Excision and a Theorem of Popa
英文摘要:
We give an elementary C*-algebraic proof of a result of Sorin Popa which is of fundamental importance to Elliott's Classification Program.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
Nathanial P. Brown
We give an elementary C*-algebraic proof of a result of Sorin Popa which is of fundamental importance to Elliott's Classification Program.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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These notes contain an Ergodic-theoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of ``rigid'' groups) satisfy the assumptions of Popa's theorem.
We discuss the relative K-theory for a $C^{*}$-algebra, $A$, together with a $C^{*}$-subalgebra, $A' \subseteq A$. The relative group is denoted $K_{i}(A';A), i = 0, 1$, and is due to Karoubi. We present a situation of two pairs $A' \subseteq A$ and $B' \subseteq B$ are related so that there is a natural isomorphism between their respective relative K-theories. We also discuss applications to the case where $A$ and $B$ are $C^{*}$-algebras of a pair of locally compact, Hausdorff topological groupoids, with Haar systems.
We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.
We extend Cuntz-Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.The excision theorems for the bivariant periodic cyclic cohomology of discrete,topological and bornological algebras and pro-algebras follow from this.
Using various finite dimensional approximation properties, four convex subsets of the tracial space of a unital C*-algebra are defined. Applications of these tracial invariants include: (1) An analogue of Szego's limit theorem for arbitrary self adjoint operators. (2) A McDuff factor embeds into the ultrapower of the hyperfinite II_1 factor if and only if it contains a weakly dense operator system which is injective. (3) There exists a simple, quasidiagonal, real rank zero C*-algebra with non-hyperfinite II_1 factor representations and which is not tracially AF. This answers negatively questio…
Weak relative Dixmier property and Popa's intertwining technique for type III subfactors
本文证明了含算子值权重的von Neumann代数包含满足弱相对Dixmier性质,并由此得到III型因子的新结构定理,包括Popa交织准则的无迹重述、相对固态定理的推广及交叉积的Galois对应。
We survey Sorin Popa's recent work on Bernoulli actions. The paper was written on the occasion of the Bourbaki seminar. Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra, yielding the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II_1 factors with prescribed countable fundamental group.
The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators" associated to spectral triples. This formula is notably well adapted when the zeta function has multiple poles. The second part is devoted to give a concrete realization of this formula by deriving an index theorem on the simple, but significant example of Heisenberg elliptic operators on a trivial foliation, which are in general non-elliptic but hypoelliptic. The l…
We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.
An Infinite Transitivity Theorem
本文提出并证明了关于大规模C*代数的无限Kadison传递定理,并展示了Calkin代数中存在不满足无限传递性的不可约表示。
We show that if $Γ= Γ_1\times\dotsb\times Γ_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $Λ$ which is $W^*$-equivalent to $Γ$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $Λ$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II$_1$ factors.
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