On Isometries of Operator Algebras

We study the Banach space isometries of triangular subalgebras of C*-algebras that contain diagonals in the sense of Kumjian. Under a mild technical assumption, we prove that every isometry between two such algebras decomposes as a direct sum of a unitary multiple of an isometric algebra isomorphism and a unitary multiple of an isometric algebra anti-isomorphism. Moreover, each isometric algebraic isomorphism (anti-isomorphism) between two algebras of the type considered here extends to a C*-isomorphism (C*-anti-isomorphism) between the enveloping C*-algebras. Our hypotheses enable us to "coordinatize" the algebras under consideration, and the structure of the isometries between the algebras is expressed in terms of the coordinates.