A new version of the Gleason-Kahane-Żelazko theorem in complete random normed algebras

Gleason [1] and, independently, Kahane and Żelazko [2] proved the so-called Gleason-Kahane-Żelazko theorem which is a famous theorem in classical Banach algebras. There are various extensions and generalizations of this theorem [3]. The Gleason-Kahane-Żelazko theorem in an unital complete random normed algebra as a random generalization of the classical Gleason-Kahane-Żelazko theorem is given in [4].

Based on the study of [5], we will establish a new version of the Gleason-Kahane-Żelazko theorem in an unital complete random normed algebra. In this article we first present the notion of multiplicative L⁰-functions. Then, we give the new version of the Gleason-Kahane-Żelazko theorem in an unital complete random normed algebra as another random generalization of the classical Gleason-Kahane-Żelazko theorem.

The remainder of this article is organized as follows: in Section 2 we give some necessary definitions and lemmas and in Section 3 we give the main results and proofs.

Throughout this article, N denotes the set of positive integers, K the scalar field R of real numbers or C of complex numbers, R ̄ (or [-∞, +∞]) the set of extended real numbers, ( Ω , ℱ , P ) a probability space, ℒ ̄ 0 ( ℱ , R ) the set of extended real-valued ℱ-random variables on Ω, L ̄ 0 ( ℱ , R ) the set of equivalence classes of extended real-valued ℱ-random variables on Ω, ℒ 0 ( ℱ , K ) the algebra of K-valued ℱ-random variables on Ω under the ordinary pointwise addition, multiplication and scalar multiplication operations, L 0 ( ℱ , K ) the algebra of equivalence classes of K-valued ℱ-random variables on Ω, i.e., the quotient algebra of ℒ 0 ( ℱ , K ), and 0 and 1 the null and unit elements, respectively.

It is well known from [6] that L ̄ 0 ( ℱ , R ) is a complete lattice under the ordering ≤: ξ ≤ η iff ξ⁰(ω) ≤ η⁰(ω) for P-almost all ω in Ω (briefly, a.s.), where ξ⁰ and η⁰ are arbitrarily chosen representatives of ξ and η, respectively. Furthermore, every subset A of L ̄ 0 ( ℱ , R ) has a supremum, denoted by ∨A, and an infimum, denoted by ∧A, and there exist two sequences {a _n , n ∈ N} and {b _n , n ∈ N} in A such that ∨_n≥1a _n = ∨A and ∧_n≥1b _n = ∧A. If, in addition, A is directed (accordingly, dually directed), then the above {a _n , n ∈ N} (accordingly, {b _n , n ∈ N}) can be chosen as nondecreasing (accordingly, nonincreasing). Finally L 0 ( ℱ , R ), as a sublattice of L ̄ 0 ( ℱ , R ), is complete in the sense that every subset with an upper bound has a supremum (equivalently, every subset with a lower bound has an infimum).

Specially, let L ̄ + 0 ( ℱ ) = { ξ ∈ L ̄ 0 ( ℱ , R ) | ξ ≥ 0 } and L + 0 ( ℱ ) = { ξ ∈ L 0 ( ℱ , R ) | ξ ≥ 0 }.

The following notions of generalized inverse, absolute value, complex conjugate and sign of an element in L 0 ( ℱ , K ) bring much convenience to this article.

Definition 2.1. [7] Let ξ be an element in L 0 ( ℱ , K ). For an arbitrarily chosen representative ξ⁰ of ξ, define two ℱ-random variables (ξ⁰)^-1 and |ξ⁰|, respectively, by

and

Then the equivalence class of (ξ⁰)^-1, denoted by ξ^-1, is called the generalized inverse of ξ; the equivalence class of |ξ⁰|, denoted by |ξ|, is called the absolute value of ξ. When ξ ∈ L 0 ( ℱ , C ), set ξ = u + iv, where u , v ∈ L 0 ( ℱ , R ) , ξ ̄ : = u - i v is called the complex conjugate of ξ and sgn(ξ) := |ξ|^-1 · ξ is called the sign of ξ. It is obvious that ξ = ξ ̄ , ξ ⋅ sgn ( ξ ̄ ) = ξ , sgn ( ξ ) = Ĩ A , ξ - 1 ⋅ ξ = ξ ⋅ ξ - 1 = Ĩ A, where A = {ω ∈ Ω : ξ⁰(ω) ≠ 0} and Ĩ A denotes the equivalence class of the characteristic function I _A of A. Throughout this article, the symbol Ĩ A is always understood as above unless stated otherwise.

