Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces

: The class of measure spaces which can be represented as unions of Lebesgue-Rohlin spaces with continuous measures contains a lot of important examples, such as R n for any n ∈ N with the Lebesgue measure. In this work we consider symmetric functions on Banach spaces of all complex-valued integrable essentially bounded functions on such unions. We construct countable algebraic bases of algebras of continuous symmetric polynomials on these Banach spaces. The completions of such algebras of polynomials are Fréchet algebras of all complex-valued entire symmetric functions of bounded type on the abovementioned Banach spaces. We show that each such Fréchet algebra is isomorphic to the Fréchet algebra of all complex-valued entire symmetric functions of bounded type on the complex Banach space of all complex-valued essentially bounded functions on [ 0,1 ] .

In [41], the authors constructed an algebraic basis of the algebra of symmetric continuous complex-valued polynomials on the complex Banach space L ∞ [0, 1] of complex-valued Lebesgue measurable essentially bounded functions on [0, 1] and described the spectrum of the Fréchet algebra H bs (L ∞ [0, 1]) of symmetric analytic entire functions, which are bounded on bounded sets, on L ∞ [0, 1]. In [42], the authors showed that the algebra H bs (L ∞ [0, 1]) is isomorphic to the algebra of all analytic functions on the strong dual of the topological vector space of entire functions on the complex plane C. In addition in [42], it was shown that the algebra H bs (L ∞ [0, 1]) is a test algebra for the famous Michael problem (see [55]). In [49], the authors showed that the algebra H bs (L ∞ [0, 1]) is isomorphic to the algebra of symmetric entire functions on the complex Banach space of complex-valued Lebesgue integrable essentially bounded functions on the semi-axis.
In this work, we generalize the results of the work [49], replacing the semi-axis with the arbitrary union of Lebesgue-Rohlin spaces (which are also known as standard probability spaces) with continuous measures. Note that there are a lot of important measure spaces which can be represented as the abovementioned union. For example, R n for any n ∈ N with the Lebesgue measure is one such space. We consider symmetric functions on Banach spaces of all complex-valued integrable essentially bounded functions on the unions of Lebesgue-Rohlin spaces with continuous measures. We construct countable algebraic bases of algebras of continuous symmetric polynomials on these Banach spaces. The completions of such algebras of polynomials are Fréchet algebras of all complex-valued entire symmetric functions of bounded type on the abovementioned Banach spaces. We show that every such Fréchet algebra is isomorphic to the Fréchet algebra H bs (L ∞ [0, 1]).

Preliminaries
Let us denote by N and Z + the set of all positive integers and the set of all nonnegative integers, respectively.

Polynomials
Let X be a complex Banach space. Let N ∈ N. A mapping P : X → C, which is the restriction to the diagonal of some N-linear mapping A P : X N → C, i.e., P(x) = A P x, . . . , x N for every x ∈ X, is called an N-homogeneous polynomial.
A mapping P : X → C, which can be represented in the form where N ∈ N, P 0 is a constant mapping, and P n : X → C is an n-homogeneous polynomial for every n ∈ {1, . . . , N}, is called a polynomial of a degree at most N. It is known that a polynomial P : X → C is continuous if and only if its norm |P(x)| is finite. Consequently, for every continuous N-homogeneous polynomial P : X → C and for every x ∈ X we have the following inequality: (See [56] (p. 53)). The finitely open subsets of E define a translation invariant topology τ f . The balanced τ f -neighborhoods of zero form a basis for the τ f -neighborhoods of zero. On a topological vector space (E, τ), the topology τ f is finer than τ, i.e., τ f ≥ τ. [56] (Def. 2.2, p. 54)) The complex-valued function f , defined on a finitely open subset U of a complex vector space E is said to be G-holomorphic if for each a ∈ U, b ∈ E the complex-valued function of one complex variable

Definition 2. (See
is holomorphic in some neighborhood of zero. We let H G (U) denote the set of all G-holomorphic mappings from U into C.
The following proposition is a partial result of [56] for all y in some τ f -neighborhood of zero. This series is called the Taylor series of f at a. converges and defines a continuous function on some τ-neighborhood of zero. We let H(U) denote the algebra of all holomorphic functions from U into C endowed with the compact open topology (the topology of uniform convergence on the compact subsets of U). A function, which is holomorphic on E, is called entire.

