X and Y, respectively, and let Φ be a unital additive surjection from  onto . We introduce thirteen parts of the spectrum for elements in  and , and prove that if Φ preserves any one of these parts of the spectrum, then it is either an isomorphism or an anti-isomorphism.     

Additive maps between standard operator algebras compressing certain spectral functions

We characterize the surjective additive maps compressing the spectral function Δ（·） between standard operator algebras acting on complex Banach spaces, where Δ（·） stands for any one of nine spectral functions σ（·）, σl（·）, σr（·）,σl（·） ∩ σr（·）, δσ（·）, ησ（·）, σap（·）, σs（·）, and σap（·） ∩ σs（·）.     

Additive maps preserving rank-one operators on nest algebras

In this article, we give a thorough discussion of additive maps between nest algebras acting on Banach spaces which preserve rank-one operators in both directions.     

Linear maps on operator algebras that preserve elements annihilated by a polynomial

In this paper some purely algebraic results are given concerning linear maps on algebras which preserve elements annihilated by a polynomial of degree greater than 1 and with no repeated roots and applied to linear maps on operator algebras such as standard operator algebras, von Neumann algebras and Banach algebras. Several results are obtained that characterize such linear maps in terms of homomorphisms, anti-homomorphisms, or, at least, Jordan homomorphisms.

     

自伴标准算子代数上强保持k-斜交换性映射的刻画

令H是维数大于2的复Hilbert空间,A是H上自伴标准算子代数.对于给定的正整数k≥1,H上算子A与B的k-斜交换子递推地定义为_*[A,B]_k=_*[A,_*[A,B]_k-1],其中_*[A,B]₀=B,_*[A,B]₁=AB-BA^*.设k≥4,φ是A上的值域包含所有一秩投影的映射.本文证明了φ满足_*[φ（A）,φ（B）]_k=_*[A,B]_k对任意A,B∈A都成立的充分必要条件是φ（A）=A对任意A∈A都成立,或φ（A）=-A对任意A∈A都成立.当k是偶数时后一情形不出现.     

Jordan zero-product preserving additive maps on operator algebras

Let Φ:A→B be an additive surjective map between some operator algebras such that AB+BA=0 implies Φ(A)Φ(B)+Φ(B)Φ(A)=0. We show that, under some mild conditions, Φ is a Jordan homomorphism multiplied by a central element. Such operator algebras include von Neumann algebras, C^∗-algebras and standard operator algebras, etc. Particularly, if H and K are infinite-dimensional (real or complex) Hilbert spaces and A=B(H) and B=B(K), then there exists a nonzero scalar c and an invertible linear or conjugate-linear operator U:H→K such that either Φ(A)=cUAU⁻¹ for all A∈B(H), or Φ(A)=cUA^∗U⁻¹ for all A∈B(H).     

Additive maps preserving Jordan zero-products on nest algebras

Let and be nest algebras associated with the nests and on Banach Spaces. Assume that and are complemented whenever N_-=N and M_-=M. Let be a unital additive surjection. It is shown that Φ preserves Jordan zero-products in both directions, that is Φ(A)Φ(B)+Φ(B)Φ(A)=0AB+BA=0, if and only if Φ is either a ring isomorphism or a ring anti-isomorphism. Particularly, all unital additive surjective maps between Hilbert space nest algebras which preserves Jordan zero-products are characterized completely.     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

Maps preserving peripheral spectrum of generalized Jordan products of operators

Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. For k ≥ 2, let(i1, i2,..., im) be a finite sequence such that {i1, i2,..., im} = {1, 2,..., k} and assume that at least one of the terms in(i1,..., im) appears exactly once. Define the generalized Jordan product T1 o T2 o ··· o Tk= Ti1Ti2··· Tim+ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 + A2A1, and the Jordan triple A1 A2 A3 + A3 A2 A1. Let Φ : A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ(Φ(A1) o ··· o Φ(Ak)) = σπ(A1 o ··· o Ak) for all A1,..., Ak,where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.     

On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Some functional equations on standard operator algebras

The main purpose of this paper is to prove the following result. Let H be a complex Hilbert space, let (H) be the algebra of all bounded linear operators on H, and let (H) ⊂ (H) be a standard operator algebra which is closed under the adjoint operation. Suppose that T: (H) → (H) is a linear mapping satisfying T(AA* A) = T(A)A* A − AT(A*)A + AA*T(A) for all A ∈ (H). Then T is of the form T(A) = AB + BA for all A ∈ (H), where B is a fixed operator from (H). A result concerning functional equations related to bicircular projections is proved      

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

An elementary proof of the characterization of isomorphisms of standard operator algebras

This study provides an elementary proof of the well-known fact that any isomorphism of standard operator algebras on normed spaces , respectively, is spatial; i.e., there exists a topological isomorphism such that for any . In particular, is continuous.

     

On the cohomology of joins of operator algebras

By analogy with the join in topology, the join A*B for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith (Amer. J. Math. 116 (1994) 541-561). Assuming that K is finite dimensional, they calculated the Hochschild cohomology groups for A*B with coefficients in L(K⊕H). We assume that A is a maximal abelian von Neumann algebra acting on H, A is a subalgebra of , and B is an ultraweakly closed subalgebra of M_n(A) containing A⊗1_n. We show that B may be decomposed into a finite sum of free modules. In this context, we redefine the join of A and B, generalize the calculations of Gilfeather and Smith, and calculate , for all m?0.     

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.