ON PRE-HILBERT ALGEBRAS CONTAINING A NONZERO CENTRAL IDEMPOTENT SUCH THAT ‖FA‖=‖A‖ and ‖A^2‖≤‖A‖^2

This paper aims to prove that every two-dimensional real absolute valued algebra is isomorphic C〖,C〗^*,*C, or C┴*, and if the A power commutative, then third associative. Let be pre-Hilbert without divisors of zero; we has dimension two satisfying,‖a^2 ‖=‖a‖^2 for all a ∈ new classes algebras. We also characterize algebraic algebras zero containing nonzero central idempotent f such ‖fa‖=‖a‖ ‖a^2 ‖≤‖a‖^2, flexible Furthermore, contains ‖≤‖a‖^2 in A, these statements are equivalent: associative, degree two.