• #### quasi-central nets in C*-algebras

From vrunde@ualberta.ca@21:1/5 to All on Mon Jul 31 15:15:30 2017
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.

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• From David Cullen@21:1/5 to vru...@ualberta.ca on Tue Aug 8 06:39:43 2017
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.

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