On Commutative and Non-Commutative C* -Algebras with the Approximate n-th Rott Property

We say that a C *-algebra X has the approximate n-th root property (n = 2) if for every a ? X with kak = 1 and every " > 0 there exits b ? X such that kbk = 1 and ka - bnk < ". Some properties of commutative and non-commutative C *-algebras having the approximate nth root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unital C *-algebra X such that any other (commutative) separable unital C *-algebra is a quotient of X. Also we illustrate a commutative C *-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first ?Cech cohomology.