Besides the equivalence classes of ℱ-random variables, we also use the equivalence classes of ℱ-measurable sets. Let A ∈ ℱ, then the equivalence class of A, denoted by Ã, is defined by à = { B ∈ ℱ : P ( A Δ B ) = 0 }, where A ΔB = (A\B)∪(B\A) is the symmetric difference of A and B, and P ( à ) is defined to be P(A). For two ℱ-measurable sets G and D, G ⊂ D a.s. means P(G\D) = 0, in which case we also say G ̃ ⊂ D ̃; G ̃ ∩ D ̃ denotes the the equivalence class determined by G ⋂ D. Other similar notations are easily understood in an analogous manner.

As usual, we also make the following convention: for any ξ , η ∈ L 0 ( ℱ , R ) , ξ > η means ξ ≥ η and ξ ≠ η; [ξ > η] stands for the equivalence class of the ℱ-measurable set {ω ∈ Ω : ξ⁰(ω) > η⁰(ω)} (briefly, [ξ⁰ > η⁰]), where ξ⁰ and η⁰ are arbitrarily selected representatives of ξ and η, respectively, and I_[ξ>η]stands for Ĩ [ ξ 0 > η 0 ]. If A ∈ ℱ, then ξ > η on à means ξ⁰(ω) > η⁰(ω) a.s. on A, similarly ξ ≠ η on à means that ξ⁰(ω) ≠ η⁰(ω) a.s. on A, also denoted by à ⊂ [ ξ ≠ η ].

Definition 2.2. [7] An ordered pair (S, || · ||) is called a random normed module (briefly, an RN module) over K with base ( Ω , ℱ , P ) if S is a left module over the algebra L 0 ( ℱ , K ) and || · || is a mapping from S to L + 0 ( ℱ ) such that the following conditions are satisfied:

(RNM-1) ||ξx|| = |ξ|||x||, ∀ ξ ∈ L 0 ( ℱ , K ), x ∈ S;

(RNM-2) ||x + y|| ≤ ||x|| + ||y||, ∀x, y ∈ S;

(RNM-3) ||x|| = 0 implies x = 0(the zero element in S).

Where ||x|| is called the L⁰-norm of the vector x in S.

In this article, given an RN module (S, || · ||) over K with base ( Ω , ℱ , P ) it is always assumed that (S, || · ||) is endowed with its (ϵ, λ)-topology: for any ϵ > 0, 0 < λ < 1, let N(ϵ, λ) = {x ∈ S | P{ω ∈ Ω : ||x||(ω) < ϵ} > 1 - λ}, then the family U 0 = { N ( ε , λ ) | ε > 0 , 0 < λ < 1 } forms a local base at the null element 0 of some metrizable linear topology for S, called the (ϵ, λ)-topology for S. It is well known that a sequence {x _n , n ≥ 1} in S converges in the (ϵ, λ)-topology to some x in S if {||x _n - x||, n ≥ 1} converges in probability P to 0, and that S is a topological module over the topological algebra L 0 ( ℱ , K ), namely the module multiplication · : L 0 ( ℱ , K ) × S → S is jointly continuous (see [7] for details). Besides, let L 0 ( ℱ , K ) be the RN module of equivalence classes of X-valued ℱ-random variables on ( Ω , ℱ , P ), where X is an ordinary normed space, then it is easy to see that the (ϵ, λ)-topology on L 0 ( ℱ , K ) is exactly the topology of convergence in probability and L 0 ( ℱ , K ) is complete iff X is complete, in particular L 0 ( ℱ , K ) is complete.

Definition 2.3. [5] An ordered pair (S, || · ||) is called a random normed algebra(briefly, an RN algebra) over K with base ( Ω , ℱ , P ) if (S, || · ||) is an RN module over K with base ( Ω , ℱ , P ) and also a ring such that the following two conditions are satisfied:

Furthermore, the RN algebra is said to be unital if it has the identity element e and ||e|| = 1. As usual, the RN algebra (S, || · ||) is said to be complete if the RN module (S, || · ||) is complete.

Example 2.1. [5] Let (X, ||·||) be a normed algebra over C and L 0 ( ℱ , X ) be the RN module of equivalence classes of X-valued ℱ-random variables on ( Ω , ℱ , P ). Define a multiplication · : L 0 ( ℱ , X ) × L 0 ( ℱ , X ) → L 0 ( ℱ , X ) by x·y = the equivalence class determined by the ℱ-random variable x⁰y⁰, which is defined by (x⁰y⁰)(ω) = (x⁰(ω)) · (y⁰(ω)), ∀ω ∈ Ω, where x⁰ and y⁰ are arbitrarily chosen representatives of x and y in L 0 ( ℱ , X ), respectively. Then ( L 0 ( ℱ , X ) , ⋅ ) is an RN algebra, in particular L 0 ( ℱ , C ) is a unital RN algebra with identity 1.

Example 2.2. [5] It is easy to see that L ℱ ∞ ( ε , C ) is a unital RN algebra with identity 1 (see [8, 9] for the construction of L ℱ ∞ ( ε , C ).

