1. Introduction and Preliminaries
Let
X be a complex Banach space. A (continuous) map
is said to be a (continuous)
n-homogeneous polynomial if there exists a (continuous)
n-linear map
such that
0-homogeneous polynomial is just a constant function. A finite sum of homogeneous polynomials is a polynomial. We denote by
the space of all continuous
n-homogeneous polynomials on
X and by
the space of all polynomials on
. Note that
is a Banach space with respect to any of the norms
Let
be the topology on
of uniform convergence on bounded subsets of
This topology is generated by the countable family of norms (
1) for positive rational numbers
r and so is metrisable. We denote by
the completion of
. So
is a Fréchet algebra which consists of entire analytic functions on
X which are bounded on all bounded subsets (so-called
entire functions of bounded type). For details on polynomials and analytic functions on Banach spaces we refer the reader to [
1]. The spectra (sets of continuous complex homomorphisms = sets of characters) of
and its subalgebras were investigated by many authors (see e.g., [
2,
3,
4,
5]).
Let
G be a group of isometric operators on
. We denote by
the subalgebra of
which consists of
G-invariant analytic functions. Such algebras were considered in the general case in [
6,
7]. For some special cases of
G there is a sequence of
G-symmetric homogeneous polynomials
which forms an algebraic basis in the algebra of
G-symmetric polynomials
. For example, if
is the group of all permutations of the basis vectors in
, then the functions
form an algebraic basis in
[
8]. The following bases in
also are important
and
Let
,
and
be formal series
and
which also are called generating functions. From combinatorial considerations it is known ([
9] p. 3) that
and
where the equality holds for every
and every
t in the common domain of convergence. In [
10] it is shown that every complex homomorphism
of
is completely defined by its value on
and
is a function of exponential type with
. Moreover, if
is the point evaluation functional at
(that is
,
)), then
Note that (
5) is an absolutely convergent Hadamard Product—the entire function defined by its zeros
for
. Also [
10,
11], there is a family
,
in the spectrum of
such that
In [
12] it is shown that there is a function of exponential type
with
but which cannot be represented as in (
4). Spectra of algebras
were investigated also in [
13,
14]. Polynomials which are symmetric with respect to some other representations of the group of permutations of natural numbers were considered in [
15,
16,
17].
In this paper we consider a subalgebra of entire functions of bounded type which is generated by so-called supersymmetric polynomials. Algebras of supersymmetric polynomials on finite-dimensional spaces were considered in [
18,
19,
20]. In
Section 2.1 we consider some important bases in the algebra of supersymmetric polynomials.
Section 2.2 is devoted to the spectrum of the algebra of supersymmetric analytic functions of bounded type. In particular, we show that the set of point evaluation functionals on the algebra can be described as a metric ring which is not a linear space. Some operators on this ring are investigated.
2. Results
2.1. Bases of Supersymmetric Polynomials
We will use
for natural numbers and
for integers. Also, we set
and denote by
the Banach space of all absolutely summing complex sequences indexed by numbers in
. The symbol
means the classical Banach space of absolutely summing complex sequences. Any element
z in
has the representation
with
where
and
are in
,
,
for
and
are natural isometric embeddings of the copies of
into
.
Let us define the following polynomials on
:
Definition 1. A polynomial P on is said to be supersymmetric if it can be represented as an algebraic combination of polynomials . In other words, P is a finite sum of finite products of polynomials in and constants. We denote by the algebra of all supersymmetric polynomials on .
Note first that polynomials are algebraically independent because are so. Hence forms an algebraic basis in .
We say that , for some if for every . Let us denote by the quotient set which is a natural domain for supersymmetric polynomials. For a given , let be the class of equivalence which contains .
Similarly like in [
10] we introduce an operation “•” on
:
where
and
. Also, we denote
. Clearly,
and
. These operations can be naturally defined on
by
Theorem 1. The following statements hold:
- 1.
for every .
- 2.
The operations in (6) are well defined, that is, they do not depend on the choice of representatives. - 3.
is a commutative group with zero .
- 4.
if and only if there are such that and for all . Equivalently, all nonzero coordinates of d coincides with nonzero coordinates of s up to a permutation.
