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

- Yuehan Tang
^{1, 2}Email author

**2012**:85

**DOI: **10.1186/1029-242X-2012-85

© Tang; licensee Springer. 2012

**Received: **21 December 2011

**Accepted: **13 April 2012

**Published: **13 April 2012

## Abstract

In this article we first present the notion of multiplicative *L*^{0}-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.

### Keywords

random normed module random normed algebra multiplicative*L*

^{0}-linear function Gleason-Kahane-Żelazko theorem.

## 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 *L*^{0}-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, $\stackrel{\u0304}{R}$ (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 real-valued ℱ-random variables on Ω, ${\stackrel{\u0304}{L}}^{0}\left(\mathcal{F},R\right)$ the set of equivalence classes of extended real-valued ℱ-random variables on Ω, ${\mathcal{L}}^{0}\left(\mathcal{F},K\right)$ the algebra of *K*-valued ℱ-random 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 *K*-valued ℱ-random 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 *P*-almost 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 ℱ-random variables (

*ξ*

^{0})

^{-1}and |

*ξ*

^{0}|, respectively, by

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 $u,v\in {L}^{0}\left(\mathcal{F},R\right),\stackrel{\u0304}{\xi}:=u-iv$ is called the complex conjugate of *ξ* and sgn(*ξ*) := |*ξ*|^{-1} · *ξ* is called the sign of *ξ*. It is obvious that $\left|\xi \right|=\left|\stackrel{\u0304}{\xi}\right|,\xi \cdot \mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\left(\stackrel{\u0304}{\xi}\right)=\left|\xi \right|,\left|\mathsf{\text{sgn}}\phantom{\rule{1em}{0ex}}\left(\xi \right)\right|={\u0128}_{A},\phantom{\rule{2.77695pt}{0ex}}{\xi}^{-1}\cdot \xi =\xi \cdot {\xi}^{-1}={\u0128}_{A}$, 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 ℱ-random variables, we also use the equivalence classes of ℱ-measurable 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* Δ*B* = (*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 ℱ-measurable 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 ℱ-measurable 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:

(RNM-1) ||*ξx*|| = |*ξ*|||*x*||, $\forall \xi \in {L}^{0}\left(\mathcal{F},K\right)$, *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 *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 *X*-valued ℱ-random 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*|| ≤ ||*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 ${L}^{0}\left(\mathcal{F},X\right)$ be the RN module of equivalence classes of *X*-valued ℱ-random 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 ℱ-random 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

*x, y*∈

*S*. So it remains to verify that

*f*(

*xy*) =

*f*(

*yx*). For

*a, b*∈

*S*, the identity

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

*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 ${\u0128}_{A}\left(\xi -f\left(x\right)\right)=f\left[{\u0128}_{A}\cdot \left(\xi \cdot e-x\right)\right]\ne 0$ 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

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

_{ i }. This implies that λ

_{ i }∈

*σ*(

*x, S*) and hence |λ

_{ i }| <

*r*

_{ p }(

*x*) by Lemma 2.1. Note that

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 Gleason-Kahane-Żelazko theorem automatically degenerates to the classical case.

## Declarations

### Acknowledgements

This work was supported by the NSF of China under Grant No. 10871016.

## Authors’ Affiliations

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