Unitary dilations and the \(C^ *\)-algebra \({\mathcal O}_ 2\)
Unitary \(2\times 2\) matrices of operators are studied. Unitary dilations of isometries T reduce to the case where the upper left corner is T and the lower left corner is 0. A unitary dilation is the minimal unitary power dilation of the given isometry if and only if the lower right corner is the adjoint of a unilateral shift. Also the order properties of the set of all unitary dilations of a given nonunitary isometry are considered. Let \({\mathcal A}\) be the \(C^*\)- algebra generated by the upper right and lower right corners in a \(2\times 2\) matrix which is a unitary dilation of a given isometry T. Let I be the ideal generated by the complementary projection to the defect projection of T. Then either \(I=A\) or the quotient A/I is isomorphic to the Cuntz algebra \({\mathcal O}_ 2\).
scientific article; zbMATH DE number 4036496
scientific article