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.

     

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.     

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.     

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.     

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.     

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.     

Super-biderivations and super-commuting maps on the super Galilean conformal algebras

Let 𝔤 be the super Galilean conformal algebra. In this paper, we first determine all the super-skewsymmetric super-biderivations of 𝔤. In particular, we find that there exist non-inner super-biderivations of 𝔤. Based on the result of super-biderivations, we show that all the linear super-commuting maps on 𝔤 are standard.     

Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.     

Cuntz-Krieger algebras representations from orbits of interval maps

Let f be an expansive Markov interval map with finite transition matrix Af. Then for every point, we yield an irreducible representation of the Cuntz-Krieger algebra OAf and show that two such representations are unitarily equivalent if and only if the points belong to the same generalized orbit. The restriction of each representation to the gauge part of OAf is decomposed into irreducible representations, according to the decomposition of the orbit.     

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（·）.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

All-derivable points in nest algebras

Suppose that A is an operator algebra on a Hilbert space H. An element V in A is called an all-derivable point of A for the strong operator topology if every strong operator topology continuous derivable mapping φ at V is a derivation. Let N be a complete nest on a complex and separable Hilbert space H. Suppose that M belongs to N with {0}≠M≠H and write for M or M^⊥. Our main result is: for any with , if is invertible in , then Ω is an all-derivable point in for the strong operator topology.