Unbounded derivations in AT algebras
Unbounded derivations in AT algebras
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DOI：
10.1006/jfan.1998.3333
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发表时间：
1998-12
影响因子：
1.7
通讯作者：
A. Kishimoto
中科院分区：
文献类型：
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作者：
A. Kishimoto
设Abe是一个真实的秩为零的单酉AT代数，使得它有唯一的迹态τ且K1（A）既不是0也不是Z.对R中稠密值域的每个k ∈Hom（K_1（A），R），构造A中的一个闭导子δ，它生成A的一个单参数自同构群α，使得对任意酉u ∈D（δ），τ（δ（u）u ~*）=2πi_n（[u]）.进一步，我们构造了这样一个α，它具有Kishimoto（Comm.Math.Phys.179（1996），599-622）中定义的Rohlin性质，在这种情况下，交叉积A ×α R是一个真实的秩为零的单AT代数.作为应用，我们得到了这样一个C ~*-代数A的近似内自同构群In（A）到Ext（K_1（A），K_0（A））的自然同态的核是与内自同构同伦的自同构群HIn（A）.结合Kishimoto和Kumjian（trans. Amer. Math. Soc.，（A）是一个简单的函数，它与上面的直接和同构。作为导子构造的另一个应用，我们证明了：如果A是上述类型的C *-代数，且α∈HInn（A）具有Rohlin性质，且来自于Kishimoto和Kumjian（预印本）中的稠密值域的H ∈Hom（K1（A），R），则交叉积A ×α Z也是同一类型的，特别是A ×α Z是AT代数. (The其它性质从Kishimoto [J. Operator Theory 40（1998）]中已知。
LetAbe a simple unital AT algebra of real rank zero such that it has a unique tracial stateτandK1(A) is neither 0 norZ. For eachϕ∈Hom(K1(A), R) with dense range inRwe construct a closed derivationδinAwhich generates a one-parameter automorphism groupαofAsuch thatτ(δ(u) u*)=2πiϕ([u]) for any unitaryu∈D(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 groupInn(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),Inn(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 andα∈HInn(A) has the Rohlin property and comes fromϕ∈Hom(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)].)