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A new version of the Gleason-Kahane-Żelazko 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 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, (or [-∞, +∞]) the set of extended real numbers, a probability space, the set of extended real-valued ℱ-random variables on Ω, the set of equivalence classes of extended real-valued ℱ-random variables on Ω, the algebra of K-valued ℱ-random variables on Ω under the ordinary pointwise addition, multiplication and scalar multiplication operations, the algebra of equivalence classes of K-valued ℱ-random variables on Ω, i.e., the quotient algebra of , and 0 and 1 the null and unit elements, respectively.
It is well known from [6] that is a complete lattice under the ordering ≤: ξ ≤ η 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 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 , as a sublattice of , 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 and .
The following notions of generalized inverse, absolute value, complex conjugate and sign of an element in bring much convenience to this article.
Definition 2.1. [7] Let ξ be an element in . For an arbitrarily chosen representative ξ0 of ξ, define two ℱ-random variables (ξ0)-1 and |ξ0|, respectively, by
and
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 , set ξ = u + iv, where is called the complex conjugate of ξ and sgn(ξ) := |ξ|-1 · ξ is called the sign of ξ. It is obvious that , where A = {ω ∈ Ω : ξ0(ω) ≠ 0} and denotes the equivalence class of the characteristic function I A of A. Throughout this article, the symbol 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 , then the equivalence class of A, denoted by Ã, is defined by , where A ΔB = (A\B)∪(B\A) is the symmetric difference of A and B, and 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 ; 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 means ξ ≥ η and ξ ≠ η; [ξ > η] stands for the equivalence class of the ℱ-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 . If , then ξ > η on à means ξ0(ω) > η0(ω) a.s. on A, similarly ξ ≠ η on à means that ξ0(ω) ≠ η0(ω) a.s. on A, also denoted by .
Definition 2.2. [7] An ordered pair (S, || · ||) is called a random normed module (briefly, an RN module) over K with base if S is a left module over the algebra and || · || is a mapping from S to such that the following conditions are satisfied:
(RNM-1) ||ξx|| = |ξ|||x||, , 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 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 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 , namely the module multiplication · : is jointly continuous (see [7] for details). Besides, let be the RN module of equivalence classes of X-valued ℱ-random variables on , where X is an ordinary normed space, then it is easy to see that the (ϵ, λ)-topology on is exactly the topology of convergence in probability and is complete iff X is complete, in particular is complete.
Definition 2.3. [5] An ordered pair (S, || · ||) is called a random normed algebra(briefly, an RN algebra) over K with base if (S, || · ||) is an RN module over K with base and also a ring such that the following two conditions are satisfied:
-
(1)
(ξ · x)y = x(ξ · y) = ξ · (xy), for all and all x, y ∈ S;
-
(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 be the RN module of equivalence classes of X-valued ℱ-random variables on . Define a multiplication · : by x·y = the equivalence class determined by the ℱ-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 , respectively. Then is an RN algebra, in particular is a unital RN algebra with identity 1.
Example 2.2. [5] It is easy to see that is a unital RN algebra with identity 1 (see [8, 9] for the construction of .
Definition 2.4. [5] Let (S, ||·||) be an RN algebra with identity e over C with base , 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 . Clearly, is unique and called the inverse on A of x, denoted by . Let G(S, A) denote the set of elements of S which are invertible on A. Then is also a group, and for any x and y in . 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 and 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 . For any x ∈ S, r(x) = ∨{|ξ| : ξ ∈ σ(x, S)} is called the random spectral radius of x.
Besides, is denoted by r p (x), for any x in an RN algebra over K with base .
Lemma 2.1. [5] Let (S, ||·||) be a unital complete RN algebra with identity e over C with base . 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, and f be an L0-linear function on S, i.e., a mapping from S to such that f(ξ · x + η · y) = ξf(x) + ηf(y) for all 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 .
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
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 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)
f is nonzero and multiplicative.
-
(2)
f(e) = 1 and f(x) ≠ 0 on à for any 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(e2) = f(e)f(e). Since f is nonzero, we have f(e) = 1 and hence for any 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 with P(A) > 0 such that 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
of degree n. Therefore we can find 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(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 of the RN module is a trivial probability space, i.e., , 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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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
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DOI: https://doi.org/10.1186/1029-242X-2012-85