On σ-algebras related to the measurability of compositions

Given a measurable space (T, F), a set X, and a map ?: T → X, the σ-algebras N _Ф = ?_?∈Φ N _?, and M _Φ = ?_?∈Φ N _?, where G _?(t) = (t, ?(t)) and Φ ? X ^T , are considered. These σ-algebras are used to characterize the (F, B, ?)-measurability of the compositions g ○ ? and f о G _?, where g: X → Y, f: T × X → Y, and (Y, ?) is a measurable space. Their elements are described without using the operations ? ^?1 and G _? ^?1 .     

Some examples and counterexamples of convergence of σ-algebras and filtrations

We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of σ-algebras to filtrations generated by different discretizations of a single process. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 295–306, July–September, 2000. Translated by V. Mackevičius     

Characterization of numerical radius parallelism in $$C^*$$C∗-algebras

Positivity - Let v(x) denote the numerical radius of an element x in a $$C^*$$-algebra $$\mathfrak {A}$$. First, we prove several numerical radius inequalities in $$\mathfrak {A}$$. Particularly,...     

Completely Positive Definite Maps on σ-C*-algebras

Recently,there has been increased interest[1-8]in topological*-algebras that are inverse     

Tracial Limit of C^＊-algebras

A new limit of C^*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C^*-algebra A is a tracial limit of C^*-algebras in Z-(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.     

K-theory of Continuous Deformations of C＊-algebras

We study K-theory of continuous deformations of C＊-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.     

Comparison of topologies on ⁎-algebras of locally measurable operators

We consider the local measure topology ${t(\mathcal{M})}$ on the ?-algebra ${LS(\mathcal{M})}$ of all locally measurable operators and on the ?-algebra ${S(\mathcal{M},\tau)}$ of all τ-measurable operators affiliated with a von Neumann algebra ${\mathcal{M}}$ . If τ is a semifinite but not a finite trace on ${\mathcal{M},}$ then one can consider the τ-local measure topology t _τ l and the weak τ-local measure topology t _{w τ l} . We study relationships between the topology ${t(\mathcal{M})}$ and the topologies t _τ l , t _{w τ l} , and the (o)-topology ${t_o(\mathcal{M})}$ on ${LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^\ast=T\}}$ . We find that the topologies ${t(\mathcal{M})}$ and t _τ l (resp. ${t(\mathcal{M})}$ and t _{w τ l} ) coincide on ${S(\mathcal{M},\tau)}$ if and only if ${\mathcal{M}}$ is finite, and ${t(\mathcal{M})=t_o(\mathcal{M})}$ on ${LS_h(\mathcal{M})}$ holds if and only if ${\mathcal{M}}$ is a σ-finite and finite. Moreover, it turns out that the topology t _{τ l} (resp. t _{w τ l} ) coincides with the (o)-topology on ${S_h(\mathcal{M},\tau)}$ only for finite traces. We give necessary and sufficient conditions for the topology ${t(\mathcal{M})}$ to be locally convex (resp., normable). We show that (o)-convergence of sequences in ${LS_h(\mathcal{M})}$ and convergence in the topology ${t(\mathcal{M})}$ coincide if and only if the algebra ${\mathcal{M}}$ is an atomic and finite algebra.     

Cardinalities of σ-complete OML's

In this paper we show that the Comfort-Hager result on cardinalities of-complete Boolean algebras is also true for-complete OML's having a bound on the number of complements. Using the Kaplansky theorem on continuous geometries we get a result on modular ortho-lattices. We also get a result on cardinalities of saturated OML's.Presented by R. McKenzie.Supported by the NSF of Srbija through Math. Inst., Grant #401A.     

On σ-Embedded and σ-n-Embedded Subgroups of Finite Groups

Siberian Mathematical Journal - Let G be a finite group, and let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σi | σi ∩...     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

On 𝒪ℒ∞ structures of nuclear C * -algebras

 We study the local operator space structure of nuclear C ^* -algebras. It is shown that a C ^* -algebra is nuclear if and only if it is an 𝒪ℒ_∞,λ space for some (and actually for every) λ>6. The 𝒪ℒ_∞ constant λ provides an interesting invariant

Motivic Milnor Fibre of Cyclic L_∞-algebras

We define the motivic Milnor fiber of cyclic L_∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L_∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.     

On σ-images of Metric Spaces

In this paper the relation between the σ-images of metrical spaces and spaces with σ-locally finite cs-network, or spaces with σ-locally finite cs^*-network, or spaces with σ-locally finite sequence neighborhood network, or spaces with σ-locally finite sequence open network are established by use of σ-mapping.     

Diagonal Invariant Ideals of Topologically Graded C*-algebras

We study diagonal invariant ideals of topologically graded C~*-algebras over discrete groups. Since all Toeplitz algebras defined on discrete groups are topologically graded, the results in this paper have improved the first author's previous works on this topic.