Irreducible positive linear maps on operator algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given.

     

Strong 3-commutativity preserving maps on standard operator algebras

Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ :A →A is said to be strong 3-commutativity preserving if [Φ(A), Φ(B)]3 = [A,B]3 for all A,B∈ A, where[A,B]3 is the 3-commutator of A,B defined by[A, B]3 = [[[A, B],B],B] with [A,B] = AB-BA. The main result in this paper is shown that.,if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h : A →F and a scalar λ∈ F with λ~4 = 1 such that Φ(A)=λ A+h(A)I for all A ∈ A.     

Additivity of Jordan maps on standard Jordan operator algebras

Let A be a standard Jordan operator algebra on a Hilbert space of dimension >1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ?:A→B satisfies

     

Regular abelian Banach algebras of linear maps of operator algebras

With H a complex Hilbert space we study regular abelian Banach subalgebras of the Banach algebra of bounded linear maps of B(H) into itself. If a ? b denotes the map x → axb, a, b, x ? B(H), it is shown that normalized positive maps in algebras of the form A ? A with A an abelian C¹-algebra, can be described by a generalized Bochner theorem.     

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（·）.     

Biseparating maps between operator algebras

We prove that a biseparating map between spaces B(E), and some other Banach algebras, is automatically continuous and a multiple of an algebra isomorphism.     

Multilinear maps on products of operator algebras

Let A₁,A₂,…,A_r be C∗-algebras with second duals A₁″,A₂″,…,A_r″, and let X be an arbitrary Banach space. Let be a bounded r-linear map, and denote by the Johnson-Kadison-Ringrose extension (i.e., the separately to continuous r-linear extension) of Γ. The problem of characterising those Γ for which Γ″ takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C∗-algebras, the ‘obvious’ extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C∗-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra.     

Additive maps derivable or Jordan derivable at zero point on nest algebras

Let AlgN be a nest algebra associated with the nest N on a (real or complex) Banach space X. Assume that every N∈N is complemented whenever N_-=N. Let δ:AlgN→AlgN be an additive map. It is shown that the following three conditions are equivalent: (1) δ is derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B) whenever AB=0; (2) δ is Jordan derivable at zero point, i.e., δ(AB+BA)=δ(A)B+Aδ(B)+Bδ(A)+δ(B)A whenever AB+BA=0; (3) δ has the form δ(A)=τ(A)+cA for some additive derivation τ and some scalar c. It is also shown that δ is generalized derivable at zero point, i.e., δ(AB)=δ(A)B+Aδ(B)-Aδ(I)B whenever AB=0, if and only if δ is an additive generalized derivation. Finer characterizations of above maps are given for the case dimX=∞.     

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      

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).     

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.     

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 .

     

Invariants of linear maps on matrix algebras

Let L be a linear map on the space of n×n matrices over a field. We determine the structure of the maps L that preserve commutativity. We also determine the structure of all linear maps on complex matrices that preserve the higher numerical range. The main tool is the Fundamental Theorem of Projective Geometry.     

Nonlinear *-Lie-type derivations on standard operator algebras

Let \(\mathcal{H}\) be an infinite dimensional complex Hilbert space and \(\mathcal{A}\) be a standard operator algebra on \(\mathcal{H}\) which is closed under the adjoint operation. It is shown that each nonlinear *-Lie-type derivation δ on \(\mathcal{A}\) is a linear *-derivation. Moreover, δ is an inner *-derivation as well.     

Completely rank nonincreasing linear maps on nest algebras

In this paper, the completely rank nonincreasing bounded linear maps on nest algebras acting on separable Hilbert spaces are characterized, and an affirmative answer to a problem posed by Hadwin and Larson is given for the case of such nest algebras.

     

Banach algebras associated with linear operator pencils

A direct relationship between the theory of pseudoresolvents and the spectral theory of linear operator pencils is established.     

Jordan higher derivable maps on triangular algebras by commutative zero products

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.