122,356 research outputs found
Sous-facteurs de L(F∞) d'indice 4cos2π/n,n≥3
No full textLet Q be a factor of type II1, λ a number in the Jones discrete series {4cosπ/m:m≥3}, and {ei} the Jones projections associated with λ. Denote by A2n and A1n the finite-dimensional von Neumann algebras generated, respectively, by {1,e2,⋯,en} and {1,e1,⋯,en}, with the corresponding traces. The author shows that, for n sufficiently large, the index of the inclusion An=(Q⊗A2n)∗A2nA1n⊂(Q⊗A2n+1)∗A2n+1A1n+1=An+1 is equal to λ (here ∗ denotes the reduced, amalgamated free product of the algebras in question). Using the random matrix model of Voiculescu, he proves that if Q is the von Neumann algebra L(F∞) of the free group with infinitely many generators, then An is isomorphic to L(F∞).
The two facts together imply the existence, for any λ in the Jones discrete series, of an irreducible subfactor of L(F∞) of index λ. This constitutes the first example of a nonhyperfinite, non-Γ II1 factor such that its Jones invariant is fully computable (the existence of nonirreducible subfactors of L(F∞) for any index ≥4 is a simple consequence of known results)
A non-commutative, analytic version of Hilbert's 17th problem in type II1 von Neumann algebras
No full textWe prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices
SUBFACTORS OF L (F-INFINITY) WITH INDEX 4-COS2-PI/N, N-GREATER-THAN-OR-EQUAL-TO 3
No full textWe introduce a noncommutative probability model (in the sense of Voiculescu) for the (reduced) amalgamated free product (L (F(N))X A) A*B, where A subset-or-equal-to B is an inclusion of finite dimensional algebras (with trace). Using this model we prove that A(lambda)n = (L (F(infinity))X A(n)) A(n)*A(n+1) is isomorphic to L (F(infinity)) where A(n) = {e2,..., e(n)}", A(n+1) = {e1,..., e(n)}", (e(i))i being the Jones projections associated to an index value lambda--1. For lambda--1 in Jones' discrete series {4 cos2 pi/m\m greater-than-or-equal-to 3} and n big enough, A(lambda)n is-approximately-equal-to L (F(infinity) is an irreducible subfactor of index lambda--1 in A(lambda)n+1 is-approximately-equal-to L (F(infinity)) of index lambda--1
Singularity of the radial subalgebra of ℒ(FN) and the Pukánszky invariant
No full textLet ℒ(F N ) be the von Neumann algebra of the free group with N generators x 1 ,⋯,x N , N≥2 and let A be the abelian von Neumann subalgebra generated by x 1 +x 1 -1 +⋯+x N +x N -1 acting as a left convolutor on ℓ 2 (F N ). The radial algebra A appeared in the harmonic analysis of the free group as a maximal abelian subalgebra of ℒ(F N ), the von Neumann algebra of the free group. The aim of this paper is to prove that A is singular (which means that there are no unitaries u in ℒ(F N ) except those coming from A such that u * Au⊆A). This is done by showing that the Pukánszky invariant of A is infinite, where the Pukánszky invariant of A is the type of the commutant of the algebra A in B(ℓ 2 (F N )) generated by A and x 1 +x 1 -1 +⋯+x N +x N -1 regarded also as a right convolutor on ℓ 2 (F N
A comparison between the max and min norms on C∗(Fn)⊗C∗(Fn).
No full textAbstract. Let Fn, n > 2, be the free group on n generators, denoted by U1,U2, . . . ,Un. Let C¤(Fn) be the full C¤-algebra of Fn. Let X be the
vector subspace of the algebraic tensor product C(Fn) C¤(Fn), spannedby 1 1,U1 1, . . . ,Un 1, 1 U1, . . . , 1 Un. Let k · kmin and k · kmax be
the minimal and maximal C¤ tensor norms on C¤(Fn)C¤(Fn), and use the
same notation for the corresponding (matrix) norms induced on Mk(C)X,
k 2 N.
