We consider the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that these spaces have the uniform Opial property with respect to convergence ρ a.e. for both the Luxemburg norm and the Amemiya norm. The above geometric properties will enable us to obtain some results in noncommutative Orlicz spaces.
The Opial property originates from the fixed point theorem proved by Opial (1967). The uniform Opial property with respect to the weak topology was defined by Prus (1992). It is well known that the Opial property of a Banach space X play an important role in metric fixed point theory for nonexpansive mappings, as well as in the theory of differential and integral equations. The Opial property also plays an important role in the study of weak convergence of iterates, random products of nonexpansive mappings.
Definition We say that a Banach space X has the Opial property w.r.t. a topology τ if for any τ-null sequence {x n } in X and every x 0 in X there holds lim inf n x n < lim inf n x n + x. Opial proved that l p (1 < p < ) has this property, but L p [0, 2π] does not have it if (1 < p < ), p 2, w.r.t. weak topology.
Definition We say that X has the uniform Opial property with respect a topology τ if for every ε > 0 there exists δ > 0 such that for any x X with x ε and τ-null sequence {x n } in the unit sphere S(X ) of X there holds 1 + δ lim inf n x n + x. It is obvious that the uniform Opial property implies the Opial property.
Associated to the uniform Opial property, we can define the following modules: o τ (α) = inf{lim inf n x n + x 1}, where the infimum is taken over all x X with x α and all sequence {x n } such that τ lim n x n = 0 and lim inf n x n 1. It is clear that X has the uniform Opial property with respect to τ iff o τ (α) > 0 for every α > 0.
A trace τ : M + [0, ] is called : i) faithful if τ(x) > 0 whenever 0 < x M; ii) semi finite if for every x M + with τ(x) > 0 there exists 0 y x such that 0 < τ(y) < ; iii) normal if τ(x α ) τ(x) whenever x α x in M +. Where x α x means that α x α is an increasing net converging to x in the strong operator topology. A von Neumann algebra equipped with a semi finite, faithful, normal trace is called a semi finite von Neumann algebra.
Example Let H be a Hilbert space and M = B(H). Given an orthonormal basis {e i } in H we define τ(x) = Σ i I xe i, e i, x B(H) +. It is clear that τ : B(H) + [0, ] is a semi finite faithful normal trace on B(H). This is called the standard trace on B(H).
A linear operator x : D(x) H with domain D(x) H is called affiliated with M, if ux = xu for all unitaries u in the commutant M of M. This is denoted by xηm. Note that the equality ux = xu involves the equality of the domains of the operators ux and xu, that is, D(x) = u 1 (D(x)). If x is in the algebra B(H) of all bounded linear operators on the Hilbert space H, then x is affiliated with M if and only if x M. If x is a self adjoint operator affiliated with M, then the spectral projection χ B (x) is an element of M for any Borel set B R.
Here χ denotes the usual indicator function. The closed and densely defined operator x, affiliated with M, is called τ measurable if and only if there exist a number s 0 such that τ ( χ (s, ) ( x ) ) <. The collection of all τ measurable operators is denoted by M.
Measure topology Given 0 < ε, δ R, we define V(ε, δ) to be the set of all x M for which there exists p P(M) such that xp B(H) ε and τ(1 p) δ. An alternative description of this set is given by { V(ε, δ) = x M ; τ ( χ (ε, ) ( x ) ) } < δ. The collection {V(ε, δ)} ε,δ>0 is a neighborhood base at 0 for a vector space topology τ m on M.
For x M, the generalized singular value function µ(x,.) = µ( x,.) is defined by { ( ) } µ(x; t) = inf λ 0; τ e x (λ, ) t, t 0. It follows directly that the generalized singular value function µ(x) is a decreasing right continuous function on the positive half-line [0, ).
In the following proposition, we list some properties of the rearrangement mapping µ(.; t). Theorem Let x, y and z be τ measurable operators. (i) the map t (0, ) µ(x; t) is non increasing and continuous from the right. Moreover, lim µ(x; t) = x [0, ]. t 0 (ii) µ(x; t) = µ( x ; t) = µ(x ; t). (iii) µ(x; t) µ(y; t), t > 0, if 0 x y. (iv) µ(x + y; t + s) µ(x; t) + µ(y; s), t, s > 0. (v) µ(zxy; t) z y µ(x; t), t > 0. (vi) µ(xy; t + s) µ(x; t)µ(y; s) t, s > 0.
Let Ω be a nonempty set and Σ be a nontrivial σ algebra of subsets of Ω. Let P be a δ ring of subsets of Ω, such that E A P for any E P and A Σ. Let us assume that there exists an increasing sequence of sets K n P such that Ω = K n. By E we denote the linear space of all simple functions with supports from P. By M we will denote the space of all measurable functions, i.e., all functions f : Ω R such that there exists a sequence {s n } E, s n f and s n (ω) f (ω) for all ω Ω. Let us recall that a set function ν : Σ [0, + ] is called a σ subadditive measure if ν( ) = 0, ν(a) ν(b) for any A B and ν( n A n ) Σ n ν(a n ) for any sequence of sets {A n } Σ.
