Infinite Dimensional Central Simple Regular Algebras with Outer Derivations

We give an example of infinite dimensional central simple (von Neumann) regular algebra with outer derivations. Given the algebra $$S(M)$$ of all measurable operators affiliated with the type II $${}_{1}$$ hyperfinite factor $$M$$ we construct its $$\ast$$ -subalgebra $$\mathcal{R}_{\infty}$$ which is dense in the measure topology. We prove that this algebra has a derivation which is discontinuous in the measure topology and hence is non-inner. We show that the algebra $$\mathcal{R}_{\infty}$$ admits also a continuous (in the measure topology) derivation, implemented by a measurable operator, but which is still an outer derivation.

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