When locally contractive representations are completely contractive
The representation theory of non-self-adjoint matrix algebras is explored. A representation is called locally contractive if each matrix unit is sent to a contraction. The class of finite dimensional algebras for which every locally contractive representation of the enveloping \(C^*\)-algebra coincides with the class of finite dimensional algebras for which locally contractive representations are contractive. A lattice theoretic characterization of this class is obtained. A matrix algebra containing the diagonal algebra is classified by a directed graph depending on the matrix units in the algebra. There is given also a lattice theoretic characterization of algebras for which the commutant lifting theorem is valid. The results are generalized to infinite dimensional commutative subspace lattice algebras.
infinite dimensional commutative subspace lattice algebras
scientific article; zbMATH DE number 764005