A Banach Algebra Similar to Cameron-Storvick’s One with Its Equivalent Spaces

Abstract

Let denote an analogue of a generalized Wiener space, that is, the space of continuous, real-valued functions on the interval . In this paper, we introduce a Banach algebra on which generalizes Cameron-Storvick’s one, the space of generalized Fourier-Stieltjes transforms of the -valued, and finite Borel measures on . We also investigate properties of the Banach algebra on and equivalence between the Banach algebra and the Fresnel class which plays a significant role in Feynman integration theories and quantum mechanics.

1. Introduction

Let denote the Wiener space, that is, the space of continuous, real-valued functions on the interval with . The (generalized) Paley-Wiener-Zygmund (PWZ) stochastic integrals on the (generalized) Wiener space have been used in various papers, in particular, concerning Feynman integration theories [1–4]. In particular, the PWZ integral is used in the definition of Cameron-Storvick’s Banach algebra of functions on which is the space of generalized Fourier-Stieltjes transforms of the -valued and finite Borel measures on [1]. Johnson [4] showed that is isometrically isomorphic to the Banach algebra of the Fresnel integrable functions given by Albeverio and Høegh-Krohn [5]. Further work for relationships between the Banach algebra and the Fresnel class was studied by Chang et al. [6] on infinite dimensional Hilbert spaces, and the same work was done by Chang et al. [7] on the space which is a generalized Wiener space with mean function and variance function , where and are appropriate functions on . We note that every Wiener path in both and starts at the origin; that is, .

On the other hand, let denote an analogue of a generalized Wiener space, that is, the space of continuous, real-valued functions on the interval . On the space , Ryu [8, 9] introduced a finite measure and investigated its properties, where are continuous functions such that is strictly increasing, and is arbitrary finite measure on the Borel class of . On this space , the author [10] introduced an Itô type integral and a generalized PWZ integral with their relation. The relation says that is reduced to the generalized PWZ integral if is a probability measure on and the generalized PWZ integral exists. In this paper, we will introduce a Banach algebra on by using , which generalizes Cameron-Storvick’s Banach algebra with the mean function and the variance function determined by and , respectively. We also investigate properties of , and relationships between and the Fresnel class [5] which plays a significant role in Feynman integration theories and quantum mechanics. We note that every path in starts at arbitrary point so that generalizes both and .

When the generalized PWZ integral on is defined, one of difficulties encountered is the existence of a complete orthonormal basis of functions in such that these functions are of bounded variation and orthogonal in , where and are the -spaces with respect to the Lebesgue-Stieltjes measures induced by and [10]. In order to avoid this difficulty, we will use instead of the generalized PWZ integral so that we can define the functions in regardless of the existence of the orthonormal basis of satisfying the orthogonality in as described above.

2. An Analogue of a Generalized Wiener Space

In this section, we introduce an analogue of a generalized Wiener space with the Itô type integral as described in Section 1.

Let denote the Lebesgue measure on . Let denote the space of continuous, real-valued functions on the interval . Let be two continuous functions, where is strictly increasing. Let be a positive finite measure on . For with , let be the function given by For a rectangle in , the subset of is called an interval and let be the set of all such intervals . Define a premeasure on by where for and , The Borel -algebra of with the supremum norm coincides with the smallest -algebra generated by and there exists a unique positive finite measure on with for all . This measure is called an analogue of a generalized Wiener measure on according to [8, 9].

For further work, we give additional conditions for and . Let and be functions defined on such that is absolutely continuous and is continuous, positive, and bounded away from 0. We observe that the functions and induce a Lebesgue-Stieltjes measure on by where and for a Lebesgue measurable subset of . Define to be the space of functions on that are square integrable with respect to the measure ; that is, The space is in fact a Hilbert space (as our notation suggests) and has the obvious inner product [11]

Let denote the collection of all step functions on . For in , let be a sequence of the step functions in with . Define the Itô type integral of by the -limitfor all for which this limit exists, where denotes the Riemann-Stieltjes integral of with respect to . We note that exists for a.e. and the limit in (7) is independent of choice of the sequence in . Moreover, one can show that is an injective, bounded linear operator from into . The following lemma is due to a result in [10].

