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A new version of the GleasonKahaneŻelazko theorem in complete random normed algebras
Journal of Inequalities and Applications volume 2012, Article number: 85 (2012)
Abstract
In this article we first present the notion of multiplicative L^{0}linear function. Moreover, we establish a new version of the GleasonKahaneŻelazko theorem in unital complete random normed algebras.
Mathematics Subject Classification 2000: 46H25; 46H05; 15A78.
1 Introduction
Gleason [1] and, independently, Kahane and Żelazko [2] proved the socalled GleasonKahaneŻelazko theorem which is a famous theorem in classical Banach algebras. There are various extensions and generalizations of this theorem [3]. The GleasonKahaneŻelazko theorem in an unital complete random normed algebra as a random generalization of the classical GleasonKahaneŻelazko theorem is given in [4].
Based on the study of [5], we will establish a new version of the GleasonKahaneŻelazko theorem in an unital complete random normed algebra. In this article we first present the notion of multiplicative L^{0}functions. Then, we give the new version of the GleasonKahaneŻelazko theorem in an unital complete random normed algebra as another random generalization of the classical GleasonKahaneŻelazko 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.
2 Preliminary
Throughout this article, N denotes the set of positive integers, K the scalar field R of real numbers or C of complex numbers, $\stackrel{\u0304}{R}$ (or [∞, +∞]) the set of extended real numbers, $\left(\Omega ,\mathcal{F},P\right)$ a probability space, ${\stackrel{\u0304}{\mathcal{L}}}^{0}\left(\mathcal{F},R\right)$ the set of extended realvalued ℱrandom variables on Ω, ${\stackrel{\u0304}{L}}^{0}\left(\mathcal{F},R\right)$ the set of equivalence classes of extended realvalued ℱrandom variables on Ω, ${\mathcal{L}}^{0}\left(\mathcal{F},K\right)$ the algebra of Kvalued ℱrandom variables on Ω under the ordinary pointwise addition, multiplication and scalar multiplication operations, ${L}^{0}\left(\mathcal{F},K\right)$ the algebra of equivalence classes of Kvalued ℱrandom variables on Ω, i.e., the quotient algebra of ${\mathcal{L}}^{0}\left(\mathcal{F},K\right)$, and 0 and 1 the null and unit elements, respectively.
It is well known from [6] that ${\stackrel{\u0304}{L}}^{0}\left(\mathcal{F},R\right)$ is a complete lattice under the ordering ≤: ξ ≤ η iff ξ^{0}(ω) ≤ η^{0}(ω) for Palmost all ω in Ω (briefly, a.s.), where ξ^{0} and η^{0} are arbitrarily chosen representatives of ξ and η, respectively. Furthermore, every subset A of ${\stackrel{\u0304}{L}}^{0}\left(\mathcal{F},R\right)$ 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≥1}a_{ n }= ∨A and ∧_{n≥1}b_{ 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 ${L}^{0}\left(\mathcal{F},R\right)$, as a sublattice of ${\stackrel{\u0304}{L}}^{0}\left(\mathcal{F},R\right)$, 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 ${\stackrel{\u0304}{L}}_{+}^{0}\left(\mathcal{F}\right)=\left\{\xi \in {\stackrel{\u0304}{L}}^{0}\left(\mathcal{F},R\right)\xi \ge 0\right\}$ and ${L}_{+}^{0}\left(\mathcal{F}\right)=\left\{\xi \in {L}^{0}\left(\mathcal{F},R\right)\xi \ge 0\right\}$.
The following notions of generalized inverse, absolute value, complex conjugate and sign of an element in ${L}^{0}\left(\mathcal{F},K\right)$ bring much convenience to this article.
Definition 2.1. [7] Let ξ be an element in ${L}^{0}\left(\mathcal{F},K\right)$. For an arbitrarily chosen representative ξ^{0} of ξ, define two ℱrandom variables (ξ^{0})^{1} and ξ^{0}, respectively, by
and