Definition A functional ρ : E Σ [0, ] is called a function modular if (i) ρ(0, A) = 0 for any A Σ; (ii) ρ(f, A) ρ(g, A) whenever f (ω) g(ω) for anr ω Ω, f, g E, A Σ; (iii) ρ(f, ) : Σ [0, + ] is a σ subadditive measure for every f E; (iv) ρ(α, A) 0 as α decreases to 0 for every A P, where ρ(α, A) = ρ(αχ A, A); (v) If there exists α > 0 such that ρ(α, A) = 0, then ρ(β, A) = 0 for every β > 0; (vi) For any α > 0, ρ(α, ) is order continuous on P, i.e., ρ(α, A n ) 0 if {A n } P and decreases to.
When ρ satisfies (iii) ρ(f, ) : Σ [0, + ] is a σ subadditive measure, we say that ρ is a additive if ρ(f, A B) = ρ(f, A) + ρ(f, B) whenever A, B Σ such that A B = and f M. The definition of ρ is extended to all f M by ρ(f, A) = sup{ρ(s, A); s E, s(x) f (x) for every ω Ω}.
In the previous conditions, we define the functional ρ : M [0, + ] given by ρ(f ) = ρ(f, Ω). Then it is easy to check that ρ is a modular. A function modular ρ defines a corresponding modular function space, i.e., the vector space L ρ given by L ρ = {x M : ρ(λx) 0 as λ 0}. When ρ is convex, the formulas { f ρ = inf λ > 0 : and f A ρ = inf ( ) } f ρ 1, λ { } 1 ρ (1 + αf ) : α > 0, α define two complete norms on L ρ which are called the Luxemburg norm and Amemiya norm, respectively..
Example Orlicz Musielak spaces: Let (Ω, Σ, ν) be a measure space, where ν is a positive σ finite measure. Let us denote by P the δ ring of all sets of finite measure. Define the modular ρ by the formula ρ ϕ (f, E) = ϕ(t, f (t) )dν(t), where ϕ is an Orlicz function. E
Definition Similarly as in the case of measure spaces, we say that a set A Σ is ρ null if ρ(α, A) = 0 for every α > 0. A property p(ω) is said to hold ρ almost everywhere (ρ a.e.) if the set {ω Ω : p(ω) does not hold} is ρ null. For example, we will frequently say f n f ρ-a.e., which means that {ω Ω : f (ω) lim n f n (ω)} is ρ null. As usual we identify any pair of measurable sets whose symmetric difference is ρ null as well as any pair of measurable functions differing only on a ρ null set.
Definition A function modular ρ is said to satisfy the 2 condition if there exists k > 0 such that for any f L ρ we have ρ(2f ) kρ(f ). Notice that the 2 condition implies that 0 < ρ(kf ) < for every k > 0 provided 0 < ρ(f ) <
In this section, by L 0 (ν) we denote the linear space of all (equivalence class of) real valued Lebesgue measurable functions on the Ω. Let ρ be a convex additive function modular on L 0 (ν) with the 2 condition such that for every f, g L 0 (ν), if f is submajorized by g (f g) in the sense of Hardy, Littlewood and Polya implies that ρ(f ) ρ(g) and L ρ denoted by the modular function space associated to ρ on L 0 (ν).
The noncommutative modular function space L ρ ( M, τ) is defined by { } L ρ ( M, τ) = x M : ρ(λx) 0 as λ 0, where the functional ρ on M is defined by ρ(x) = ρ(µ( x )). The vector space L ρ ( M, τ) can be equipped with the Luxemburg norm defined by { ( x } x ρ = inf λ > 0 : ρ 1. λ)
Similar to the commutative case one can define the Amemiya norm as follows: { } 1 x à ρ = inf (1 + ρ(λx)) : λ > 0. λ We have also the following relation between these norms x ρ x à ρ 2 x ρ.
Theorem The noncommutative modular function space L ρ ( M, τ) is a Banach space.
Definition Let {x n } and x be in M. Then the sequence {x n } is said to be ρ a.e. convergent to x if µ(x n x) 0 ρ a.e.
Theorem The noncommutative modular function space L ρ ( M, τ) equipped with the Luxemburg norm ρ (Amemiya norm A ρ ) has the uniform Opial property with to respect convergence ρ a.e.
Corollary Let ϕ satisfy the 2 condition. Then the noncommutative Orlicz space L ϕ ( M, τ) equipped with the Luxemburg norm (Amemiya norm A ρ ) has the uniform Opial property with respect to convergence in measure.
Corollary The noncommutative L p space L p ( M, τ) has the uniform Opial property with respect to convergence in measure.
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