Identifying X with the subspace of C¤(F2n) obtained by mapping U1
1, . . . , 1Un into the 2n generators and the identity into the identity, we get
a matrix norm k · kC¤(F2n) which dominates the k · kmax norm on Mk(C)X.
In this paper we prove that, with N = 2n + 1 = dimX, we have
kXkmax 6 kXkC¤(F2n) 6 (N2 − N)1/2kXkmin, X 2 Mk(C)
B. Lortat-Jacob, Musiques en fête. Maroc, Sardaigne, Roumanie
No full textRadulescu Speranta. B. Lortat-Jacob, Musiques en fête. Maroc, Sardaigne, Roumanie. In: L'Homme, 1996, tome 36 n°138. pp. 174-176
Convex sets associated with von~Neumann algebras and Connes' approximate embedding problem
No full textAbstract
Connes' approximate embedding problem, asks whether any countably generated type II1
II1 factor M
Mcan be approximately embedded in the hyperfinite type II1
factor. Solving this problem in the affirmative, amounts to showing that given any integers N,p
, any elements x1,…,xN
in Mand any ϵ>0 one can find k
and matrices X1,…,XN
in the algebra Mk(ℂ), endowed with the normalized trace \ tr
, such that for every i1,…,ip∈{1,…,N}
and for every s
with 1≤s≤p
s that |τ(xi1…xis)−\ tr\ (Xi1…Xis)|<ϵ.
In this paper we show that this is always possible if s
s
is 2 and 3
AN INVARIANT FOR SUBFACTORS IN THE VONNEUMANN ALGEBRA OF A FREE GROUP
No full textIn this Note we are considering a new invariant for subfactors in the von Neumann algebra L(F(k)) of a free group. This invariant is obtained by computing the Connes's chi invariant for the enveloping von Neumann algebra in the iteration of the Jone's basic construction for the given inclusion. In the case of the subfactors considered in [22], [24] this invariant is easily computed as a relative chi invariant, in the form considered in [14].
One considers the inclusion L (F(n)) subset-or-equal-to A = L (F(n)) x (theta)Z(k)2, where theta is an injective homomorphism from Z(k) into Out (L (F(n))) (i. e. a Z(n)-kernel) with minimal period k2 [in Aut (L (F(n)))]. Then there exists a canonical copy theta of Z(k) in chi(A) which can be lifted to Aut(A) [4]. The decomposition of the generator of the dual action of Z(k) on A x (theta)Z(k) as the product of a centrally trivial automorphism and an almost inner automorphism, gives an action of Z(k)2 + Z(k)2 on the algebra A x (theta)Z(k). The algebraic invariants [9] for this last action give a more subtle invariant for theta.
As an application we show that, contrary to the case of finite group actions (or more general G-kernels) on the hyperfine II1 factor (settled in [2], [9], [18]), there exists non outer conjugate, injective homomorphisms (i.e. two Z2-kernels) from Z, into Out(L(F(k))), with non-trivial obstruction to lifting to an action on L (F(k)). Moreover the algebraic invariants [3] do not distinguish between these two Z2-kernels. Also, there exists two non-outer conjugate, outer actions of Z2 on L(F(k)) x R that are neither almost inner or centrally trivial
Free Group Factors and Hecke Operators
No full textThis is a note on Radulescu’s work taken by N. Ozawa. We first review the
construction of the classical Hecke operator and recall the Ramanujan–Petersson conjecture
about its spectrum. We then relate it to the study of a type II1 factor via Berezin
calculus
Finite generation properties for fuchsian group von Neumann algebras tensor
No full textWe prove that the algebra , a free group with finitely many generators, contains a subnormal operator such that the linear span of the set is weakly dense in . This is the analogue for the factor , finite, of a well known fact about the unilateral shift on a Hilbert space : the linear span of all the monomials is weakly dense in .
We also show that for a suitable space of square summable analytic functions, if is the projection from the Hilbert space of all square summable functions onto and is the unbounded operator of multiplication by on , then the (unbounded) operator is nonzero (with nonzero domain)
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