Jordan Maps On Operator Algebras And Jordan Ideals

Recently many people have investigated the structure of Jordan maps and Jordan ideals in operator algebras vastly.When it refers to Jordan algebra a special operator algebra,the additivity of Jordan maps is valuable to be considered too.Let B(H)_s be the real linear special of all linear bounded self-adjoint operators on a Hilbert space of dimension＞1 with Jordan product o and B be an arbitrary Jordan algebra.The first part of this paper aims to investigate the Jordan maps from B(H)_s to B.It shows that if mapφfrom B(H)_s onto B is bijective and is Jordan map,thenφis additive.Both the structure of Jordan algebra and purely algebra method called Pierce decomposition are used in the proof.Many people have been studying the connections of Jordan ideals and associative ideals for some operator algebras,because it is very import to reveal the structure of various operator algebras.Making use of the special structure for triangular algebra inⅡ₁ type hyperfinte factor,the second part of this paper shows every weakly closed Jordan ideal in triangular algebra inⅡ₁ type hyperfinte factor is an associative ideal.