  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.

Operator Algebras