arXiv subjects
Publications and source records attributed to A. Kishimoto.
When $α$ is an approximately inner flow on a C$^*$-algebra $A$ and commutes with an automorphism $γ$ of $A$ we may extend $α$ to a flow $\barα$ on the crossed product $A\times_γZ$ by setting $\barα_t(U)=U$ where $U$ is the canonical unitary implementing $γ$ in $A\times_γZ$ and ask whether $\barα$ is also approximately inner or not. We will consider very specific examples of this type; some of which we can answer affirmatively.
We introduce two notions for flows on quasi-diagonal C*-algebras, quasi-diagonal and pseudo-diagonal flows; the former being apparently stronger than the latter. We derive basic facts about these flows and give various examples. In addition we extend results of Voiculescu from quasi-diagonal C*-algebras to these flows.
We prove that the pure state space is homogeneous under the action of the automorphism group (or the subgroup of asymptotically inner automorphisms) for all the separable simple C*-algebras. The first result of this kind was shown by Powers for the UHF algebras some 30 years ago.
We prove that the pure state space is homogeneous under the action of the automorphism group (or a certain smaller group of approximately inner automorphisms) for a fairly large class of simple separable nuclear C*-algebras, including the approximately homogeneous C*-algebras and the class of purely infinite C*-algebras which has been recently classified by Kirchberg and Phillips. This extends the known results for UHF algebras and AF algebras by Powers and Bratteli.
We prove that the pure state space is homogeneous under the action of the group of asymptotically inner automorphisms for all the separable simple nuclear C*-algebras. If simplicity is not assumed for the C*-algebras, the set of pure states whose GNS representations are faithful is homogeneous for the above action.
A UHF flow is an infinite tensor product type action of the reals on a UHF algebra $A$ and the flip automorphism is an automorphism of $A\otimes A$ sending $x\otimes y$ into $y\otimes x$. If $α$ is an inner perturbation of a UHF flow on $A$, there is a sequence $(u_n)$ of unitaries in $A\otimes A$ such that $α_t\otimes α_t(u_n)-u_n$ converges to zero and the flip is the limit of $\Ad u_n$. We consider here whether the converse holds or not and solve it with an additional assumption: If $A\otimes A\cong A$ and $α$ absorbs any UHF flow $β$ (i.e., $α\otimesβ$ is cocycle conjugate to $α$), then the converse holds; in this case $α$ is what we call a universal UHF flow.
Let A be a simple unital AT algebra of real rank zero and Inn(A) the group of inner automorphisms of A. In the previous paper we have shown that the natural map of the group of approximately inner automorphisms into Ext(K_1(A),K_0(A)) oplus Ext(K_0(A),K_1(A)) is surjective; the kernel of this map includes the subgroup of automorphisms which are homotopic to Inn(A). In this paper we consider the quotient of the group of approximately inner automorphisms by the smaller normal subgroup AInn(A) which consists of asymptotically inner automorphisms and describe it as OrderExt(K_1(A),K_0(A)) oplus Ext(K_0(A),K_1(A)), where OrderExt(K_1(A),K_0(A)) is a kind of extension group which takes into account the fact that K_0(A) is an ordered group and has the usual Ext as a quotient.
Trace scaling automorphisms of stable AF algebras with dimension group totally ordered are outer conjugate if the scaling factors are the same (not equal to one). This is an adaptation of a similar result for the AFD type II_infty factor by Connes and extends the previous result for stable UHF algebras.