Lemma 1. If and is absolutely continuous on with , then exists and it is given by for any sequence of the step functions in with .

For satisfying the assumption in Lemma 1, we redefine as . Nevertheless, we have (7) for a.e. . Also, is still an injective, bounded linear operator from into .

Throughout the remainder of this paper, unless otherwise specified, we assume that ; that is, is a probability measure on . We now have the following generalized PWZ theorem [10].

Theorem 2. Let be a set of functions in which are nonzero and orthogonal in . Then are independent random variables and each has the normal distribution with the mean and the variance . Moreover, if is Borel measurable, then where means that if either side exists, then both sides exist and they are equal.

Let be a measurable function and suppose that the integral exists as a finite number for all . If there exists a function analytic in such that for all , then is defined to be a generalized analytic Wiener -integral of over with parameter and it is denoted by for . Let be a nonzero real number. If has a limit as approaches through , then we call it a generalized analytic Feynman -integral of over with parameter and it is denoted by

3. A Banach Algebra with Its Applications

In this section, we introduce a Banach algebra which generalizes the Banach algebra given by Cameron and Storvick [1]. To define it, we need the following lemmas.

Lemma 3. Let be a separable, real Hilbert space with the inner product . Let be the -algebra generated by the class of sets of the formwhere and range over all elements in and over all real numbers, respectively. Then we have , where is the Borel -algebra of .

Proof. Since is continuous for each , contains all sets of the form given by (13) so that we have . To prove , it suffices to show that for each and , the closed ball belongs to since is separable. Let be a dense subset of . For each , take such that and which can be justified by Corollary 14.13 of [12] as an application of the Hahn-Banach theorem and the Riesz representation theorem. For each , let Then for all positive integer , we have which belongs to . We will prove so that also belongs to . Indeed, if , then we have for all by the Schwarz inequality, so that ; that is, . Conversely, let and let arbitrary. Since is a dense subset of , we can take with . Then we have by the Schwarz inequality since . Since is arbitrary, we have so that ; that is, . Now the proof is completed as desired.

Remark 4. By the Riesz representation theorem and Lemma 3, is in fact the smallest -algebra satisfying that all bounded linear functionals on are measurable.
Applying the method used in the proof of [13, Theorem 4.2, p. 74], we can also prove Lemma 3.

Lemma 5. We have as vector spaces, where denotes the Lebesgue space. Moreover, the two norms and are equivalent so that .
   if and only if as vector spaces. In this case, the two norms and are equivalent.

Proof. Since is bounded away from 0 and continuous so that it is bounded on , the two norms and are equivalent which implies . Moreover, the identity operator from to is bicontinuous so that the topologies induced by and are equal. We now have which proves . If , then clearly we have . Conversely, if , then by the open mapping theorem, the identity operator is a bounded operator and an open map since for all ; that is, is bicontinuous. Now the two norms and are equivalent, and the topologies induced by and are equal so that which completes the proof of .

Example 6. It is not difficult to show that is a constant function on if and only if for all if and only if for all . In this case, for all so that isometrically which implies , by Lemma 5. In particular, if for all , then isometrically.
If for some constant , for all , which is the condition suggested by Yoo et al. [14], then so that isometrically. Since , as vector spaces, we have as vector spaces, but they need not be equal isometrically. In this case, and are equivalent and by Lemma 5.
It is obvious that . Suppose that is bounded on . For , we have for some  since is bounded and bounded away from 0. We now have and hence as vector spaces, but they need not be equal isometrically. In this case, and are equivalent and by Lemma 5.

