COORDINATE SYSTEMS AND BOUNDED ISOMORPHISMS

ABSTRACT. For a BanachD-bimoduleM over an abelian unital C∗-algebraD, we define E1(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual M#. Equip E1(M) with the weak- ∗ topology. We develop general properties of E1(M). It is properly viewed as a coordinate system for M when M ⊆ C, where C is a unital C∗-algebra containing D as a regular MASA with the extension property; moreover, E1(C) coincides with Kumjian’s twist in the context of C∗-diagonals. We identify the C∗-envelope of a subalgebraA of a C∗-diagonal whenD ⊆ A ⊆ C. For triangular subalgebras, each containing the MASA, a bounded isomorphism induces an algebraic iso-morphism of the coordinate systems which can be shown to be continuous in certain cases. For subalgebras, each containing the MASA, a bounded isomor-phism 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. Ourmethods simplify and extend various known results; for example, isometric isomorphisms of the tri-angular algebras extend to isometric isomorphisms of the C∗-envelopes, and the conditional expectation E: C → 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.