A mapping between projections of C^*-algebras preserving the orthogonality, is called an orthoisomorphism. We define the order-isomorphism mapping on C^*-algebras, and using Dye's result, we prove in the case of commutative unital C^*-algebras that the concepts; order-isomorphism and the orthoisomorphism coincide. Also, we define the equipotence relation on the projections of C(X); indeed, new concepts of finiteness are introduced. The classes of projections are represented by constructing a special diagram, we study the relation between the diagram and the topological space X. We prove that an order-isomorphism, which preserves the equipotence of projections, induces a diagram-isomorphism; also if two diagrams are isomorphic, then the C^*-algebras are isomorphic.
AL-RAWASHDEH, AHMED S. and AL-SULEIMAN, SULTAN M.
"Order-isomorphism and a projection's diagram of C(X),"
Turkish Journal of Mathematics: Vol. 34:
4, Article 9.
Available at: https://journals.tubitak.gov.tr/math/vol34/iss4/9