Throughout the remainder of this paper, we assume , as sets, unless otherwise specified. In this case, we have as vector spaces and by Lemma 5, but they need not be equal isometrically as Hilbert spaces. We also note that is separable since is separable [10] and the two norms and are equivalent by Lemma 5. Let be the class of complex measures of finite variation on with as its -algebra of measurable sets. If , then we set , the total variation of over . We also note that and they are Banach algebras under convolution and with the total variation norm, since and are separable, real, infinite dimensional Hilbert spaces [15].

We now have the following lemma.

Lemma 7. For , is a -measurable function on . Moreover, for a.e. , exists for a.e. .

Proof. Note that each term of the right-hand side of (7) is -measurable so that it is a -measurable function on . Since , is also measurable with respect to the measure on . Moreover, we have that, for a.e. , exists for a.e. .

Let be the space of functions of the formfor all for which the integral exists, where . We note that is well-defined for a.e. by Lemma 7 and it is an integrable function of on . Moreover, is a linear space over .

We have the following uniqueness theorem.

Theorem 8. For and , let and be related by (20). Then is uniquely determined by ; that is, there is a unique such that (20) holds for a.e. so that there is an one-to-one correspondence between and .

Proof. Suppose that there are two measures and in such that (20) holds with for a.e. . Let . Then for a.e. , we have Let be any real number and let with . Let . If , then let , and if , then let be the trapezoidal approximation and let for . Then is a bounded continuous function of class having a piecewise continuous derivative of bounded variation and vanishing outside a finite interval. Consequently, its Fourier transform is a bounded continuous function of class and we have For , letfor a.e. . Then (24) holds for a.e. which implies and Suppose that in . Then in . By Theorem 2, it is not difficult to show that is an integrable function of with respect to over . Now we have by Theorem 2 and the linearity of which still holds for in since in implies in both and . We now have by the Fubini theoremwhere is the measure on defined byfor a.e. . Note that for all and . Letting , we have by the dominated convergence theorem Also, letting , we have by the dominated convergence theorem again so that if then we have . If , then is either or . Let be the measure defined by (29) with replacing by 1. If , then . If , then by the above argument since . Solving (29) for , we have for all Since , is generated by the sets of the form so that we have for all , that is, , which completes the proof.

Remark 9. A difference between the proof of Theorem 8 and the proof of Theorem 2.1 in [1] is to use the additional condition as sets in Theorem 8. In order to apply Theorem 2 to the proof of Theorem 8, we need an orthonormalization process in the Hilbert space for the functions in the Hilbert space . See of Lemma 5.

Example 10. For some constant , let for all . Then we have isometrically. Replacing and by the zero function and , respectively, in the proof of Theorem 8, we can show that is uniquely determined by .

Corollary 11. Suppose that is a constant function on . Let   for each , where is a measure space, and let be a measurable function of on . Let be the family of measures corresponding to so that for each and , for a.e. . Then, for each , is measurable as a function of on .

Proof. This corollary follows from the fact that the method of proof of Theorem 8 has provided a method for explicitly constructing in terms of . We will use the same assumptions and notations in the proof of Theorem 8. Indeed, for with , let It is a measurable function of by the assumption. Then we have where for   a.e.  . Letting , we have by the dominated convergence theorem which is still a measurable function of . Letting , we have by the dominated convergence theorem again so that if then is a measurable function of on . If , is either or . If , then which is a measurable function of . If , then we have which is also a measurable function of by the above argument. Since is generated by the sets of the form , is a measurable function of on for all . Solving for , we have for all which is a measurable function of on , since the integrand can be expressed by a limit of simple functions on .

Definition 12. If and are related by (20), we define a norm of by . We note that is well defined by Theorem 8.

Using the same process used in the proof of Theorem 2.3 in [1], we can prove the following theorem.

Theorem 13. is a normed algebra. Moreover, the correspondence given by (20) for is an algebra isometric isomorphism between and , so that is a Banach algebra.

