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Let A be a factor von Neumann algebra and φ be the * -Jordan2010 Mathematics Subject Classification. 46J10 , 47B48.

Let 𝒜 and ℬ be two prime C *-algebras. In this paper, we investigate the additivity of map Φ from 𝒜 onto ℬ that are bijective unital and satisfies Φ ( A P + λ P A ∗ ) = Φ ( A ) Φ ( P ) + λ Φ ( P ) Φ ( A ) ∗ , $$\Phi(AP+\lambda PA^{*})=\Phi(A)\Phi(P)+\lambda \Phi(P)\Phi(A)^{*}, $$ for all A ∊ 𝒜 and P ∊ {P 1, I 𝒜 − P 1} where P 1 is a nontrivial projection in 𝒜 and λ∊ {−1, +1}. Then, Φ is *-additive.

Abstract. We introduce the concept of operator h-convex functions for positive linear maps, and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain several trace inequalities for operators.