Characterizing isomorphisms in terms of completely preserving invertibility or spectrum

A Survey of Invertibility and Spectrum Preserving Linear Maps

a survey of invertibility and spectrum preserving linear maps

Linear Maps Preserving Invertibility or Spectral Radius on Some $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra which has a faithful state. If $varphi:Arightarrow A$ is a unital linear map which is bijective and invertibility preserving or surjective and spectral radius preserving, then $varphi$ is a Jordan isomorphism. Also, we discuss other types of linear preserver maps on $A$.

Isomorphisms Preserving Invariants

Let V and W be finite dimensional real vector spaces and let G ⊂ GL(V ) and H ⊂ GL(W ) be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to R[V ] and R[W ] , respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L se...

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. ...