关于C^＊-代数中的可逆元和酉元的凸组合

Invertible elements and convex combination of unitary elements in a C*-algebra

讨论了Hilbert空间上的C 代数A中的可逆群和酉群的一些关系,证明了C 代数A中的元素A是可逆的充要条件是存在两个非负实数λ1和λ2,且λ1≠λ2以及酉群中的两个元素U1和U2使得A=λ1U1+λ2U2,给出了C 代数A中范数不大于1的可逆元的全体闭包和酉群的一些关系.

Let A be a C~*-algebra acting on a Hilbert space H, G(A) and U(A) be the invertible group and the unitary group of A, respectively. Some relations of the elements of G(A) and U(A) are considered. It is proved that A is invertible in C~*-algebra A if and only if there exist two nonnegative real numbers λ_1 and λ_2 with λ_1≠λ_2 and two unitary U_1 and U_2 in A such that A=λ_1U_1+λ_2U_2. Moreover, some relations between G_1(A)d{A∈G(A): ‖A‖≤1} and U(A) are discussed.