Proof. Assertions (1)–(3) are straightforward consequences of definitions. In [
13] is proved that for given
for all
if and only if all nonzero coordinates of
d coincides with nonzero coordinates of
s up to a permutation. □
Let and be some algebras of polynomials on linear spaces X and Y respectively such that is generated by an algebraic basis and is generated by an algebraic basis with Then the map, defined on the basic vectors by and extended to by linearity and multiplicativity, obviously is an algebraic isomorphism from onto which preserves degrees of polynomials.
Let us denote by
the isomorphism from
to
such that
Proposition 1. If is an algebraic basis in then is an algebraic basis in
Proof. The proof follows from the general fact that the range of any algebraic basis under an isomorphism is an algebraic basis. Indeed,
are algebraically independent because
are so and
is injective. Also, every
belongs to the algebraic combination of
because
belongs to the algebraic combination of
and
is surjective (cf. [
13]). □
Let be the completion of with respect to the topology of uniform convergence on bounded subsets. In other words, is the minimal closed subspace of which contains . Elements of will be called supersymmetric analytic or entire functions on .
Proposition 2. The map is continuous and can be extended to a continuous homomorphism from to with a dense range. The map Λ is discontinuous and densely defined on .
Proof. Let us observe first that
is the restriction of
onto the closed subspace
The operator of the restriction is obviously continuous on
and is the extension of
The range of
is dense because it contains all symmetric polynomials on
. On the other hand, in [
10] it is proved that the homomorphism
such that
is discontinuous on
Moreover, in [
21] a function
such that
was constructed. If
is continuous, it can be extended to the whole space
and so
It leads to a contradiction because on the left side we have a bounded function on all bounded subsets but on the right side, it is not so. □
For a given
we denote by
It is easy to see that
Theorem 2. Let Thenandwhere the equality is true on the common domains of convergence. Proof. In [
10] it is proved that
Hence, for a fixed
From (
10) we have
and so (
8) holds. Taking into account Formula (
2) we have
Proof. The required statement immediately follows if we combine Formulas (
9) and (
8). □
Corollary 2. For every and we haveand Proof. Taking coefficients of we have the first equality. The second and thirds equalities we can obtain by the same reasoning. □
It is clear that for all . We say that is an irreducible representative of if and for every and every ,
Proposition 3. is irreducible if and only if and have no common zeros.
Proof. According to (
5), for nonzero elements
and
the numbers
and
are zeros of
and
respectively. □
Corollary 3. Let . Then u is completely defined by and is a meromorphic functions of the form such that are entire functions of exponential type with and where Moreover, let and be zeros of f and g respectively. Then both and belong to andis an irreducible representation of Let
We denote by
the support of
that is,
Corollary 4. Let and be two irreducible representatives of Then there are bijections and such that and for all and
Proposition 4. For every the following equality holds on the common domain of convergencewhere Proof. From (
8) and (
5) it follows that
converges for every
if
and in the ball
where
if
Since
is a continuous homomorphism, from (
3) we have that for each
such that
converges
Since
the series
converges if
Also,
and the series
converges if
So in the common domain of convergence
is in the domain of
and
Theorem 3. Let and For a given there is such that if and only if λ is an integer number.
Proof. Let If then If then If then
Let now
According to (
11)
But it contradicts representation (
8) for
. □
2.2. The Spectrum of and the Nonlinear Normed Ring
2.2.1. The Spectrum
Let us denote by the spectrum of that is, the set of all continuous nonzero complex homomorphisms (characters) of Clearly for every point there is a character (so-called point evaluation functional) such that Moreover, if then In this sense, we can say that
Since polynomials
form an algebraic basis in
any character
is completely defined by its values on
In other words, every character
can be represented by the function
Note that if
for some
then
can be described by Corollary 3. Using ideas in [
11,
13] it is possible to construct a character which is not a point-evaluation functional. Let
and
be complex numbers. Consider
From the compactness reasons, we have that
has a cluster point
in
So
Taking into account [
10] that
we have
Comparing the representation with Corollary 3, we can see that cannot be equal to a point evaluation functional.