Proposition 2.
If U is an open subset of a complex locally convex space E and f : U → C is G-holomorphic, then f ∈ H(U) if and only if f is locally bounded.
The following proposition is a partial result of [56] (Cor. 2.9, p. 59). (See [56] (p. 166)). Let U be an open subset of a complex locally convex space E, and let B be a balanced closed subset of E. We let d B (a, U) = sup |λ| : λ ∈ C, a + λB ⊂ U for every a ∈ U. If E is a complex normed linear space and B is the unit ball of E, then d B (a, U) is the usual distance of a to the complement of U in E.
Let f ∈ H(U). The B-radius of boundedness of f at a ∈ U, is defined as The B-radius of uniform convergence of f at a ∈ U is defined as R f (a, B) = sup |λ| : λ ∈ C, a + λB ⊂ U, and the Taylor series of f at a converges to f uniformly on a + λB .
where m ∈ Z + , y ∈ E and r > 0. Equation (2) is called the Cauchy Integral Formula. Let E be a complex Banach space. Let H b (E) be the Fréchet algebra of all entire functions of bounded type f : E → C endowed with the topology of the uniform convergence on bounded subsets. Let f r = sup x ≤r | f (x)| for f ∈ H b (E) and r ∈ (0, +∞). The topology of the Fréchet algebra H b (E) is generated by any set of norms { · r : r ∈ I}, where I is an arbitrary unbounded subset of (0, +∞). For details on holomorphic functions on Banach spaces, we refer the reader to [57] or [56,58] .

1.
For every measurable set A ⊂ Ω, there exists a set B such that A ⊂ B ⊂ Ω, B is identical with A modulo zero, and B is an element of the σ algebra generated by G.

2.
For every pair of points x, y ∈ Ω, there exists a set G ⊂ G such that either x ∈ G, y ∈ G, or x ∈ G, y ∈ G.
Every countable system G of measurable sets satisfying conditions (1) and (2) is called a basis of the space (Ω, F , ν).
Let (Ω, F , ν) be a separable measure space, and let B = {B n } ∞ n=1 be an arbitrary basis in (Ω, F , ν). If all intersections of the form ∩ ∞ n=1 A n , where A n is one of the two sets B n and Ω \ B n , are nonempty, then the space (Ω, F , ν) is called complete with respect to the basis B. By [59] ( §2, No. 2), if the space (Ω, F , ν) is complete modulo zero (i.e., isomorphic modulo zero to some complete measure space) with respect to some basis, then it is complete modulo zero with respect to every other basis. Separable measure spaces which are complete modulo zero with respect to their bases are called Lebesgue-Rohlin spaces or standard probability spaces. By [59] ( §2, No. 4), every Lebesgue-Rohlin space with continuous measure (i.e., there are no points of positive measure) is isomorphic modulo zero to [0, 1] with Lebesgue measure. The following simple lemma shows that every such space is isomorphic to [0, 1] with Lebesgue measure.

Symmetric Functions
In general, symmetric functions are defined in the following way.

Definition 4.
Let A be an arbitrary nonempty set, and let S be a nonempty set of mappings acting from A to itself. A function f , defined on A, is called symmetric with respect to the set S if f (s(a)) = f (a) for every s ∈ S and a ∈ A.
Let us describe the partial case of Definition 4, which we will use in this work. The set of all measurable automorphisms of some measure space (Ω, F , ν) we will denote by Ξ Ω . A complex Banach space X of measurable functions x : Ω → C such that x • σ belongs to X for every x ∈ X and σ ∈ Ξ Ω will be in the role of the set A from Definition 4. The set of for every x ∈ X and σ ∈ Ξ Ω .

