An order characterization of commutativity for 𝐶*-algebras

Abstract

Abstract. In this paper, we investigate the problem of when a C * -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function e t , t ∈ [0, ∞), is operator-monotonic increasing on a C * -algebra A if and only if A is commutative. Therefore, C * -algebra A is commutative if and only if e x+y = e x e y in A+C for all positive elements x, y in A.

On characterizations of commutativity of C*-algebras

“…[2,3]). Very recently, Wu in [6] gave another characterization for commutativity based on the function exp x. We observe that both x p and exp x are monotone increasing functions on the positive axis but not operator monotone on M 2 , the matrix algebra of all complex 2 × 2 matrices.…”

“…If we make use of the operator monotone property of the log function, we can deduce from monotone assumption for exp the monotone property of the function x t for any t > 1. To discuss on a C * -algebra, this will also considerably shorten the proof in [6], where Wu reduces his whole proof to the monotone property of the function x 2 and uses the old Ogasawara's result [2]. On the matrix algebra M 2 however, we prefer the above elementary approach.…”

A Few Conditions for aC*-Algebra to Be Commutative

“…We recall that in the literature one can find several conditions characterizing commutativity of * -algebras most of which are related to the order structure; see, for example, [4][5][6][7][8]. A characterization of algebraic character can be found in [9].…”

On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators

“…A few comments on this modified conjecture follow. First, we refer to the paper [10] where it was shown that if the exponential function on the interval [0, ∞[ is monotone increasing relative to a given * -algebra A (meaning that for any , ∈ A with 0 ≤ ≤ we have exp( ) ≤ exp( )), then the algebra A is commutative. In [11]…”