Linear maps of holomorphic functions on commutative algebras

Generalized function theory on finite dimensional linear associative algebras over the fieldo freals has pro- duced many forms o f generalized regular1ty Of particularin - terest are the forms o fregularity known as S-regularity and F- regularity. It is well known that functions regular in one sense need not be regular in the other. Here we prove a transformation theorem which allows the formation o f F-regular functions from S- regular functions. Furthermore, a classi flcation theorem is proved that determines the algebras for which this transformation property holds.     

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.

     

Linear maps preserving zero products on nest subalgebras of von Neumann algebras

In this paper, it is proved that every surjective linear map preserving identity and zero products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is an isomorphism; and that every surjective weakly continuous linear map preserving identity and zero Jordan products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is either an isomorphism or an anti-isomorphism.     

Elementary maps on nest algebras

Let A₁, A₂ be algebras and let M:A₁→A₂, M^∗:A₂→A₁ be maps. An elementary map of A₁×A₂ is an ordered pair (M,M^∗) such that

     

k-Commuting maps on triangular algebras

In this paper, k-commuting maps on certain triangular algebras are determined. As an application we show that every k-commuting map on an upper triangular matrix algebra over a unital commutative ring of 2-torsion free or a nest algebra is proper.     

Jordan maps on triangular algebras

Let T be a triangular algebra and R′ be an arbitrary ring. Suppose that M:T→R^′ and M^∗:R^′→T are surjective maps such that

     

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.     

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.

     

Additivity of Jordan maps on triangular algebras

The aim of this article is to prove a result on the additivity of Jordan maps on triangular algebras. As a consequence the additivity of Jordan maps on upper triangular matrix algebras over a faithful commutative ring of 2-torsion free is determined.     

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.     

Polynomial preserving maps on certain Jordan algebras

LetB andQ be associative algebras and letS be a Jordan subalgebra ofB. Letf(x ₁,…,x _m ) be a (noncommutative) multilinear polynomial such thatS is closed underf. Letα:S→Q be anf-homomorphism in the sense that it is a linear map preservingf. Under suitable conditions it is shown thatα is essentially given by a ring homomorphism. An analogous theorem forf-derivations is also proved. The proofs rest heavily on results concerning functional identities andd-freeness. The second author was partially supported by a grant from the Ministry of Science of Slovenia.     

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.     

Interval maps from Cuntz-Krieger algebras

Markov interval maps f naturally produce transition aperiodic 0-1 matrices with extra features. We characterize the 0-1 matrices that can be realized as Markov transition matrices of interval maps, and parametrize the orbit representations (yielded in Correia Ramos et al. (2008) [2]) of the Cuntz-Krieger algebra OA obtained from interval maps with the same matrix A.     

Lipschitz algebras and peripherally-multiplicative maps

Let X be a compact metric space and let Lip（X） be the Banach algebra of all scalar- valued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip（X）, σπ（f） denotes the peripheral spectrum of f. We state that any map Φ from Lip（X） onto Lip（Y） which preserves multiplicatively the peripheral spectrum：
σπ（Φ（f）Φ（g）） = σπ（fg）, A↓f, g ∈ Lip（X）
is a weighted composition operator of the form Φ（f） = τ· （f °φ） for all f ∈ Lip（X）, where τ ： Y → {-1, 1} is a Lipschitz function and φ ： Y→ X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip（X）-algebras is of the form above.     

Automorphisms and involutions on incidence algebras

This article consists of a survey of known results on automorphisms and involutions (anti-automorphisms of order two) of incidence algebras and some new results on involutions of these algebras.     

On the number of $$k$$ k -compositions $$n$$ n satisfying certain coprimality conditions

Acta Mathematica Hungarica - We generalize the asymptotic estimates by Bubboloni, Luca and Spiga [2] on the number of $$k$$ -compositions of n satisfying some coprimality conditions. We...     

Characterization of incidence algebras

We develop an internal characterization of the incidence algebra of a lower anite quasi-ordered set and dualize it to obtain a characterization of the corresponding coalgebra. The algebra characterization consists of two axioms concerning the homomorphic images of the algebra and another axiom stipulating the existence of a faithful module over the algebra which has a distributive lattice of submodules.