We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan ⁎-isomorphisms. In particular, we prove ...that two von Neumann algebras without type I1 direct summands are Jordan ⁎-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Gehér and P. Šemrl, which is a generalization of Wigner's theorem.
Let
be a factor von Neumann algebra with
For any
define
and
for all integers
In this article, we prove that a map
satisfies
for all
if and only if L is an additive *-derivation.
In this paper we extend the classical Bohr’s inequality to the setting of the non-commutative Hardy space H^1 associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr’s ...inequality for operators in the von Neumann-Schatten class \mathcal C_1 and square matrices of any finite order. Interestingly, we establish that the optimal bound for r in the above mentioned Bohr’s inequality concerning von Neumann-Schatten class is 1/3 whereas it is 1/2 in the case of 2\times 2 matrices and reduces to \sqrt {2}-1 for the case of 3\times 3 matrices. We also obtain a generalization of our above-mentioned Bohr’s inequality for finite matrices where we show that the optimal bound for r, unlike above, remains 1/3 for every fixed order n\times n,\ n\ge 2.
A Banach algebra A is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols when A⁎=WAP(A). To identify the opposite behaviour, Granirer called a Banach algebra ...extremely non-Arens regular (enAr, for short) when the quotient A⁎/WAP(A) contains a closed subspace that has A⁎ as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra A is r-enAr, with r≥1, when there is an isomorphism with distortion r of A⁎ into A⁎/WAP(A). When r=1, we obtain an isometric isomorphism and we say that A is isometrically enAr. We then identify sufficient conditions for the predual V⁎ of a von Neumann algebra V to be r-enAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be r-enAr:(i)the weighted semigroup algebra of any weakly cancellative discrete semigroup, when the weight is diagonally bounded with diagonal bound c≥r. When the weight is multiplicative, i.e., when c=1, the algebra is isometrically enAr,(ii)the weighted group algebra of any non-discrete locally compact infinite group and for any weight,(iii)the weighted measure algebra of any locally compact infinite group, when the weight is diagonally bounded with diagonal bound c≥r. When the weight is multiplicative, i.e., when c=1, the algebra is isometrically enAr. The Fourier algebra A(G) of a locally compact infinite group G is shown to be isometrically enAr provided that (1) the local weight of G is greater or equal than its compact covering number, or (2) G is countable and contains an infinite amenable subgroup.
We prove that the regular von Neumann subalgebras B of the hyperfinite II1 factor R satisfying the condition B′∩R=Z(B) are completely classified (up to conjugacy by an automorphism of R) by the ...associated discrete measured groupoid G=GB⊂R. We obtain a similar classification result for triple inclusions A⊂B⊂R, where A is a Cartan subalgebra in R and the intermediate von Neumann algebra B is regular in R. A key step in proving these results is to show the vanishing cohomology for the associated cocycle actions (αB⊂R,uB⊂R) of G on B. We in fact prove two very general vanishing cohomology results for free cocycle actions (α,u) of amenable discrete measured groupoids G on arbitrary tracial von Neumann algebras B, resp. Cartan inclusions A⊂B. Our work provides a unified approach and generalizations to many known vanishing cohomology and classification results 6, 25, 35, 3, 10, 29, etc.
Nous démontrons que les sous-algèbres de von Neumann régulières B du facteur hyperfini II1R vérifiant la condition B′∩R=Z(B) sont entièrement classifiées (à conjugaison près par un automorphisme de R) par le groupoïde discret mesuré G=GB⊂R associé. Nous obtenons un résultat de classification similaire pour les inclusions triples A⊂B⊂R, où A est une sous-algèbre de Cartan de R et la sous-algèbre de von Neumann intermédiaire B est régulière dans R. Une étape clé dans la démonstration de ces résultats est la preuve de l'annulation de cohomologie pour les actions à cocycle associées, (αB⊂R,uB⊂R) de G sur B. Nous démontrons deux résultats très généraux d'annulation de cohomologie pour les actions à cocycle libres (α,u) de groupoïdes discrets mesurés moyennables G sur des algèbres de von Neumann traciales arbitraires B, resp. inclusions Cartan A⊂B. Notre article donne une approche unifiée et des généralisations de plusieurs résultats connus d'annulation de cohomologie et de classification 6, 25, 35, 3, 10, 29, etc.
We identify the optimal range of the Calderòn operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the ...optimal range of the triangular truncation operator, operator Lipschitz functions and commutator estimates in ideals of compact operators.
In this paper, we show that if the reduced Fourier–Stieltjes algebra Bρ(G) of a second countable locally compact group G has either weak* fixed point property or asymptotic center property, then G is ...compact. As a result, we give affirmative answers to open problems raised by Fendler and et al. in 2013. We then conclude that a second countable group with a discrete reduced dual must be compact. This generalizes a theorem of Baggett. We also construct a compact scattered Hausdorff space Ω for which the dual of the scattered C*-algebra C(Ω) lacks weak* fixed point property. The C*-algebra C(Ω) provides a negative answer to a question of Randrianantoanina in 2010. In addition, we prove a variant of Bruck's generalized fixed point theorem for the preduals of von Neumann algebras. Furthermore, we give some examples emphasizing that the conditions in Bruck's generalized conjecture (BGC) can not be weakened any more.
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