Spectrum Preserving Linear Maps Between Banach Algebras

Polynomially Spectrum-preserving Maps between Commutative Banach Algebras

Let A and B be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials p which satisfy the following property: a map T from A onto B such that the equality σ(p(Tf, T g)) = σ(p(f, g)), f, g ∈ A holds is an algebra isomorphism.

Invertibility Preserving Linear Maps of Banach Algebras

This talk discusses a conjecture of R. V. Kadison and myself. Our conjecture is that each one-to-one linear map of one unital C*-algebra onto another that preserves the identity is a Jordan isomorphism if it maps the invertible elements of the first C*-algebra onto the invertible elements of the other C*-algebra. Connections are shown between this conjecture and Cartan’s uniqueness theorem. 1. ...

Multiplicatively Spectrum-preserving and Norm-preserving Maps between Invertible Groups of Commutative Banach Algebras

Let A and B be unital semisimple commutative Banach algebras and T a map from the invertible group A onto B. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on T : (1) σ(TfTg) = σ(fg); (2) σπ(TfTg−α)∩σπ(fg−α) 6= ∅; (3) r(TfTg−α) = r(fg−α) hold for some non-zero complex number α and for every f, g ∈ A, where σ(·) (resp. σπ(·)) denotes the (resp. peripheral) ...

A Note on Spectrum Preserving Additive Maps on C*-Algebras

Mathieu and Ruddy proved that if be a unital spectral isometry from a unital C*-algebra Aonto a unital type I C*-algebra B whose primitive ideal space is Hausdorff and totallydisconnected, then is Jordan isomorphism. The aim of this note is to show that if be asurjective spectrum preserving additive map, then is a Jordan isomorphism without the extraassumption totally disconnected.

A Survey of Invertibility and Spectrum Preserving Linear Maps

On Preserving Properties of Linear Maps on $C^{*}$-algebras

Let $A$ and $B$ be two unital $C^{*}$-algebras and $varphi:A rightarrow B$ be a linear map. In this paper, we investigate the structure of linear maps between two $C^{*}$-algebras that preserve a certain property or relation. In particular, we show that if $varphi$ is unital, $B$ is commutative and $V(varphi(a)^{*}varphi(b))subseteq V(a^{*}b)$ for all $a,bin A$, then $varphi$ is a $*$-homomorph...