Between *-closed sets and ℐ-<Emphasis Type=Italic>g</Emphasis>-closed sets in ideal topological spaces

Dontchev et al. [4] introduced the notion of ℐ-g-closed sets. In [16], the further properties of ℐ-g-closed sets are investigated. In this paper, we introduce the notion of ℐ-mg-closed sets and obtain the unified characterizations for certain families of subsets between *-closed sets and ℐ-g-closed sets in an ideal topological space.     

Nuclear multilinear operators with respect to a partition

We introduce the notion of multilinear nuclear operator with respect to a partition and extend Grothendieck??s characterizations of nuclear linear operators into $l_{1},$ resp. $L_{1}( \mu ),$ in this new setting. We give the necessary and sufficient conditions for a natural operator to be nuclear with respect to a partition.     

Infinite divisibility and a non-commutative Boolean-to-free Bercovici–Pata bijection

We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue for the Bercovici–Pata bijection. An important tool is Voiculescu?s subordination property for operator-valued free convolution.     

Hyers–Ulam stability of a first order partial differential equation

We prove that the existence of a global prime integral leads, in appropriate conditions, to the Hyers–Ulam stability of a linear partial differential equation of first order.     

An uncountable family of nonorbit equivalent actions of

For each , we construct an uncountable family of free ergodic measure preserving actions of the free group on the standard probability space such that any two are nonorbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid' (in the sense of Popa), with the IIfactors mutually nonisomorphic (even nonstably isomorphic) and in the class

     

Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras and of a type II₁ factor . For a map φ on , we define

     

Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

We consider amalgamated free product II₁ 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 II₁ factors, with B equal to either C, to a Cartan subalgebra A in M _i , or to a regular hyperfinite II₁ 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 , with {t _i } _i⩾1 = S a given countable subgroup of R ₊ ^*, we obtain continuously many non-stably isomorphic factors M with fundamental group equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying and Out(M) abelian and calculable. Taking B = R, we get examples of factors with , Out(M) = K, for any given separable compact abelian group K.     

On the best constant in Hyers–Ulam stability of some positive linear operators

We obtain the infimum of the Hyers–Ulam stability constants for Stancu, Bernstein and Kantorovich operators and prove that in a class of certain positive linear operators this infimum for Bernstein operator has a minimality property.     

Almost summing and multiple almost summing multilinear operators on l p spaces

We use the Maurey–Rosenthal factorization theorem to give a characterization of almost summing and multiple almost summing multilinear operators on a cartesian product of l _p spaces. As applications we give the necessary and sufficient conditions for the multiplication operator on l _p spaces be almost summing and multiple almost summing.     

The Hyers-Ulam stability of nonlinear recurrences

We show some Hyers-Ulam type stability results for some nonlinear recurrences in metric spaces.