Jordan Mapping Between Operator Algebra And Kowalski - Slodkowski Theorem Of Promotion

In this paper,we first introduce Kaplansky’s conjecture, and then we prove that if a linear operator φ between C*algebra of real rank zero and semi-simple Banach algebra satisfy the conditions of Kaplansky’s conjecture and also φ is bounded below and preserves spectrum of all the idempotents, then φ is Jordan mapping. And then we study almost Jordan mapping. Next we introduce a new concept named AJNJ algebra pair and prove some properties about it. And we obtain Riesz representation theorem about almost multiplicative functions on C(X). At last, we generalize the concept of spectrum. Using these results we obtain an improvement of Kowalski-Slodkowski theorem.