Open Access

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

Journal of Inequalities and Applications20122012:85

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

Received: 21 December 2011

Accepted: 13 April 2012

Published: 13 April 2012

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.

Keywords

random normed module random normed algebra multiplicative L0-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 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 , , P ) a probability space, ̄ 0 ( , R ) the set of extended real-valued -random variables on Ω, L ̄ 0 ( , R ) the set of equivalence classes of extended real-valued -random variables on Ω, 0 ( , K ) the algebra of K-valued -random variables on Ω under the ordinary pointwise addition, multiplication and scalar multiplication operations, L 0 ( , K ) the algebra of equivalence classes of K-valued -random variables on Ω, i.e., the quotient algebra of 0 ( , K ) , and 0 and 1 the null and unit elements, respectively.

It is well known from [6] that L ̄ 0 ( , 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 ( , 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 ( , R ) , as a sublattice of L ̄ 0 ( , 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 ( ) = { ξ L ̄ 0 ( , R ) | ξ 0 } and L + 0 ( ) = { ξ L 0 ( , R ) | ξ 0 } .

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

Definition 2.1. [7] Let ξ be an element in L 0 ( , 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 ( , C ) , set ξ = u + iv, where u , v L 0 ( , 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 -random variables, we also use the equivalence classes of -measurable sets. Let A , then the equivalence class of A, denoted by Ã, is defined by à = { B : 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 -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 ( , 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 , then ξ > η on à means ξ0(ω) > η0(ω) a.s. on A, similarly ξη 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 , , P ) if S is a left module over the algebra L 0 ( , K ) and || · || is a mapping from S to L + 0 ( ) such that the following conditions are satisfied:

(RNM-1) ||ξx|| = |ξ|||x||, ξ L 0 ( , 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 , , 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 ( , K ) , namely the module multiplication · : L 0 ( , K ) × S S is jointly continuous (see [7] for details). Besides, let L 0 ( , K ) be the RN module of equivalence classes of X-valued -random variables on ( Ω , , P ) , where X is an ordinary normed space, then it is easy to see that the (ϵ, λ)-topology on L 0 ( , K ) is exactly the topology of convergence in probability and L 0 ( , K ) is complete iff X is complete, in particular L 0 ( , 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 , , P ) if (S, || · ||) is an RN module over K with base ( Ω , , 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 ( , 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. [5] Let (X, ||·||) be a normed algebra over C and L 0 ( , X ) be the RN module of equivalence classes of X-valued -random variables on ( Ω , , P ) . Define a multiplication · : L 0 ( , X ) × L 0 ( , X ) L 0 ( , 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 ( , X ) , respectively. Then ( L 0 ( , X ) , ) is an RN algebra, in particular L 0 ( , C ) is a unital RN algebra with identity 1.

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

Definition 2.4. [5] Let (S, ||·||) be an RN algebra with identity e over C with base ( Ω , , 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 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
σ ( x , S , A ) = ξ L 0 ( , C ) : Ĩ A ( ξ e - x ) Ĩ A G ( S , A ) , σ ( x , S ) = A 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 ( , C ) \ σ ( x , S , A ) and ρ ( x , S ) = L 0 ( , 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 , , 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 , , P ) .

Lemma 2.1. [5] Let (S, ||·||) be a unital complete RN algebra with identity e over C with base ( Ω , , 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 and f be an L0-linear function on S, i.e., a mapping from S to L 0 ( , C ) such that f(ξ · x + η · y) = ξf(x) + ηf(y) for all ξ , η L 0 ( , 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 à for any A 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 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 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
p ( λ ) = f ( ( λ e - x ) n )
of degree n. Therefore we can find λ i L 0 ( , 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 , , P ) of the RN module is a trivial probability space, i.e., = { Ω , } , 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

(1)
LMIB and School of Mathematics and Systems Science, Beihang University
(2)
College of Mathematics Physics and Information Engineering, Jiaxing University

References

  1. Gleason AM: A characterization of maximal ideals. J Anal Math 1967, 19: 171–172.MathSciNetView ArticleGoogle Scholar
  2. Kahane JP, Żelazko W: A characterization of maximal ideals in commutative Banach algebras. Studia Math 1968, 29: 339–343.MathSciNetGoogle Scholar
  3. Jarosz K: Generalizations of the Gleason-Kahane-Żelazko theorem. Rocky Mount J Math 1991, 21(3):915–921.MathSciNetView ArticleGoogle Scholar
  4. Tang YH: The Gleason-Kahane-Żelazko theorem in a complete random normed algebra. Acta Anal Funct Appl 2011.Google Scholar
  5. Tang YH, Guo TX: Complete random normed algebras. to appear
  6. Dunford N, Schwartz JT: Linear Operators 1. In Interscience. New York; 1957.Google Scholar
  7. Guo TX: Some basic theories of random normed linear spaces and random inner product spaces. Acta Anal Funct Appl 1999, 1: 160–184.MathSciNetGoogle Scholar
  8. Guo TX: Recent progress in random metric theory and its applications to conditional risk measures. Sci China Ser A 2011, 54: 633–660.View ArticleGoogle Scholar
  9. Guo TX: Relations between some basic results derived from two kinds of topologies for a random locally convex module. J Funct Anal 2010, 258: 3024–3047.MathSciNetView ArticleGoogle Scholar

Copyright

© Tang; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.