2.2.2. The Normed Ring Structure of
We consider the set
more detailed. Let
According to [
12] we introduce an operation ‘⋄’ on
and extend it to
Let
Then
we mean the resulting sequence of ordering the set
with one single index in some fixed order. If
and
then
From [
12,
22] we know that the operation on
is commutative, associative and
Finally, let
and
are in
We define
Proposition 5. For every
Proof. From [
12] we know that for all
Let
and
Then
□
Theorem 4. is a commutative ring with zero and unity
Proof. Note first that
is a commutative group and if
then
is the inverse of
The associativity and commutativity of the multiplication and the distributive low were proved in [
12] for the case
and can be checked for the general case by simple computations. □
Note that there is an operation of multiplication by a constant on
:
But, in the general case,
So is not a linear space over Hence is not an algebra. In order to topologise we can use the standard norm on
Definition 2. Let We define a norm of u by the following way:where is an irreducible representative of From Corollary 4 it follows that the definition of norm does not depend on the irreducible representative. The next proposition shows that, like in a linear space, the norm has natural properties.
Proposition 6. Let The following properties hold:
- 1.
and if and only if
- 2.
- 3.
- 4.
- 5.
- 6.
Proof. We need to prove just item (6). Let
be a representation of
We can write up to a permutation that
for some
and irreducible
So
for ever
□
We define a metric
on
associated with the norm by the natural way. Let
Set
It is easy to check that is a metric using the same arguments as in the classical case of linear normed spaces.
Proposition 7. The multiplication by for a fixed is discontinuous in general at each nonzero point in and continuous at zero. Here we consider the standard topology on and the topology on generated by
Proof. Let
be a sequence in
such that
as
and
where
or
and
Then
while
as
Let now
and
be an irreducible representation of
Then
□
Theorem 5. The operations ‘•’ and ‘⋄’ are jointly continuous on
Proof. It is easy to check that if
and
then
and
□
Proposition 8. The metric space is nonseparable.
Proof. Let us consider the following set
So the unit sphere of contains an uncountable set such that the distance between each pair of distinct points of is equal to □
Theorem 6. The metric space is complete.
Proof. Let
u and
v be in
and
and
be an irreducible representations of
Then there is an irreducible representation
of
v such that
Indeed the inequality
implies that there is
such that
and
Let us consider a representation
of
v such that the element
w in
is represented by the same vector that in
Let
be the irreducible representation of
in
and
be the irreducible representation of
in
Then
Let
be a Cauchy sequence in
Taking a subsequence, if necessary, we can assume that if
and
then
Let us chose irreducible representations
of
such that
So if
and
then
Hence, is a Cauchy sequence in and so it has a limit point Let be the ith coordinate of that is, if and if Clearly that as We claim that if then there is a number N such that for every Indeed, it it is not so, then for every and we have a contradiction.
For a given
we denote by
a vector in
such that
has a finite support,
or
and
Note that for this case
Let
N be a number such that for every
for all
and
So
2.2.3. Invertibility and Homomorphisms
If
has an inverse with respect to the multiplication ‘⋄’ we denote it by
, that is,
Proposition 9. Let and Then is invertible in
Proof. It is easy to check that the proof for classical Banach algebras can be literally repeated for this case. In particular,
where
and the series on the right converges in
□
Next we consider ring homomorphisms and subrings of In sequel we do not assume that ring homomorphisms preserve the multiplication by constants. Note that an element x of a commutative Banach algebra A is invertible if and only if for every character of The situation in is different. Let
Proposition 10. Let φ be a nonzero ring homomorphism from to Then but is non invertible for
Proof. Clearly, On the other hand, for every □
Example 1. The following maps are ring homomorphisms from to
- 1.
Polynomials are (continuous) complex valued ring homomorphism of but only preserves the multiplication by constants.
- 2.
Let We define Clearly, Θ is well defined. The additivity and multiplicativity will be proved for more general case.
