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

## Abstract

In this article we first present the notion of multiplicative L0-linear function. Moreover, we establish a new version of the Gleason-Kahane-Żelazko 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 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.

## 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, $\stackrel{̄}{R}$ (or [-∞, +∞]) the set of extended real numbers, $\left(\Omega ,ℱ,P\right)$ a probability space, ${\stackrel{̄}{ℒ}}^{0}\left(ℱ,R\right)$ the set of extended real-valued -random variables on Ω, ${\stackrel{̄}{L}}^{0}\left(ℱ,R\right)$ the set of equivalence classes of extended real-valued -random variables on Ω, ${ℒ}^{0}\left(ℱ,K\right)$ the algebra of K-valued -random variables on Ω under the ordinary pointwise addition, multiplication and scalar multiplication operations, ${L}^{0}\left(ℱ,K\right)$ the algebra of equivalence classes of K-valued -random variables on Ω, i.e., the quotient algebra of ${ℒ}^{0}\left(ℱ,K\right)$, and 0 and 1 the null and unit elements, respectively.

It is well known from [6] that ${\stackrel{̄}{L}}^{0}\left(ℱ,R\right)$ is a complete lattice under the ordering ≤: ξη 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 ${\stackrel{̄}{L}}^{0}\left(ℱ,R\right)$ 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≥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 ${L}^{0}\left(ℱ,R\right)$, as a sublattice of ${\stackrel{̄}{L}}^{0}\left(ℱ,R\right)$, 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 ${\stackrel{̄}{L}}_{+}^{0}\left(ℱ\right)=\left\{\xi \in {\stackrel{̄}{L}}^{0}\left(ℱ,R\right)|\xi \ge 0\right\}$ and ${L}_{+}^{0}\left(ℱ\right)=\left\{\xi \in {L}^{0}\left(ℱ,R\right)|\xi \ge 0\right\}$.

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

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

${\left({\xi }^{0}\right)}^{-1}\left(\omega \right)=\left\{\begin{array}{cc}\frac{1}{{\xi }^{0}\left(\omega \right)}\hfill & \mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}{\xi }^{0}\left(\omega \right)\ne 0,\hfill \\ 0,\hfill & \mathsf{\text{otherwise}},\hfill \end{array}\right\$

and

$\left|{\xi }^{0}\right|\left(\omega \right)=\left|{\xi }^{0}\left(\omega \right)\right|,\phantom{\rule{1em}{0ex}}\forall \omega \in \Omega .$

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 $\xi \in {L}^{0}\left(ℱ,C\right)$, set ξ = u + iv, where $u,v\in {L}^{0}\left(ℱ,R\right),\stackrel{̄}{\xi }:=u-iv$ is called the complex conjugate of ξ and sgn(ξ) := |ξ|-1 · ξ is called the sign of ξ. It is obvious that $\left|\xi \right|=\left|\stackrel{̄}{\xi }\right|,\xi \cdot \mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\left(\stackrel{̄}{\xi }\right)=\left|\xi \right|,\left|\mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\left(\xi \right)\right|={Ĩ}_{A},\phantom{\rule{2.77695pt}{0ex}}{\xi }^{-1}\cdot \xi =\xi \cdot {\xi }^{-1}={Ĩ}_{A}$, 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 -random variables, we also use the equivalence classes of -measurable sets. Let $A\in ℱ$, then the equivalence class of A, denoted by Ã, is defined by $Ã=\left\{B\in ℱ:P\left(A\Delta B\right)=0\right\}$, where A ΔB = (A\B)(B\A) is the symmetric difference of A and B, and $P\left(Ã\right)$ is defined to be P(A). For two -measurable sets G and D, G D a.s. means P(G\D) = 0, in which case we also say $\stackrel{̃}{G}\subset \stackrel{̃}{D}$; $\stackrel{̃}{G}\cap \stackrel{̃}{D}$ 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 $\xi ,\phantom{\rule{2.77695pt}{0ex}}\eta \in {L}^{0}\left(ℱ,R\right),\xi >\eta$ means ξη and ξη; [ξ > η] stands for the equivalence class of the -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 ${Ĩ}_{\left[{\xi }^{0}>{\eta }^{0}\right]}$. If $A\in ℱ$, then ξ > η on Ã means ξ0(ω) > η0(ω) a.s. on A, similarly ξη on Ã means that ξ0(ω) ≠ η0(ω) a.s. on A, also denoted by $Ã\subset \left[\xi \ne \eta \right]$.

