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Sorin Popa - One of the best experts on this subject based on the ideXlab platform.
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Amalgamated free products of weakly rigid factors and calculation of their symmetry groups
Acta Mathematica, 2008Co-Authors: Adrian Ioana, Jesse Peterson, Sorin PopaAbstract:We consider amalgamated free product II_1 factors M = M _1* B M _2* B … and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the M _ i ’s. We apply this to the case where the M _ i ’s are w-rigid II_1 factors, with B equal to either C , to a Cartan subalgebra A in M _ i , or to a regular hyperfinite II_1 subfactor R in M _ i , to obtain the following type of unique decomposition results, àla Bass–Serre: If M = ( N _1 * CN_2* C …)^ t , for some t > 0 and some other similar inclusions of algebras C ⊂ N _ i then, after a permutation of indices, ( B ⊂ M _ i ) is inner conjugate to ( C ⊂ N _ i )^ t , for all i . Taking B = C and $ M_{i} = {\left( {L{\left( {Z^{2} \rtimes F_{2} } \right)}} \right)}^{{t_{i} }} $ , with { t _ i }_ i ⩾1 = S a given Countable Subgroup of R _+ ^*, we obtain continuously many non-stably isomorphic factors M with fundamental group $ {\user1{\mathcal{F}}}{\left( M \right)} $ equal to S . For B = A , we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ and Out(M) abelian and calculable. Taking B = R , we get examples of factors with $ {\user1{\mathcal{F}}}{\left( M \right)} = {\left\{ 1 \right\}} $ , Out( M ) = K , for any given separable compact abelian group K .
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Amalgamated Free Products of $w$-Rigid Factors and Calculation of their Symmetry Groups
arXiv: Operator Algebras, 2005Co-Authors: Adrian Ioana, Jesse Peterson, Sorin PopaAbstract:We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$ and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\subset M$ can be intertwined into one of the $M_i$'s. We apply this to the case $M_i$ are w-rigid II$_1$ factors, with $B$ equal to either $\Bbb C$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, \`a la Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other similar inclusions of algebras $C\subset N_j$ then, after a permutation of indices, $(B\subset M_i)$ is inner conjugate to $(C\subset N_i)^t$, $\forall i$. Taking $B=\Bbb C$ and $M_i = (L(\Bbb Z^2 \rtimes \Bbb F_{2}))^{t_i}$, with $\{t_i\}_{i\geq 1}=S$ a given Countable Subgroup of $\Bbb R_+^*$, we obtain continuously many non stably isomorphic factors $M$ with fundamental group $\mycal F(M)$ equal to $S$. For $B=A$, we obtain a new class of factors $M$ with unique Cartan subalgebra decomposition, with a large subclass satisfying $\mycal F(M)=\{1\}$ and Out$(M)$ abelian and calculable. Taking $B=R$, we get examples of factors with $\mycal F(M)=\{1\}$, Out$(M)=K$, for any given separable compact abelian group $K$.
Cyril Houdayer - One of the best experts on this subject based on the ideXlab platform.
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construction of type rm ii_1 factors with prescribed Countable fundamental group
arXiv: Operator Algebras, 2007Co-Authors: Cyril HoudayerAbstract:In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type ${\rm II_1}$ with prescribed fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC $w$-rigid group $G$ such that $\mathcal{F}(L(G)) = \{1\}$ (e.g. $G = \Z^2 \rtimes \SL(2, \Z)$). For any Countable Subgroup $S \subset \R^*_+$, we show that there exists an action of $G$ on $L(\F_\infty)$ such that $L(\F_\infty) \rtimes G$ is a type ${\rm II_1}$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H), \psi)$ any factor of type ${\rm I}$ endowed with a faithful normal state and denote by $\Gamma \subset \R^*_+$ the Subgroup generated by the point spectrum of $\psi$. We show that the centralizer $(L(G) \ast B(H))^{\tau \ast \psi}$ is a type ${\rm II_1}$ factor and its fundamental group is $\Gamma$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.
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Construction of type ${\rm II_1}$ factors with prescribed Countable fundamental group
arXiv: Operator Algebras, 2007Co-Authors: Cyril HoudayerAbstract:In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type ${\rm II_1}$ with prescribed fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC $w$-rigid group $G$ such that $\mathcal{F}(L(G)) = \{1\}$ (e.g. $G = \Z^2 \rtimes \SL(2, \Z)$). For any Countable Subgroup $S \subset \R^*_+$, we show that there exists an action of $G$ on $L(\F_\infty)$ such that $L(\F_\infty) \rtimes G$ is a type ${\rm II_1}$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H), \psi)$ any factor of type ${\rm I}$ endowed with a faithful normal state and denote by $\Gamma \subset \R^*_+$ the Subgroup generated by the point spectrum of $\psi$. We show that the centralizer $(L(G) \ast B(H))^{\tau \ast \psi}$ is a type ${\rm II_1}$ factor and its fundamental group is $\Gamma$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.
Slawomir Solecki - One of the best experts on this subject based on the ideXlab platform.