As usual is a subring of if it is a subset of and a ring with respect to ‘•’ and ‘⋄’. For example, let consists of all elements such that if is irreducible, then and are finite sets. Then is a dense subring of We consider some nontrivial examples of closed subrings of
Example 2. Let and be defined by Clearly, and are subrings of and Also, is isomorphic to the ring of integer numbers and the restriction of the topology of to and coincides with the discrete topology. In the general case, let U be a subset of We denote by Then is a subring of if U is closed with respect to the multiplication in and
Proposition 11. Let be a function of one variable which is well defined and multiplicative on a subset We definewhere If U is closed with respect to the multiplication and then is a complex valued ring homomorphism of Proof. Note first that
does not depend of the choice of a representative. Thus
By the multiplicativity of
we have
and
So
□
Example 3. Let us consider some examples of complex valued homomorphisms of subrings of
- 1.
Let g be a multiplicative function from In Number Theory such functions are called completely multiplicative arithmetic functions. Then for is a complex valued ring homomorphisms of and
- 2.
Let and be the closed disk in of radius centered at zero. Then is an ideal in Letthen is a complex valued ring homomorphisms of Note that if and then From here we have that is continuous.
We do not know whether or not every complex valued homomorphism of or its closed subring is continuous.
2.2.4. Additive Operator Calculus
Let
be an additive map. Since it is a homomorphism of the additive group
to itself,
is continuous at every point if and only if it is continuous at a point in
Let
be an arbitrary function. Then it is well defined the following additive map from
to itself:
Proposition 12. If there are constants and such that then is continuous, additive and well defined on
Proof. If then and so is continuous at zero. Thus it is continuous. □
Example 4. (Power operators.) Let Then satisfies Proposition 12 and so the map where andis a continuous additive operator on Let andbe the multi-valued kth power root function. Let us consideras an element in Then, for every such that for an irreducible representation of uwe can define The map for is a discontinuous additive operator, defined on a dense subset of But if then we can define an additive operatorwhich is continuous on Note that if because is the identical operator while We say that a map is a linear operator if it is additive, preserves multiplications by constants, that is, and if for all From Proposition 12 it follows that there are a lot of additive operators. Linear operators, in contrast, can be described in a simple way.
Theorem 7. Let A be a continuous linear operator from to itself. Then there exists an element such that Proof. Let
and
Set
Let now
u be an element in
which can be represented by a vector
with finite support
Since the set of elements with finite supports is dense in and A is continuous, for every □
We denote by the operator Let us prove some natural properties of operators
Proposition 13. - 1.
The operator is bijective if and only if v is invertible in
- 2.
If the operator is surjective, then it is bijective.
- 3.
The operator is injective if and only if
- 4.
If for some then for some
- 5.
If for some then is not surjective.
Proof. (1) If
v is invertible, then
so
is a bijection. Let now
Then from the Open Map theorem for complete metric groups (see [
23]) it follows that
B is continuous. From Theorem 7 we have that
for some
Since
(2) Let be surjective. Then there exists such that So v is invertible and
(3) If is injective, then Conversely, If there are such that then and so is nontrivial.
(4) If
then
and so
Since there exists a number such that So
(5) If
is surjective, then it is bijective and so
v is invertible. But
a contradiction. □
Note that for
the operator
is not surjective but it is injective because
and
for every
On the other hand for
and so
is not surjective but it is injective. Indeed, it is easy to check that
for
So, if
for some
then
for
But from [
13] it follows that also
and so
for all
that is
Finally, for
has a nontrivial kernel which contains
3. Discussion
According to Gelfand’s theory, every commutative semi-simple algebra Fréchet
can be represented as an algebra of continuous functions on its spectrum
(see e.g., [
24] p. 217, p. 231). If
consists of analytic functions on a Banach space
then for every
the point evaluation functional
belongs to
The map
is one-to-one if and only if
separates points of
for example, if
is the algebra of all analytic functions of bounded type on
Investigations of the spectrum of
were started by Aron, Cole and Gamelin in their fundamental work [
2]. Note that, in the general case,
has complicated topological and algebraic structures (see [
5,
25]) which can be described only implicitly involving such tools as the Aron-Berner extension, topological tensor products, StoneČech compactification, ect. On the other hand, it is convenient for applications to have algebras of analytic functions of infinite many variables whose spectra admit explicit descriptions. If a subalgebra
of
has an algebraic basis of polynomials
then every
is completely defined by its values on this basis,
So we can describe
as a subset of a sequence space
Moreover, if
and
then it is not difficult to check that sequences
should satisfy the following condition
Note that for the algebra of symmetric analytic functions of bounded type on
condition (
14) is sufficient [
14] but for the algebra
is not [
12]. In the present paper we use this approach for
which is a subalgebra of
generated by polynomials
. We can see that
is quite different than
For example, the homomorphism defined by
is continuous in
while
is discontinuous in
On the other hand, the homomorphism defined by
is discontinuous for
and so the set of sequences
does not support multiplications by constants. From here we have that condition (
14) is not sufficient for description of
The results of
Section 2.2 show that the spectrum of
admits an interesting algebraic structure of commutative ring with respect to operations ‘•’ and ‘⋄’ which play roles of addition and multiplication. Using these operations and the
-norm we introduced a natural metric
on
and proved that
is a complete metric space. We studied homomorphisms of
and described all linear operators of
to itself. So obtained results may be interesting in the theory of commutative topological algebras and for algebras of analytic functions on Banach spaces as well.
