On the commutativity of C* -algebras

A Few Conditions for aC*-Algebra to Be Commutative

“…In accordance with Example 5 in [9], as an immediate application of the above result, we mention that a unital *algebra A is commutative if and only if any of the following relations sin ( + ) − (sin cos + cos sin ) ∈ Z…”

“…Concerning our final proposition we recall that in [9] Jeang and Ko presented the following result. Assume A is a * -algebra and there are nonconstant continuous functions and defined on intervals , in the real line such that for any pair , ∈ A of self-adjoint elements with ( ) ⊂ , ( ) ⊂ ( (⋅) stands for the spectrum) we have ( ) ( ) = ( ) ( ).…”

“…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]. In this line we also mention that in the recent paper [10] we have investigated the standard -loop structure of the positive definite cone of a * -algebra and obtained some results showing that the commutativity, associativity (even in certain weak senses), or distributivity of that structure are each equivalent to the commutativity of the underlying *algebra.…”

Untitled

“…Moreover, as a corollary to this result, it turns out that A is abelian if and only if e a+b = e a e b for each a, b in (the unitization of) A. Later, this result was generalized by Jeang and Ko [10]. Meanwhile, there exists a characterization of the Stinespring type.…”