The Experts below are selected from a list of 291 Experts worldwide ranked by ideXlab platform
Sorin Popa - One of the best experts on this subject based on the ideXlab platform.
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Cartan SubalgebraS AND BIMODULE DECOMPOSITIONS OF II1 FACTORS
2020Co-Authors: Sorin PopaAbstract:Let A ⊂M be a MASA in a II1 factor M. We describe the von Neumann Subalgebra of M generated by A and its normalizer N (A) as the set Nw q (A) consisting of those elements m ∈M for which the bimodule AmA is discrete. We prove that two MASAs A and B are conjugate by a unitary u ∈ Nw q (A) iff A is discrete over B and B is discrete over A in the sense defined by Feldman and Moore [5]. As a consequence, we show that A is a Cartan Subalgebra of M iff for any MASA B⊂M, B = uAu∗ for some u ∈M exactly when A is discrete over B and B is discrete over A.
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on a class of ii1 factors with at most one Cartan Subalgebra
Annals of Mathematics, 2010Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:We prove that the normalizer of any diffuse amenable Subalgebra of a free group factor L(F-r) generates an amenable von Neumann Subalgebra. Moreover, any II1 factor of the form Q (circle times) over barL(F-r), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan Subalgebras. We also prove that if a free ergodic measure-preserving action of a free group F-r, 2 <= r <= infinity, on a probability space (X, mu) is profinite then the group measure space factor L-infinity (X) F-r has unique Cartan Subalgebra, up to unitary conjugacy.
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On a class of II1 factors with at most one Cartan Subalgebra
Annals of Mathematics, 2010Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:We prove that the normalizer of any diffuse amenable Subalgebra of a free group factor L(F-r) generates an amenable von Neumann Subalgebra. Moreover, any II1 factor of the form Q (circle times) over barL(F-r), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan Subalgebras. We also prove that if a free ergodic measure-preserving action of a free group F-r, 2
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on a class of ii1 factors with at most one Cartan Subalgebra ii
American Journal of Mathematics, 2010Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:This is a continuation of our previous paper studying the structure of Cartan Subalgebras of von Neumann factors of type ${\rm II}_1$. We provide more examples of ${\rm II}_1$ factors having either zero, one, or several Cartan Subalgebras. We also prove a rigidity result for some group measure space ${\rm II}_1$ factors.
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on a class of ii_1 factors with at most one Cartan Subalgebra ii
arXiv: Operator Algebras, 2008Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:This is a continuation of our previous paper studying the structure of Cartan Subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan Subalgebras. We also prove a rigidity result for some group measure space II_1 factors.
Willem A De Graaf - One of the best experts on this subject based on the ideXlab platform.
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Computing with real Lie algebras: Real forms, Cartan decompositions, and Cartan Subalgebras☆
Journal of Symbolic Computation, 2013Co-Authors: Heiko Dietrich, Paolo Faccin, Willem A De GraafAbstract:Abstract We describe algorithms for performing various tasks related to real simple Lie algebras. These algorithms form the basis of our software package CoReLG , written in the language of the computer algebra system GAP4 . First, we describe how to efficiently construct real simple Lie algebras up to isomorphism. Second, we consider a real semisimple Lie algebra g . We provide an algorithm for constructing a maximally (non-)compact Cartan Subalgebra of g ; this is based on the theory of Cayley transforms. We also describe the construction of a Cartan decomposition g = k ⊕ p . Using these results, we provide an algorithm to construct all Cartan Subalgebras of g up to conjugacy; this is a constructive version of a classification theorem due to Sugiura.
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Using Cartan Subalgebras to calculate nilradicals and Levi Subalgebras of Lie algebras
Journal of Pure and Applied Algebra, 1999Co-Authors: Willem A De GraafAbstract:Abstract In this paper we investigate the structure of a non-semisimple Lie algebra of characteristic 0, by using the action of a Cartan Subalgebra. We give an algorithm for calculating the nilradical and an algorithm for finding a Levi Subalgebra. At the end of the paper these algorithms are put to practical tests.
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An algorithm for the decomposition of semisimple Lie algebras
Theoretical Computer Science, 1997Co-Authors: Willem A De GraafAbstract:We consider the problem of decomposing a semisimple Lie algebra defined over a field of characteristic zero as a direct sum of its simple ideals. The method is based on the decomposition of the action of a Cartan Subalgebra. An implementation of the algorithm in the system ELIAS is discussed at the end of the paper.
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calculating the structure of a semisimple lie algebra
Journal of Pure and Applied Algebra, 1997Co-Authors: Willem A De GraafAbstract:Abstract First we briefly describe two previously published algorithms: one that constructs a Cartan Subalgebra and one that decomposes a semisimple Lie algebra L as a direct sum of simple ideals. Then, by reducing L modulo a prime we derive an algorithm to obtain the type of L (thereby solving the isomorphism problem for semisimple Lie algebras over Q having structure constants in Q ).
Apoorva Khare - One of the best experts on this subject based on the ideXlab platform.