Definition 2.4. [5] Let (S, ||·||) be an RN algebra with identity e over C with base ( Ω , ℱ , P ), and A be any given element in ℱ such that P(A) > 0. An element x ∈ S is invertible on A if there exists y ∈ S such that Ĩ A ⋅ x y = Ĩ A ⋅ y x = Ĩ A ⋅ e. Clearly, Ĩ A ⋅ y is unique and called the inverse on A of x, denoted by x A - 1. Let G(S, A) denote the set of elements of S which are invertible on A. Then Ĩ A ⋅ G ( S , A ) is also a group, and ( x y ) A - 1 = y A - 1 x A - 1 for any x and y in Ĩ A ⋅ G ( S , A ). For any x ∈ S, the sets

are called the random spectrum on A of x in S and the random spectrum of x in S, respectively, and further their complements ρ ( x , S , A ) = L 0 ( ℱ , C ) \ σ ( x , S , A ) and ρ ( x , S ) = L 0 ( ℱ , C ) \ σ ( x , S ) are called the random resolvent set on A of x and the random resolvent set of x, respectively.

Definition 2.5. [5] Let (S, ||·||) be an RN algebra with identity e over C with base ( Ω , ℱ , P ). For any x ∈ S, r(x) = ∨{|ξ| : ξ ∈ σ(x, S)} is called the random spectral radius of x.

Besides, ∧ x n 1 n | n ∈ N is denoted by r _p (x), for any x in an RN algebra over K with base ( Ω , ℱ , P ).

Lemma 2.1. [5] Let (S, ||·||) be a unital complete RN algebra with identity e over C with base ( Ω , ℱ , P ). Then for any x ∈ S, σ(x, S) is nonempty and r(x) = r _p (x).

Definition 3.1. Let S be a random normed algebra, A ∈ ℱ and f be an L⁰-linear function on S, i.e., a mapping from S to L 0 ( ℱ , C ) such that f(ξ · x + η · y) = ξf(x) + ηf(y) for all ξ , η ∈ L 0 ( ℱ , C ) and x, y ∈ S. Then f is called multiplicative if f(xy) = f(x)f(y) for all x, y ∈ S and is called nonzero if there exists x ∈ S such that [ f ( x ) ≠ 0 ] = Ω ̃.

Lemma 3.1. Let S be a random normed algebra with identity e, and let f be an L⁰-function on S satisfying f(e) = 1 and f(x²) = f(x)² for all x ∈ S. Then f is multiplicative.

Proof. By assumption we obtain

and hence

for all x, y ∈ S. So it remains to verify that f(xy) = f(yx). For a, b ∈ S, the identity

implies

Taking a = x - f(x) · e, so that f(a) = 0, and b = y we get f(ay) = f(ya) and hence f(xy) = f(yx). This completes the proof of Lemma 3.1.

The following theorem is a new version of the Gleason-Kahane-Żelazko theorem.

Theorem 3.1 Let S be an unital complete random normed algebra with identity e, and let f be an L⁰-linear function on S. Then the following conditions are equivalent.

Proof If f is multiplicative, then f(e) = f(e²) = f(e)f(e). Since f is nonzero, we have f(e) = 1 and hence Ĩ A = Ĩ A f ( e ) = f ( x x A - 1 ) = f ( x ) f ( x A - 1 ) for any A ∈ ℱ with P(A) > 0 and x ∈ G(S, A). Thus (1)⇒(2). (2)⇒(3) is clear since if ξ ∈ ρ(x, S), then there exists A ∈ ℱ with P(A) > 0 such that Ĩ A ( ξ - f ( x ) ) = f [ Ĩ A ⋅ ( ξ ⋅ e - x ) ] ≠ 0 on à and hence f(x) ∈ σ(x, S). Assume (3), then f(e) = 1 since f(e) ∈ σ(e, S). Now, let n ≥ 2 and consider the random polynomial

of degree n. Therefore we can find λ i ∈ L 0 ( ℱ , C ) ( i = 1 , 2 … n ) such that

for each λ _i . This implies that λ _i ∈ σ(x, S) and hence |λ _i | < r _p (x) by Lemma 2.1. Note that

Comparing coefficients we can see that

On the other hand, by the second equation,

Combining these equalities yields

Hence

Letting n → ∞, we then obtain f(x²) = f(x)² for all x ∈ S. It follows from Lemma 3.1 that f is multiplicative. Clearly, f is nonzero. Thus (3)⇒(1). This completes the proof of Theorem 3.1.

Remark 3.1. When the base space ( Ω , ℱ , P ) of the RN module is a trivial probability space, i.e., ℱ = { Ω , 0̸ }, the new version of the Gleason-Kahane-Żelazko theorem automatically degenerates to the classical case.