Derivations with Values in the Ideal of $$\tau $$-Compact Operators Affiliated with a Semifinite von Neumann Algebra
Abstract
Abstract Let $${{\mathcal {M}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> be a semifinite von Neumann algebra with a faithful normal semifinite trace $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> and let $${{\mathcal {A}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> be an arbitrary von Neumann subalgebra of $${{\mathcal {M}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> . We characterize the class of symmetric ideals $${{\mathcal {E}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> in $${{\mathcal {M}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> such that derivations $$\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {E}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> are necessarily inner, which is a unification and far-reaching extension of the results due to Johnson and Parrott (J Funct Anal 11:39–61, 1972), due to Kaftal and Weiss (J Funct Anal 62:202–220, 1985), and due to Popa (J Funct Anal 71:393–408, 1987). In particular, we show that every derivation from $${{\mathcal {A}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> into the ideal $${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of all $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -compact operators is inner, establishing a semifinite version of the Johnson–Parrott–Popa Theorem which is different from Popa and Rădulescu (Duke Math J 57(2):485–518, 1988, Theorem 1.1) and contrasts to the example of a non-inner derivation established in Popa and Rădulescu (1988, Theorem 1.2).