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# 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  and, independently, Kahane and Żelazko  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 . 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 .

Based on the study of , 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, $R ̄$ (or [-∞, +∞]) the set of extended real numbers, $( Ω , ℱ , P )$ a probability space, $ℒ ̄ 0 ( ℱ , R )$ the set of extended real-valued -random variables on Ω, $L ̄ 0 ( ℱ , R )$ the set of equivalence classes of extended real-valued -random variables on Ω, $ℒ 0 ( ℱ , K )$ the algebra of K-valued -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 -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  that $L ̄ 0 ( ℱ , R )$ 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 $L ̄ 0 ( ℱ , R )$ 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.  Let ξ be an element in $L 0 ( ℱ , K )$. For an arbitrarily chosen representative ξ0 of ξ, define two -random variables (ξ0)-1 and |ξ0|, respectively, by

$( ξ 0 ) - 1 ( ω ) = 1 ξ 0 ( ω ) if ξ 0 ( ω ) ≠ 0 , 0 , otherwise ,$

and

$ξ 0 ( ω ) = ξ 0 ( ω ) , ∀ ω ∈ Ω .$

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 $u,v∈ L 0 ( ℱ , R ) , ξ ̄ :=u-iv$ 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 -random variables, we also use the equivalence classes of -measurable sets. Let $A∈ℱ$, then the equivalence class of A, denoted by Ã, is defined by $Ã= { B ∈ ℱ : P ( A Δ B ) = 0 }$, where A ΔB = (A\B)(B\A) is the symmetric difference of A and B, and $P ( Ã )$ 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 $G ̃ ⊂ D ̃$; $G ̃ ∩ D ̃$ 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 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 $Ĩ [ ξ 0 > η 0 ]$. If $A∈ℱ$, 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 $Ã⊂ [ ξ ≠ η ]$.

Definition 2.2.  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  for details). Besides, let $L 0 ( ℱ , K )$ be the RN module of equivalence classes of X-valued -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.  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:

1. (1)

(ξ · x)y = x(ξ · y) = ξ · (xy), for all $ξ∈ L 0 ( ℱ , K )$ 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.  Let (X, ||·||) be a normed algebra over C and $L 0 ( ℱ , X )$ be the RN module of equivalence classes of X-valued -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 -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.  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.  Let (S, ||·||) be an RN algebra with identity e over C with base $( Ω , ℱ , P )$, 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 $Ĩ A ⋅xy= Ĩ A ⋅yx= Ĩ A ⋅e$. 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

$σ ( x , S , A ) = ξ ∈ L 0 ( ℱ , C ) : Ĩ A ⋅ ( ξ ⋅ e - x ) ∉ Ĩ A ⋅ G ( S , A ) , σ ( x , S ) = ⋂ A ∈ ℱ σ ( x , S , A )$

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.  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.  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).

## 3 Main results and proofs

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

$f ( x 2 ) + f ( x y + y x ) + f ( y 2 ) = f ( x 2 + x y + y x + y 2 ) = f ( ( x + y ) 2 ) = f ( x + y ) 2 = f ( x ) 2 + 2 f ( x ) f ( y ) + f ( y ) 2 ,$

and hence

$f ( x y + y x ) = 2 f ( x ) f ( y )$

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

$( a b - b a ) 2 + ( a b + b a ) 2 = 2 [ a ( b a b ) + ( b a b ) a ]$

implies

$f ( a b - b a ) 2 + 4 f ( a ) 2 f ( b ) 2 = f ( ( a b - b a ) 2 ) + f ( a b + b a ) 2 = f ( ( a b - b a ) 2 + ( a b + b a ) 2 ) = f ( ( a b - b a ) 2 + ( a b + b a ) 2 ) = 2 f ( a ( b a b ) + ( b a b ) a ) = 4 f ( a ) f ( b a b ) .$

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 $A∈ℱ$ 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 $Ĩ A = Ĩ A f ( e ) = f ( x x A - 1 ) = f ( x ) f ( x A - 1 )$ 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 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

$p ( λ ) = f ( ( λ ⋅ e - x ) n )$

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

$0 = p ( λ i ) = f ( ( λ i ⋅ e - x ) n ) ∈ σ ( ( λ i ⋅ e - x ) n , S )$

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

$∏ i = 1 n ( λ - λ i ) = p ( λ ) = λ n - n f ( x ) λ n - 1 + C n 2 f ( x 2 ) λ n - 2 + ⋯ + ( - 1 ) n f ( x n ) .$

Comparing coefficients we can see that

$∑ i = 1 n λ i = n f ( x ) , ∑ 1 ≤ i < j ≤ n λ i λ j = C n 2 f ( x 2 ) .$

On the other hand, by the second equation,

$∑ i = 1 n λ i 2 = ∑ i = 1 n λ i 2 + 2 ∑ 1 ≤ i < j ≤ n λ i λ j = ∑ i = 1 n λ i 2 + n ( n - 1 ) f ( x 2 ) .$

Combining these equalities yields

$n 2 f ( x ) 2 - f ( x 2 ) = - n f ( x 2 ) + ∑ i = 1 n λ i 2 ≤ n f ( x ) 2 + n r p ( x ) 2 .$

Hence

$f ( x ) 2 - f ( x 2 ) ≤ 1 n [ f ( x 2 ) + r p ( x ) 2 ] .$

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.

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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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The author declares that they have no competing interests.

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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 