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

(RNM-1) ||ξx|| = |ξ|||x||, $\forall \xi \in {L}^{0}\left(ℱ,K\right)$, 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 $\left(\Omega ,ℱ,P\right)$ 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 ${L}^{0}\left(ℱ,K\right)$, namely the module multiplication · : ${L}^{0}\left(ℱ,K\right)×S\to S$ is jointly continuous (see [7] for details). Besides, let ${L}^{0}\left(ℱ,K\right)$ be the RN module of equivalence classes of X-valued -random variables on $\left(\Omega ,ℱ,P\right)$, where X is an ordinary normed space, then it is easy to see that the (ϵ, λ)-topology on ${L}^{0}\left(ℱ,K\right)$ is exactly the topology of convergence in probability and ${L}^{0}\left(ℱ,K\right)$ is complete iff X is complete, in particular ${L}^{0}\left(ℱ,K\right)$ is complete.

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

1. (1)

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

2. (2)

the L0-norm || · || is submultiplicative, that is, ||xy|| ≤ ||x||||y||, 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 ${L}^{0}\left(ℱ,X\right)$ be the RN module of equivalence classes of X-valued -random variables on $\left(\Omega ,ℱ,P\right)$. Define a multiplication · : ${L}^{0}\left(ℱ,X\right)×{L}^{0}\left(ℱ,X\right)\to {L}^{0}\left(ℱ,X\right)$ by x·y = the equivalence class determined by the -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 ${L}^{0}\left(ℱ,X\right)$, respectively. Then $\left({L}^{0}\left(ℱ,X\right),∥\cdot ∥\right)$ is an RN algebra, in particular ${L}^{0}\left(ℱ,C\right)$ is a unital RN algebra with identity 1.

Example 2.2. [5] It is easy to see that ${L}_{ℱ}^{\infty }\left(\epsilon ,C\right)$ is a unital RN algebra with identity 1 (see [8, 9] for the construction of ${L}_{ℱ}^{\infty }\left(\epsilon ,C\right)$.

Definition 2.4. [5] Let (S, ||·||) be an RN algebra with identity e over C with base $\left(\Omega ,ℱ,P\right)$, and A be any given element in such that P(A) > 0. An element x S is invertible on A if there exists y S such that ${Ĩ}_{A}\cdot xy={Ĩ}_{A}\cdot yx={Ĩ}_{A}\cdot e$. Clearly, ${Ĩ}_{A}\cdot y$ 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 ${Ĩ}_{A}\cdot G\left(S,A\right)$ is also a group, and ${\left(xy\right)}_{A}^{-1}={y}_{A}^{-1}{x}_{A}^{-1}$ for any x and y in ${Ĩ}_{A}\cdot G\left(S,A\right)$. For any x S, the sets

$\begin{array}{ll}\hfill \sigma \left(x,S,A\right)& =\left\{\xi \in {L}^{0}\left(ℱ,C\right):{Ĩ}_{A}\cdot \left(\xi \cdot e-x\right)\notin {Ĩ}_{A}\cdot G\left(S,A\right)\right\},\phantom{\rule{2em}{0ex}}\\ \hfill \sigma \left(x,S\right)& =\bigcap _{A\in ℱ}\sigma \left(x,S,A\right)\phantom{\rule{2em}{0ex}}\end{array}$

are called the random spectrum on A of x in S and the random spectrum of x in S, respectively, and further their complements $\rho \left(x,S,A\right)={L}^{0}\left(ℱ,C\right)\\sigma \left(x,S,A\right)$ and $\rho \left(x,S\right)={L}^{0}\left(ℱ,C\right)\\sigma \left(x,S\right)$ 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 $\left(\Omega ,ℱ,P\right)$. For any x S, r(x) = {|ξ| : ξ σ(x, S)} is called the random spectral radius of x.

Besides, $\wedge \left\{{∥{x}^{n}∥}^{\frac{1}{n}}|n\in N\right\}$ is denoted by r p (x), for any x in an RN algebra over K with base $\left(\Omega ,ℱ,P\right)$.

Lemma 2.1. [5] Let (S, ||·||) be a unital complete RN algebra with identity e over C with base $\left(\Omega ,ℱ,P\right)$. 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, $A\in ℱ$ and f be an L0-linear function on S, i.e., a mapping from S to ${L}^{0}\left(ℱ,C\right)$ such that f(ξ · x + η · y) = ξf(x) + ηf(y) for all $\xi ,\eta \in {L}^{0}\left(ℱ,C\right)$ 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 $\left[f\left(x\right)\ne 0\right]=\stackrel{̃}{\Omega }$.