Algebraic Combinations
where T is a nonempty set, k ∈ N, f 1 , . . . , f k are mappings acting from T to C and Q is a polynomial acting from C k to C, is called an algebraic combination of mappings f 1 , . . . , f k . Let A be some algebra of complex-valued mappings. Let B ⊂ A be such that every element of A can be uniquely represented as an algebraic combination of some elements of B. Then B is called an algebraic basis of A.
Note that R n is a symmetric continuous n-homogeneous polynomial such that R n = 1 for every n ∈ N.
where k 1 , . . . , k n ∈ Z + and α k 1 ,...,k n ∈ C. In other words, {R n } forms an algebraic basis in the algebra of symmetric continuous polynomials on L ∞ [0, 1]. 1] such that R n x ξ = ξ n for every n ∈ N and For every function f ∈ H bs (L ∞ [0, 1]), its Taylor series converges uniformly to f on every bounded set. The nth term, where n ∈ N, of the Taylor series is a continuous nhomogeneous polynomial, which is symmetric by the symmetry of f and by the Cauchy Integral Equation (2). Therefore, by Theorem 1, every f ∈ H bs (L ∞ [0, 1]) can be represented as where α k 1 ,...,k n ∈ C, x ∈ L ∞ [0, 1], and the series converges uniformly on every bounded subset of L ∞ [0, 1]. [60] (p. 97, Thm. 1.3), this space is complete. For n ∈ N, let us defineR n :

Entire Symmetric Functions on
For every n ∈ N,R n is a symmetric n-homogeneous polynomial and R n = 1.
Proof. Clearly, J γ is linear and injective. Let us show that J γ is isometrical. Let Hence, J γ is an isometrical mapping.
For every E ⊂ Ω, let For n ∈ N, let the polynomialR n : The symmetry and the n-homogeneity of the polynomialR n , for every n ∈ N, can be easily verified. Let us prove the continuity ofR n .
Lemma 3. For every n ∈ N, R n = 1 and, consequently,R n is continuous.
Proof. For γ ∈ Γ, let Q γ : L ∞ [0, 1] → C be defined by where W γ and J γ are defined by (5) and (6), respectively. We have the following diagram: Since W γ and J γ are linear continuous mappings and P is a continuous n-homogeneous polynomial, it follows that P • J γ and Q γ are continuous n-homogeneous polynomials.
Recall that for every index γ ∈ Γ, the mapping w γ is an isomorphism between (Ω γ , F γ , ν γ ) and [0, 1] with Lebesgue measure µ. For every index γ ∈ Γ, let us construct the isomorphism w γ between (Ω γ , F γ , ν γ ) and [0, 1) with Lebesgue measure. Choose a countable set M ⊂ Ω γ such that w −1 Since the mapping w γ is a bijection, the set N is countable. Since measures ν γ and µ are continuous, the sets M and N are null sets. Let h : M → N be an arbitrary bijection. Let us define the mapping It can be checked that the mapping w γ is an isomorphism between (Ω γ , F γ , ν γ ) and [0, 1) with Lebesgue measure.

