OnC^*-algebras with the approximaten-th root property

Abstract

We say that a C*-algebra X has the approximate n-th root property (n ~£ 2) if for every a € X with ||a|| < 1 and every e > 0 there exits b 6 X such that ||6|| < 1 and \\a -b n \\ < t. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (respectively, 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.

Root closed function algebras on compacta of large dimension

“…be the class of all compact Hausdorff spaces X such that w(X) ≤ τ and C(X) is approximate n-th root closed (C(X) is approximate n-th root closed for each n > 1 resp.). It was shown in [3,Corollary 1.3], that A τ (n) contains a universal space for any τ ≥ ω and any n > 1. Using the idea of the proof of Theorem 1.2 from [3] one can show that A τ also contains a universal space.…”

“…It was shown in [3,Corollary 1.3], that A τ (n) contains a universal space for any τ ≥ ω and any n > 1. Using the idea of the proof of Theorem 1.2 from [3] one can show that A τ also contains a universal space.…”

“…The approximate n-th root closedness of C(X) was studied by Kawamura and Miura and was proved to be equivalent to n-divisibility of H 1 (X; Z) under the additional assumption dimX ≤ 1. The universal space for metrizable compacta with the approximately n-th root closed C(X) is constructed in [3].…”