Then the equivalence class of (ξ^{0})^{1}, denoted by ξ^{1}, is called the generalized inverse of ξ; the equivalence class of ξ^{0}, denoted by ξ, is called the absolute value of ξ. When $\xi \in {L}^{0}\left(\mathcal{F},C\right)$, set ξ = u + iv, where $u,v\in {L}^{0}\left(\mathcal{F},R\right),\stackrel{\u0304}{\xi}:=uiv$ is called the complex conjugate of ξ and sgn(ξ) := ξ^{1} · ξ is called the sign of ξ. It is obvious that $\left\xi \right=\left\stackrel{\u0304}{\xi}\right,\xi \cdot \mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\left(\stackrel{\u0304}{\xi}\right)=\left\xi \right,\left\mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\left(\xi \right)\right={\u0128}_{A},\phantom{\rule{2.77695pt}{0ex}}{\xi}^{1}\cdot \xi =\xi \cdot {\xi}^{1}={\u0128}_{A}$, where A = {ω ∈ Ω : ξ^{0}(ω) ≠ 0} and ${\u0128}_{A}$ denotes the equivalence class of the characteristic function I_{ A }of A. Throughout this article, the symbol ${\u0128}_{A}$ is always understood as above unless stated otherwise.
Besides the equivalence classes of ℱrandom variables, we also use the equivalence classes of ℱmeasurable sets. Let $A\in \mathcal{F}$, then the equivalence class of A, denoted by Ã, is defined by $\xc3=\left\{B\in \mathcal{F}:P\left(A\Delta B\right)=0\right\}$, where A ΔB = (A\B)∪(B\A) is the symmetric difference of A and B, and $P\left(\xc3\right)$ is defined to be P(A). For two ℱmeasurable sets G and D, G ⊂ D a.s. means P(G\D) = 0, in which case we also say $\stackrel{\u0303}{G}\subset \stackrel{\u0303}{D}$; $\stackrel{\u0303}{G}\cap \stackrel{\u0303}{D}$ 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 $\xi ,\phantom{\rule{2.77695pt}{0ex}}\eta \in {L}^{0}\left(\mathcal{F},R\right),\xi >\eta $ means ξ ≥ η and ξ ≠ η; [ξ > η] stands for the equivalence class of the ℱmeasurable set {ω ∈ Ω : ξ^{0}(ω) > η^{0}(ω)} (briefly, [ξ^{0} > η^{0}]), where ξ^{0} and η^{0} are arbitrarily selected representatives of ξ and η, respectively, and I_{[ξ>η]}stands for ${\u0128}_{\left[{\xi}^{0}>{\eta}^{0}\right]}$. If $A\in \mathcal{F}$, then ξ > η on Ã means ξ^{0}(ω) > η^{0}(ω) a.s. on A, similarly ξ ≠ η on Ã means that ξ^{0}(ω) ≠ η^{0}(ω) a.s. on A, also denoted by $\xc3\subset \left[\xi \ne \eta \right]$.
Definition 2.2. [7] An ordered pair (S,  · ) is called a random normed module (briefly, an RN module) over K with base $\left(\Omega ,\mathcal{F},P\right)$ if S is a left module over the algebra ${L}^{0}\left(\mathcal{F},K\right)$ and  ·  is a mapping from S to ${L}_{+}^{0}\left(\mathcal{F}\right)$ such that the following conditions are satisfied:
(RNM1) ξx = ξx, $\forall \xi \in {L}^{0}\left(\mathcal{F},K\right)$, x ∈ S;
(RNM2) x + y ≤ x + y, ∀x, y ∈ S;
(RNM3) x = 0 implies x = 0(the zero element in S).
Where x is called the L^{0}norm of the vector x in S.
In this article, given an RN module (S,  · ) over K with base $\left(\Omega ,\mathcal{F},P\right)$ 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 ${\mathcal{U}}_{0}=\left\{N\left(\epsilon ,\lambda \right)\epsilon >0,0<\lambda <1\right\}$ 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 ${L}^{0}\left(\mathcal{F},K\right)$, namely the module multiplication · : ${L}^{0}\left(\mathcal{F},K\right)\times S\to S$ is jointly continuous (see [7] for details). Besides, let ${L}^{0}\left(\mathcal{F},K\right)$ be the RN module of equivalence classes of Xvalued ℱrandom variables on $\left(\Omega ,\mathcal{F},P\right)$, where X is an ordinary normed space, then it is easy to see that the (ϵ, λ)topology on ${L}^{0}\left(\mathcal{F},K\right)$ is exactly the topology of convergence in probability and ${L}^{0}\left(\mathcal{F},K\right)$ is complete iff X is complete, in particular ${L}^{0}\left(\mathcal{F},K\right)$ is complete.
Definition 2.3. [5] An ordered pair (S,  · ) is called a random normed algebra(briefly, an RN algebra) over K with base $\left(\Omega ,\mathcal{F},P\right)$ if (S,  · ) is an RN module over K with base $\left(\Omega ,\mathcal{F},P\right)$ and also a ring such that the following two conditions are satisfied:

(1)
(ξ · x)y = x(ξ · y) = ξ · (xy), for all $\xi \in {L}^{0}\left(\mathcal{F},K\right)$ and all x, y ∈ S;

(2)
the L^{0}norm  ·  is submultiplicative, that is, xy ≤ xy, for all x, y ∈ S.
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 ${L}^{0}\left(\mathcal{F},X\right)$ be the RN module of equivalence classes of Xvalued ℱrandom variables on $\left(\Omega ,\mathcal{F},P\right)$. Define a multiplication · : ${L}^{0}\left(\mathcal{F},X\right)\times {L}^{0}\left(\mathcal{F},X\right)\to {L}^{0}\left(\mathcal{F},X\right)$ by x·y = the equivalence class determined by the ℱrandom variable x^{0}y^{0}, which is defined by (x^{0}y^{0})(ω) = (x^{0}(ω)) · (y^{0}(ω)), ∀ω ∈ Ω, where x^{0} and y^{0} are arbitrarily chosen representatives of x and y in ${L}^{0}\left(\mathcal{F},X\right)$, respectively. Then $\left({L}^{0}\left(\mathcal{F},X\right),\u2225\cdot \u2225\right)$ is an RN algebra, in particular ${L}^{0}\left(\mathcal{F},C\right)$ is a unital RN algebra with identity 1.
Example 2.2. [5] It is easy to see that ${L}_{\mathcal{F}}^{\infty}\left(\epsilon ,C\right)$ is a unital RN algebra with identity 1 (see [8, 9] for the construction of ${L}_{\mathcal{F}}^{\infty}\left(\epsilon ,C\right)$.
Definition 2.4. [5] Let (S, ·) be an RN algebra with identity e over C with base $\left(\Omega ,\mathcal{F},P\right)$, 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 ${\u0128}_{A}\cdot xy={\u0128}_{A}\cdot yx={\u0128}_{A}\cdot e$. Clearly, ${\u0128}_{A}\cdot y$ 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 ${\u0128}_{A}\cdot G\left(S,A\right)$ is also a group, and ${\left(xy\right)}_{A}^{1}={y}_{A}^{1}{x}_{A}^{1}$ for any x and y in ${\u0128}_{A}\cdot G\left(S,A\right)$. 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 $\rho \left(x,S,A\right)={L}^{0}\left(\mathcal{F},C\right)\backslash \sigma \left(x,S,A\right)$ and $\rho \left(x,S\right)={L}^{0}\left(\mathcal{F},C\right)\backslash \sigma \left(x,S\right)$ 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 $\left(\Omega ,\mathcal{F},P\right)$. For any x ∈ S, r(x) = ∨{ξ : ξ ∈ σ(x, S)} is called the random spectral radius of x.
Besides, $\wedge \left\{{\u2225{x}^{n}\u2225}^{\frac{1}{n}}n\in N\right\}$ is denoted by r_{ p }(x), for any x in an RN algebra over K with base $\left(\Omega ,\mathcal{F},P\right)$.
Lemma 2.1. [5] Let (S, ·) be a unital complete RN algebra with identity e over C with base $\left(\Omega ,\mathcal{F},P\right)$. Then for any x ∈ S, σ(x, S) is nonempty and r(x) = r_{ p }(x).
3 Main results and proofs
Definition 3.1. Let S be a random normed algebra, $A\in \mathcal{F}$ and f be an L^{0}linear function on S, i.e., a mapping from S to ${L}^{0}\left(\mathcal{F},C\right)$ such that f(ξ · x + η · y) = ξf(x) + ηf(y) for all $\xi ,\eta \in {L}^{0}\left(\mathcal{F},C\right)$ 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 $\left[f\left(x\right)\ne 0\right]=\stackrel{\u0303}{\Omega}$.
Lemma 3.1. Let S be a random normed algebra with identity e, and let f be an L^{0}function on S satisfying f(e) = 1 and f(x^{2}) = 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 GleasonKahaneŻelazko theorem.
Theorem 3.1 Let S be an unital complete random normed algebra with identity e, and let f be an L^{0}linear function on S. Then the following conditions are equivalent.

(1)
f is nonzero and multiplicative.

(2)
f(e) = 1 and f(x) ≠ 0 on Ã for any $A\in \mathcal{F}$ with P(A) > 0 and x ∈ G(S, A).

(3)
f(x) ∈ σ(x, S) for every x ∈ S.
Proof If f is multiplicative, then f(e) = f(e^{2}) = f(e)f(e). Since f is nonzero, we have f(e) = 1 and hence ${\u0128}_{A}={\u0128}_{A}f\left(e\right)=f\left(x{x}_{A}^{1}\right)=f\left(x\right)f\left({x}_{A}^{1}\right)$ for any $A\in \mathcal{F}$ 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 $A\in \mathcal{F}$ with P(A) > 0 such that ${\u0128}_{A}\left(\xi f\left(x\right)\right)=f\left[{\u0128}_{A}\cdot \left(\xi \cdot ex\right)\right]\ne 0$ on Ã 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 ${\lambda}_{i}\in {L}^{0}\left(\mathcal{F},C\right)\left(i=1,2\dots n\right)$ 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(x^{2}) = 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 $\left(\Omega ,\mathcal{F},P\right)$ of the RN module is a trivial probability space, i.e., $\mathcal{F}=\left\{\Omega ,\mathrm{0\u0338}\right\}$, the new version of the GleasonKahaneŻelazko theorem automatically degenerates to the classical case.
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Acknowledgements
This work was supported by the NSF of China under Grant No. 10871016.
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Tang, Y. A new version of the GleasonKahaneŻelazko theorem in complete random normed algebras. J Inequal Appl 2012, 85 (2012). https://doi.org/10.1186/1029242X201285
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Keywords
 random normed module
 random normed algebra
 multiplicative L^{0}linear function
 GleasonKahaneŻelazko theorem.