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measurability properties of sets of vitali s type
Proceedings of the American Mathematical Society, 1993Co-Authors: Slawomir SoleckiAbstract:Assume a group G acts on a set. Given a Subgroup H of G , by an //-selector we mean a selector of the set of all orbits of H. We investigate measurability properties of //-selectors with respect to G-invariant measures. Let us fix a set A and a group G acting on it. By p we denote a G-invariant countably additive measure on A. The most common example of such a situ- ation is an invariant measure on a group acting on itself by translations. Let H be a Subgroup of G. By an H-selector (sometimes called a set of Vitali's type for H) we understand a set having exactly one point in common with each orbit of H. Measurability properties of selectors were first systematically studied by Cichon, Kharazishvili, and Weglorz in (1). Selectors are extremely useful in constructing sets nonmeasurable with respect to an invariant measure. The first example of a Lebesgue nonmeasurable set, due to Vitali (8), is just a Q-selector where Q is the group of rationals. Also for any finite invariant diffused measure on a group (acting on itself by translations) any //-selector for a Countable Subgroup H is nonmeasurable. In fact, in both cases above the constructed sets are nonmeasurable with respect to any invariant extension of a given measure. Kharazishvili in (3) and Erdos and Mauldin in (2) constructed a nonmeasurable set for any cr-finite invariant measure. Their example is the union of a family of //-selectors where H is a Subgroup of cardinality cox . Strengthening the result from (2, 3) the author constructed in (6) sets nonmeasurable with respect to any invariant extension of a given CT-finite measure. These sets are subsets of //-selectors for an appropriately chosen Countable group H. In the present paper we take a closer look at measurability properties of selectors. Putting a freeness assumption on the action of G and assuming that G is unCountable we prove that for a rr-finite measure one can always find a Countable group H such that no //-selector is measured by any invariant extension of the given measure. We show also that the situation for Subgroups of full cardinality is just the opposite. Imposing a stronger freeness condition and
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Measurability properties of sets of Vitali’s type
Proceedings of the American Mathematical Society, 1993Co-Authors: Slawomir SoleckiAbstract:Assume a group G acts on a set. Given a Subgroup H of G , by an //-selector we mean a selector of the set of all orbits of H. We investigate measurability properties of //-selectors with respect to G-invariant measures. Let us fix a set A and a group G acting on it. By p we denote a G-invariant countably additive measure on A. The most common example of such a situ- ation is an invariant measure on a group acting on itself by translations. Let H be a Subgroup of G. By an H-selector (sometimes called a set of Vitali's type for H) we understand a set having exactly one point in common with each orbit of H. Measurability properties of selectors were first systematically studied by Cichon, Kharazishvili, and Weglorz in (1). Selectors are extremely useful in constructing sets nonmeasurable with respect to an invariant measure. The first example of a Lebesgue nonmeasurable set, due to Vitali (8), is just a Q-selector where Q is the group of rationals. Also for any finite invariant diffused measure on a group (acting on itself by translations) any //-selector for a Countable Subgroup H is nonmeasurable. In fact, in both cases above the constructed sets are nonmeasurable with respect to any invariant extension of a given measure. Kharazishvili in (3) and Erdos and Mauldin in (2) constructed a nonmeasurable set for any cr-finite invariant measure. Their example is the union of a family of //-selectors where H is a Subgroup of cardinality cox . Strengthening the result from (2, 3) the author constructed in (6) sets nonmeasurable with respect to any invariant extension of a given CT-finite measure. These sets are subsets of //-selectors for an appropriately chosen Countable group H. In the present paper we take a closer look at measurability properties of selectors. Putting a freeness assumption on the action of G and assuming that G is unCountable we prove that for a rr-finite measure one can always find a Countable group H such that no //-selector is measured by any invariant extension of the given measure. We show also that the situation for Subgroups of full cardinality is just the opposite. Imposing a stronger freeness condition and
Vincent Le Prince - One of the best experts on this subject based on the ideXlab platform.
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Matrix random products with singular harmonic measure
Geometriae Dedicata, 2011Co-Authors: Vadim A. Kaimanovich, Vincent Le PrinceAbstract:Any Zariski dense Countable Subgroup of $${SL(d, \mathbb {R})}$$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $${\mathbb {R}^d}$$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and arbitrarily long Lyapunov vector.
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Matrix random products with singular harmonic measure
arXiv: Probability, 2008Co-Authors: Vadim A. Kaimanovich, Vincent Le PrinceAbstract:Any Zariski dense Countable Subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $R^d$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.
N. C. Phillips - One of the best experts on this subject based on the ideXlab platform.
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A simple separable C*-algebra not isomorphic to its opposite algebra
Proceedings of the American Mathematical Society, 2004Co-Authors: N. C. PhillipsAbstract:We give an example of a simple separable C*-algebra that is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial K 1 , and its K 0 -group is order isomorphic to a Countable Subgroup of R.
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A simple separable C*-algebra not isomorphic to its opposite algebra
arXiv: Operator Algebras, 2002Co-Authors: N. C. PhillipsAbstract:We give an example of a simple separable C*-algebra which is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial K_1, and its K_0-group is order isomorphic to a Countable Subgroup of the real numbers.