Supersymmetric polynomials and analytic functions are applicable in other branches of Mathematics and in Physics. Note first that supersymmetric polynomials of several variables were studied by many authors and in [
18,
19,
20] we can find analogs of Formulas (
7) and (
8) for these cases (with using some different notations). Here we proved such results for infinite many variables and due to
-topology we can claim that
is a rational function, where the numerator and the denominator are functions of exponential type for every fixed
But an important difference between finite- and infinite-dimensional case is that in the finite-dimensional case we can not to use the operations ‘•’ and ‘⋄’ because they do not preserve the dimension of the underlying space. Some applications of supersymmetric polynomials for Brauer groups are described in [
26]. It seems to be that
can be applied for infinite generated Brauer groups in a similar way. Another application can be obtained for Statistical Mechanics. In [
27] we can find an approach to how classical symmetric polynomials can be used to modeling the behavior of ideal gas. According to this approach and using our notations, independent variables
correspond to abstract energy levels which particles of ideal gas may occupy; symmetric monomials
correspond to occupation these energy levels by particles; generating functions
and
correspond to grand canonical partition functions for bosons and fermions respectively, and Equation (
2) is modeling the Bose-Fermi symmetry law. From this point of view and taking into account (
7), supersymmetric polynomials may be useful for the description of ideal gas consisting of both type particles: bosons, and fermions. Moreover, the Bose-Fermi symmetry in our notations means just
Note that Statistical Mechanics work with the situation when the number of particles, N tends to infinity. The fact that we consider the closure of polynomials in a metrizable topology allows us to proceed with limit values as The -topology of the underlying space is guarantying that all abstract supersymmetric polynomials are well defined on this space. For example, if we will use instead of then will be not defined. Finally, we can expect that the algebraic operations ‘•’ and ‘⋄’ may have a physical meaning in the proposed approach. But such kind of problems is outside of the topic of our article.
4. Conclusions
In this article, we considered the algebra
of analytic functions of bounded type generated by supersymmetric polynomials on
We have described some algebraic bases of the subalgebra of supersymmetric polynomials and corresponding generating functions. Such a description is important in order to study the spectrum (the set of complex homomorphisms) of
In particular, it is shown that every point evaluation complex homomorphism can be represented as a ratio of two entire functions of exponential type. Also, we constructed an example of complex homomorphism which is not a point evaluation functional. However, we have not a complete description of the spectrum of
In particular, it is unclear under which conditions a meromorphic function is of the form (
12) for some
? Note that such kind of problem is also open for the algebra
[
10,
12].
Our goal is establishing the structure of a complete metric commutative ring on the set of point evaluation functionals of The algebraic structure of is very close to the Banach algebra structure but is not a Banach algebra because it is not a linear space. So we have a natural question: which Banach algebras properties can be extended to the ring ? For example, we can see that if an element is closed to the unity, then it is invertible. But we do not know: do admits a discontinuous complex homomorphisms? Also, we investigated homomorphisms of its subrings and additive operators of The role of obtained results in the theory of algebras of analytic functions on Banach spaces and possible applications in Physics are discussed.