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weights of simple highest weight modules over a complex semisimple lie algebra
arXiv: Representation Theory, 2013Co-Authors: Apoorva KhareAbstract:In this short note we announce three formulas for the set of weights of various classes of highest weight modules $\V$ with highest weight \lambda, over a complex semisimple Lie algebra $\lie{g}$ with Cartan Subalgebra $\lie{h}$. These include, but are not restricted to, all (highest weight) simple modules L(\lambda). We also assert that these formulas are the "best possible", in that they do not hold in general for other highest weight modules in a very precise sense. The proofs of the results in this note are included in an updated copy (Version 3) of the paper arxiv:1301.1140 . The proofs involve studying the convex hull of the set of $\lie{h}$-weights $\wt(\V)$ in their own right. Thus, we show that if $\V = L(\lambda)$ is simple, or if \lambda\ is not on a simple root hyperplane and $\V$ is arbitrary, the hull of the infinite set $\wt(\V)$ is a convex polyhedron - i.e., cut out by only finitely many hyperplanes. (This extends the notion of the Weyl polytope to arbitrary simple modules L(\lambda).) It is also shown that the partially ordered set (under quotienting) of modules $\V$ with fixed convex hull, has unique "largest" and "smallest" elements.
Narutaka Ozawa - One of the best experts on this subject based on the ideXlab platform.
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on a class of ii1 factors with at most one Cartan Subalgebra
Annals of Mathematics, 2010Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:We prove that the normalizer of any diffuse amenable Subalgebra of a free group factor L(F-r) generates an amenable von Neumann Subalgebra. Moreover, any II1 factor of the form Q (circle times) over barL(F-r), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan Subalgebras. We also prove that if a free ergodic measure-preserving action of a free group F-r, 2 <= r <= infinity, on a probability space (X, mu) is profinite then the group measure space factor L-infinity (X) F-r has unique Cartan Subalgebra, up to unitary conjugacy.
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On a class of II1 factors with at most one Cartan Subalgebra
Annals of Mathematics, 2010Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:We prove that the normalizer of any diffuse amenable Subalgebra of a free group factor L(F-r) generates an amenable von Neumann Subalgebra. Moreover, any II1 factor of the form Q (circle times) over barL(F-r), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan Subalgebras. We also prove that if a free ergodic measure-preserving action of a free group F-r, 2
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on a class of ii1 factors with at most one Cartan Subalgebra ii
American Journal of Mathematics, 2010Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:This is a continuation of our previous paper studying the structure of Cartan Subalgebras of von Neumann factors of type ${\rm II}_1$. We provide more examples of ${\rm II}_1$ factors having either zero, one, or several Cartan Subalgebras. We also prove a rigidity result for some group measure space ${\rm II}_1$ factors.
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on a class of ii_1 factors with at most one Cartan Subalgebra ii
arXiv: Operator Algebras, 2008Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:This is a continuation of our previous paper studying the structure of Cartan Subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan Subalgebras. We also prove a rigidity result for some group measure space II_1 factors.
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on a class of mathrm ii _1 factors with at most one Cartan Subalgebra
arXiv: Operator Algebras, 2007Co-Authors: Narutaka Ozawa, Sorin PopaAbstract:We prove that the normalizer of any diffuse amenable Subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann Subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan Subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,\mu)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan Subalgebra, up to unitary conjugacy.
Xin Li - One of the best experts on this subject based on the ideXlab platform.
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every classifiable simple c algebra has a Cartan Subalgebra
arXiv: Operator Algebras, 2019Co-Authors: Xin LiAbstract:We construct Cartan Subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan Subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan Subalgebra.
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Cartan Subalgebras and the uct problem
Advances in Mathematics, 2017Co-Authors: Selcuk Barlak, Xin LiAbstract:We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan Subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan Subalgebras. This observation allows us to deduce, among other things, that a crossed product O2⋊αZp satisfies the UCT if there is some automorphism γ of O2 with the property that γ(D2)⊆O2⋊αZp is regular, where D2 denotes the canonical masa of O2. We prove that this condition is automatic if γ(D2)⊆O2⋊αZp is not a masa or α(γ(D2)) is inner conjugate to γ(D2). Finally, we relate the UCT problem for separable, nuclear, M2∞-absorbing C*-algebras to Cartan Subalgebras and order two automorphisms of O2.
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Cartan Subalgebras and the UCT problem, II
arXiv: Operator Algebras, 2015Co-Authors: Selcuk Barlak, Xin LiAbstract:We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan Subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan Subalgebras. This observation allows us to deduce, among other things, that a crossed product $\mathcal O_2\rtimes_\alpha \mathbb Z_p $ satisfies the UCT if there is some automorphism $\gamma$ of $\mathcal O_2$ with the property that $\gamma(\mathcal D_2)\subseteq \mathcal O_2\rtimes_\alpha \mathbb Z_p$ is regular, where $\mathcal D_2$ denotes the canonical masa of $\mathcal O_2$. We prove that this condition is automatic if $\gamma(\mathcal D_2)\subseteq \mathcal O_2\rtimes_\alpha \mathbb Z_p$ is not a masa or $\alpha(\gamma(\mathcal D_2))$ is inner conjugate to $\gamma(\mathcal D_2)$. Finally, we relate the UCT problem for separable, nuclear, $M_{2^\infty}$-absorbing C*-algebras to Cartan Subalgebras and order two automorphisms of $\mathcal O_2$.