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

$\begin{array}{ll}\hfill f\left({x}^{2}\right)+f\left(xy+yx\right)+f\left({y}^{2}\right)& =f\left({x}^{2}+xy+yx+{y}^{2}\right)\phantom{\rule{2em}{0ex}}\\ =f\left({\left(x+y\right)}^{2}\right)\phantom{\rule{2em}{0ex}}\\ =f{\left(x+y\right)}^{2}\phantom{\rule{2em}{0ex}}\\ =f{\left(x\right)}^{2}+2f\left(x\right)f\left(y\right)+f{\left(y\right)}^{2},\phantom{\rule{2em}{0ex}}\end{array}$

and hence

$f\left(xy+yx\right)=2f\left(x\right)f\left(y\right)$

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

${\left(ab-ba\right)}^{2}+{\left(ab+ba\right)}^{2}=2\left[a\left(bab\right)+\left(bab\right)a\right]$

implies

$\begin{array}{ll}\hfill f{\left(ab-ba\right)}^{2}+4f{\left(a\right)}^{2}f{\left(b\right)}^{2}& =f\left({\left(ab-ba\right)}^{2}\right)+f{\left(ab+ba\right)}^{2}\phantom{\rule{2em}{0ex}}\\ =f\left({\left(ab-ba\right)}^{2}+{\left(ab+ba\right)}^{2}\right)\phantom{\rule{2em}{0ex}}\\ =f\left({\left(ab-ba\right)}^{2}+{\left(ab+ba\right)}^{2}\right)\phantom{\rule{2em}{0ex}}\\ =2f\left(a\left(bab\right)+\left(bab\right)a\right)\phantom{\rule{2em}{0ex}}\\ =4f\left(a\right)f\left(bab\right).\phantom{\rule{2em}{0ex}}\end{array}$

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.

1. (1)

f is nonzero and multiplicative.

2. (2)

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

3. (3)

f(x) σ(x, S) for every x S.

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 ${Ĩ}_{A}={Ĩ}_{A}f\left(e\right)=f\left(x{x}_{A}^{-1}\right)=f\left(x\right)f\left({x}_{A}^{-1}\right)$ for any $A\in ℱ$ 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 $A\in ℱ$ with P(A) > 0 such that ${Ĩ}_{A}\left(\xi -f\left(x\right)\right)=f\left[{Ĩ}_{A}\cdot \left(\xi \cdot e-x\right)\right]\ne 0$ 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

$p\left(\lambda \right)=f\left({\left(\lambda \cdot e-x\right)}^{n}\right)$

of degree n. Therefore we can find ${\lambda }_{i}\in {L}^{0}\left(ℱ,C\right)\left(i=1,2\dots n\right)$ such that

$0=p\left({\lambda }_{i}\right)=f\left({\left({\lambda }_{i}\cdot e-x\right)}^{n}\right)\in \sigma \left({\left({\lambda }_{i}\cdot e-x\right)}^{n},S\right)$

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

$\prod _{i=1}^{n}\left(\lambda -{\lambda }_{i}\right)=p\left(\lambda \right)={\lambda }^{n}-nf\left(x\right){\lambda }^{n-1}+{C}_{n}^{2}f\left({x}^{2}\right){\lambda }^{n-2}+\cdots +{\left(-1\right)}^{n}f\left({x}^{n}\right).$

Comparing coefficients we can see that

$\sum _{i=1}^{n}{\lambda }_{i}=nf\left(x\right),\phantom{\rule{1em}{0ex}}\sum _{1\le i

On the other hand, by the second equation,

${\left(\sum _{i=1}^{n}{\lambda }_{i}\right)}^{2}=\sum _{i=1}^{n}{\lambda }_{i}^{2}+2\phantom{\rule{2.77695pt}{0ex}}\sum _{1\le i

Combining these equalities yields

${n}^{2}\left|f{\left(x\right)}^{2}-f\left({x}^{2}\right)\right|=\left|-nf\left({x}^{2}\right)+\sum _{i=1}^{n}{\lambda }_{i}^{2}\right|\le n\left|f{\left(x\right)}^{2}\right|+n{r}_{p}{\left(x\right)}^{2}.$

Hence

$\left|f{\left(x\right)}^{2}-f\left({x}^{2}\right)\right|\le \frac{1}{n}\left[\left|f\left({x}^{2}\right)\right|+{r}_{p}{\left(x\right)}^{2}\right].$

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 $\left(\Omega ,ℱ,P\right)$ of the RN module is a trivial probability space, i.e., $ℱ=\left\{\Omega ,0̸\right\}$, the new version of the Gleason-Kahane-Żelazko 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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Correspondence to Yuehan Tang.

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Tang, Y. A new version of the Gleason-Kahane-Żelazko theorem in complete random normed algebras. J Inequal Appl 2012, 85 (2012). https://doi.org/10.1186/1029-242X-2012-85