Dear all,

Let's call a net ( c_\alpha )_\alpha in a C^* algebra A quasi-central if

  a c_\alpha - c_\alpha a \to 0  for each a in A.

Suppose that all c_\alphas are non-negtative (so that
c_\alpha^\frac{1}{2} exists for each \alpha). Is then the net (
c_\alpha^\frac{1}{2} )_\alpha in A also quasi-central. If ( c_\alpha
)_\alpha is bounded, the answer is easily seen to be true by Gelfand
theory, but what about the unbounded case?

Thanks for any hints!

Volker Runde.

Does this follow from prop II.8.1.5 (p. 165) in
http://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf
? Applied once for each (a, alpha, epsilon > 0) with X := the closed real
interval [0, ||c_alpha||], and f(x) := sqrt(x) to obtain:
There exists delta > 0 with
||a((c_alpha)^(1/2)) - ((c_alpha)^(1/2))a|| < ||a||(epsilon)
whenever
||a(c_alpha) - (c_alpha)a|| < ||a||(delta)

On Monday, July 31, 2017 at 3:15:34 PM UTC-6, vru...@ualberta.ca wrote:
> Dear all,
>
> Let's call a net ( c_\alpha )_\alpha in a C^* algebra A quasi-central if
>
>   a c_\alpha - c_\alpha a \to 0  for each a in A.
>
> Suppose that all c_\alphas are non-negtative (so that
> c_\alpha^\frac{1}{2} exists for each \alpha). Is then the net (
> c_\alpha^\frac{1}{2} )_\alpha in A also quasi-central. If ( c_\alpha
> )_\alpha is bounded, the answer is easily seen to be true by Gelfand
> theory, but what about the unbounded case?
>
> Thanks for any hints!
>
> Volker Runde.