Let us define the mapping
is an isomorphism, it follows that the mapping V {γ n } ∞ n=1 is a linear isometric bijection.
Theorem 4 and the Cauchy Integral Equation (2) imply the following corollary.
Corollary 1. Every function f ∈ H bs ((L 1 ∩ L ∞ )(Ω)) can be uniquely represented in the form where α k 1 k 2 ...k n ∈ C, and the series converges uniformly on bounded sets.
Proof. Consider the sequence c = {c n } ∞ n=1 , where c n =R n (y) for n ∈ N. SinceR n is an n-homogeneous polynomial and R n = 1, by (1), |R n (y)| ≤ y n for every n ∈ N. Consequently, sup n∈N n |c n | ≤ y < ∞.
Therefore, by Theorem 2, there exists x c ∈ L ∞ [0, 1] such that R n (x c ) = c n for every n ∈ N and where M is defined by (3). We set x y := x c . This completes the proof.
). The inequality (21) and the fact that f is the function of bounded type imply the fact that J( f ) is the function of bounded type. By (17) and by the symmetry ofR n , the function J( f ) is symmetric. Let us show that J( f ) is entire. By Proposition 4, lim sup n→∞ P n where P 0 = α 0 and P n = ∑ k 1 +2k 2 +...+nk n =n k 1 ,...,k n ∈Z + α k 1 ,...,k n R k 1 1 · · · R k n n for n ∈ N. Consider the series whereP 0 = α 0 andP n = ∑ k 1 +2k 2 +...+nk n =n k 1 ,...,k n ∈Z + α k 1 ,...,k nR k 1 1 · · ·R k n n for n ∈ N. Note thatP n = J(P n ); therefore, by (21), P n 1 ≤ P n 2 M for every n ∈ N. By the n-homogeneity of the polynomial P n , Therefore, P n 1 ≤ 2 M n P n 1 .
Thus, J is linear. Let us show that J is continuous. Since J is a linear mapping between Fréchet algebras, it follows that for J the continuity and the boundedness are equivalent. In turn, the boundedness of J follows from (18). Thus, J is continuous.
By using (26), it can be verified the equality for arbitrary symmetric continuous polynomials P 1 , P 2 : Let us show that J( f g) = J( f )J(g). Let f = ∑ ∞ n=0 f n and g = ∑ ∞ n=0 g n be the Taylor series expansions of f and g respectively. Then Consequently, since J is linear and continuous, taking into account (26), Thus, J is multiplicative. Let us show that J is a bijection. Let γ 0 be an arbitrary element of Γ.
where W γ 0 is defined by (5), and J γ 0 is defined by (6). Since W γ 0 is a linear isometrical bijection and J γ 0 is a linear isometrical injective mapping (by Lemma 2), it follows that v is a linear isometrical injective mapping. Therefore, for every r > 0, the image of the closed ball with the center at 0 and the radius r of the space L ∞ [0, 1] under v is a subset of the closed ball with the center at 0 and the radius r of the space (L 1 ∩ L ∞ )(Ω). Therefore, sup |g(v(x))| : x ∈ L ∞ [0, 1], x ∞ ≤ r ≤ sup |g(y)| : y ∈ (L 1 ∩ L ∞ )(Ω), y ≤ r . (29) for every function of bounded type g : (L 1 ∩ L ∞ )(Ω) → C and for every r > 0. Let us prove the following auxiliary statement.
Proof of Lemma 5. Let f ∈ H bs ((L 1 ∩ L ∞ )(Ω)). Since f is a function of bounded type, it follows that the value f r is finite for every r > 0. Therefore, by (29), the value f • v r is finite for every r > 0. Thus, the function f • v is of bounded type.
By (18) and (29), for every f ∈ H bs (L ∞ [0, 1]) and for every r > 0. This inequality implies the continuity of J and J −1 . This completes the proof of Theorem 5.

Conclusions
This work is a significant generalization of the work [49]. We consider symmetric functions on Banach spaces of all complex-valued integrable essentially bounded functions on the unions of Lebesgue-Rohlin spaces with continuous measures. Note that there are a lot of important measure spaces which can be represented as the abovementioned union. For example, R n for any n ∈ N with the Lebesgue measure is one such space. We investigate algebras of symmetric polynomials and entire symmetric functions on the abovementioned spaces. In particular, we show that Fréchet algebras of all complex-valued entire symmetric functions of bounded type on these Banach spaces are isomorphic to the Fréchet algebra of all complex-valued entire symmetric functions of bounded type on the complex Banach space L ∞ [0, 1].
The next step in this investigation is to consider the case of unions of arbitrary Lebesgue-Rohlin spaces.
Author Contributions: Conceptualization, T.V.; methodology, T.V.; writing-original draft preparation, T.V. and K.Z.; writing-review and editing, T.V. and K.Z. All authors have read and agreed to the published version of the manuscript.