Ilan Hirshberg
We classify spectrum-preserving endomorphisms of stable continuous-trace C^*-algebras up to inner automorphism by a surjective multiplicative invariant taking values in finite dimensional vector bundles over the spectrum. Specializing to automorphisms, this gives a different approach to results of Phillips and Raeburn.
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We compute the number of points of projective variety V over a finite field in terms of invariants of the so-called Serre C*-algebra of V.
It is shown that the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism is purely infinite when it is simple, provided the endomorphism does not leave a trace state invariant and maps the unit to a full projection.
For any finite dimensional C*-algebra A with any trace vector {\vec s} whose components are rational numbers, we give an endomorphism Φ of the hyperfinite II_1 factor R such that: forall k in {\mathbb N} Φ^k (R)' \cap R= \otimes^k A The canonical trace τ on R extends the trace vector {\vec s} on A. As a corollary, we construct a one-parameter family of inclusions of hyperfinite II_1 factors N^λ \subset M^λ with trivial relative commutant (N^λ)' \cap M^λ= {\mathbb C} and with the Jones index [M^λ: N^λ]= λ^{-1} \in (4, \infty) \cap {\mathbb Q} This partially solves the problem of finding all pos…
For any finite dimensional C^*-algebra A with a trace vector \vec s whose entries are rational numbers, we give an endomorphism Φof the hyperfinite II_1 factor R such that: for all k \in \mathbb {N}, Φ^k (R)' \cap R= \otimes^k A. The canonical trace τon R extends the trace vector \vec s on A. Therefore the minimal projection is not necessarily equivalent to each other.
A surjective endomorphism or, more generally, a polymorphism in the sense of \cite{SV}, of a compact abelian group $H$ induces a transformation of $L^2(H)$. We study the C*-algebra generated by this operator together with the algebra of continuous functions $C(H)$ which acts as multiplication operators on $L^2(H)$. Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a uniq…
The aim of this work is to study the structure of bounded finite potent endomorphisms on Hilbert spaces. In particular, for these operators, an answer to the Invariant Subspace Problem is given and the main properties of its adjoint operator are offered. Moreover, for every bounded finite potent endomorphism we show that Tate's trace coincides with the Leray trace and with the trace defined by R. Elliott for Riesz Trace Class operators.
Using Poincare duality, we formulate a formula of Lefschetz type which computes the Lefschetz number of an endomorphism of a separable, nuclear C*-algebra satisfying Poincare duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with the complex numbers, as in the classical case.) We then consider endomorphisms of Cuntz-Krieger algebras O_A. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of th…
Two automorphisms of a simple stable AF algebra with a finite dimensional lattice of lower semicontinuous traces are shown to be outer conjugate if they act in the same way on the K-group and the extremal traces are scaled by numbers which are not equal to 1 and satisfy a certain condition (which always holds if all the scaling factors are less than 1). The proof goes via the Rohlin property. As an application we consider the problem of classifying conjugacy or outer conjugacy classes of certain actions of the circle group on a separable purely infinite C*-algebra.
Given a unital C*-algebra A, an injective endomorphism α:A --> A preserving the unit, and a conditional expectation E from A to the range of αwe consider the crossed-product of A by αrelative to the transfer operator L=α^{-1}E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a "gauge action" on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h>1 and E(ab)=E(ba) for all a,b in A (e.g. when A is commutative) t…
Metrics on trace spaces
本文针对实秩零、稳定秩一的可分类$\text{C}^*$-代数，用最优输运方法给出了$^*$-同态酉轨道距离的迹数据计算方式，还利用迹Lipschitz结构表征了自同态的统计特征。
Brown-Voiculescu entropy revisited
本文基于现代分类理论工具，重新审视核C*-代数端omorphism的布朗-维奥库勒斯熵，引入适用于有限核维度或有限分解秩C*-代数的着色非交换拓扑熵，并证明在特定条件下熵变分原理。
Traces on the uniform tracial completion of $\mathcal{Z}$-stable C*-algebras
本文研究了一致迹完备化的迹问题，证明了张量吸收 Jiang-Su 代数的 C*-代数满足迹的连续延拓性质。
Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of cross product C*-algebras $C_0(X)\cross G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X\cross G$ cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of $C_0(X)\cross G$ induced by the endomorphism, \emph{i.e.} the Lefschetz number, in terms of fixed orbits and representation-theoretic data connected with certain isotropy subgroups of the isotropy group at that point. The technique is to use non…
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.
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