1,721,018 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 comparison between the max and min norms on C∗(Fn)⊗C∗(Fn).
Full text linkAbstract. 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)
Cyclic Hilbert Spaces
Full text linkWe analyse in this paper a concept related to the Connes Embedding Problem [Co]. A type II1 algebra is an algebra with a trace, and CEP requires for the multiplication to be approximated by matrices. Here we start the analysis of four products, which is the study of cyclic Hilbert spaces
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
Convex sets associated with von~Neumann algebras and Connes' approximate embedding problem
Full text linkAbstract
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
Singularity of the radial subalgebra of ℒ(FN) and the Pukánszky invariant
Full text linkLet 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
Arithmetic Hecke operators as completely positive maps
Full text linkWe prove that the arithmetic Hecke operators are completely positive maps with respect to the Berezin's quantization deformation product of functions on H/Gamma. We then show that the associated subfactor defined by the Connes's correspondence associated to the completely positive map has integer index and graph A(infinity). The same construction for PSL(3, Z) gives a finite index subfactor of L(PSL(3, Z)) of infinite depth
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