Coordinate Systems and Bounded Isomorphisms for Triangular Algebras

For a Banach D-bimodule M over an abelian unital C*-algebra D, we define E(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual of M. Equip E(M) with the weak-* topology. We develop general properties of E(M). It is properly viewed as a coordinate system for M when M is a subset of C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E(C) coincides with Kumjian's twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal which contains D. For triangular subalgebras, each containing the MASA, a bounded isomorphism induces an algebraic isomorphism of the coordinate systems which can be shown to be continuous in certain cases. For subalgebras, each containing the MASA, a bounded isomorphism that maps one MASA to the other MASA induces an isomorphism of the coordinate systems. We show that the weak operator closure of the image of a triangular algebra in an appropriate representation is a CSL algebra and that bounded isomorphism of triangular algebras extends to an isomorphism of these CSL algebras. We prove that for triangular algebras in our context, any bounded isomorphism is completely bounded. Our methods simplify and extend various known results; for example, isometric isomorphisms of the triangular algebras extend to isometric isomorphisms of the C*-envelopes, and the conditional expectation E of C onto D is multiplicative when restricted to a triangular subalgebra. Also, we use our methods to prove that the inductive limit of C*-diagonals with regular connecting maps is again a C*-diagonal.