Unbounded Derivations in AT Algebras
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LetAbe a simple unital AT algebra of real rank zero such that it has a unique tracial stateτandK1(A) is neither 0 norZ. For each
Hom(K1(A),R) with dense range inRwe construct a closed derivationδinAwhich generates a one-parameter automorphism groupαofAsuch thatτ(δ(u)u*)=2πi([u]) for any unitaryuD(δ). Furthermore we construct such anαwith the Rohlin property, which is defined in Kishimoto (Comm. Math. Phys.179(1996), 599–622), in this case the crossed productA×αRis a simple AT algebra of real rank zero. As an application we obtain that for such aC*-algebraAthe kernel of the natural homomorphism of the group(A) of approximately inner automorphisms intoExt(K1(A),K0(A))⊕Ext(K0(A),K1(A)),is the group HInn(A) of automorphisms homotopic to inner automorphisms. Combining with the result of Kishimoto and Kumjian (Trans. Amer. Math. Soc., to appear),(A)/HInn(A) is isomorphic to the above direct sum. As another application of the construction of derivations, we show that ifAis aC*-algebra of the above type andαHInn(A) has the Rohlin property and comes fromHom(K1(A),R) with dense range as in Kishimoto and Kumjian (preprint), then the crossed productA×αZis again of the same type; in particularA×αZis an AT algebra. (The other properties are known from Kishimoto [J. Operator Theory40(1998)].)
Hom(K1(A),R) with dense range inRwe construct a closed derivationδinAwhich generates a one-parameter automorphism groupαofAsuch thatτ(δ(u)u*)=2πi([u]) for any unitaryuD(δ). Furthermore we construct such anαwith the Rohlin property, which is defined in Kishimoto (Comm. Math. Phys.179(1996), 599–622), in this case the crossed productA×αRis a simple AT algebra of real rank zero. As an application we obtain that for such aC*-algebraAthe kernel of the natural homomorphism of the group(A) of approximately inner automorphisms intoExt(K1(A),K0(A))⊕Ext(K0(A),K1(A)),is the group HInn(A) of automorphisms homotopic to inner automorphisms. Combining with the result of Kishimoto and Kumjian (Trans. Amer. Math. Soc., to appear),(A)/HInn(A) is isomorphic to the above direct sum. As another application of the construction of derivations, we show that ifAis aC*-algebra of the above type andαHInn(A) has the Rohlin property and comes fromHom(K1(A),R) with dense range as in Kishimoto and Kumjian (preprint), then the crossed productA×αZis again of the same type; in particularA×αZis an AT algebra. (The other properties are known from Kishimoto [J. Operator Theory40(1998)].)