Remark 14. One can show that is a Banach space without the isomorphism in Theorem 13. For more details, see the proof of Theorem 2.2 in [1].
If and for all , and is the Dirac measure concentrated at 0, then we can obtain all results of Section 2 in [1] from Theorems 8 and 13 so that generalizes the Banach algebra introduced by Cameron and Storvick [1].

4. The Fresnel Class with Its Equivalent Spaces

In this section, we establish that the Fresnel class [4, 5] is isometrically isomorphic to . Some ideas of the results in this section follow from [4, 7], but the detailed proofs of the results in this section are quite different from those in [4, 7].

Let be the space of real-valued functions on which are absolutely continuous with and . Define by for , and an inner product on by for , which can be justified by the fact as vector spaces.

We now have the following results.

Lemma 15. is well defined on and is a real inner product space.

Proof. By the Minkowski inequality, it is clear that is a real linear space. Let . If and for a.e. , then and for a.e. so that we have that is, is well defined. It is obvious that is a symmetric, bilinear, nonnegative definite form on . Moreover, if for , then for a.e. so that for a.e. since is bounded away from 0. Now we have for all so that is positive definite. Hence, the theorem is proved.

Lemma 16. Define by for and for . Then is a vector space isomorphism and . Moreover, and are isometric isomorphisms between and so that is a separable, infinite dimensional Hilbert space, and both and are bounded linear operators.

Proof. Clearly, and are well defined and linear. Moreover, for all and for all if . These equalities tell us that for all , which implies that is a vector space isomorphism and . For , we have by the definition of so that and are isometric isomorphisms between and . Now, is a separable, infinite dimensional Hilbert space since is. Moreover, for , we have so that is bounded. Since is onto, is an open map by the open mapping theorem so that is bounded.

Remark 17. We have as vector spaces, but we note that it need not hold isometrically as Hilbert spaces since the inner products and need not be equal. Thus, and need not be isometric isomorphisms between and in Lemma 16.
Both and are bounded linear operators by Lemma 16 so that and are - and - measurable, respectively, where denotes the Borel -algebra of .

Let be the collection of -valued, countably additive measures of finite variation on . Define by for and by for . We note that with as its -algebra of measurable sets and both and are well defined by the arguments in of Remark 17. Moreover, is a Banach algebra under convolution and with the total variation norm, since is a separable, infinite dimensional, real Hilbert space by Lemmas 15 and 16 [15].

Lemma 18. is a Banach algebra isometric isomorphism between and with .

Proof. Clearly, is linear and bijective with . To prove that preserves convolutions, let . For , we have For each , define by for . Then is continuous so that it is - measurable and . Now we have by the change of variable theorem For each , define by for . By the same argument as above, we have that is, so that is an algebra isomorphism. To complete the proof, we have to show for , where denotes the total variation norm on each space. Indeed, we have by the Riesz representation theorem and the change of variable theorem where denotes the set of all -valued, bounded, and continuous functions on . Let denote the set of all -valued, bounded, and continuous functions on . We now have that if , then and since is a continuous isomorphism from onto by Lemma 16. Similarly, if , then and by Lemma 16. Thus, we have which completes the proof.

Let denote the Fourier transform of a measure . We note that is a proper subset of . Now the following theorem holds.

Theorem 19. For , let . Then we have , where and are related by (20).

Proof. Since is continuous on , we can take such that for all . Then it follows that for all This means that for all . Thus, by Lemma 1, the integral of the right-hand side of (20) exists; that is, exists. We now have by the change of variable theorem and since as vector spaces with the same Borel -algebra , we have which completes the proof.

Let be the Fresnel class [4, 15] defined by Then the correspondence is injective and carries convolution to point-wise multiplication. Letting , we have that is isometrically isomorphic to and is a Banach algebra [15]. The Fresnel integral is defined for by the formula

Now we have the following isomorphism theorem.

Theorem 20. The map defined by for , where , is an isometric isomorphism as Banach algebras. The inverse of is given by for , where .

Proof. Let with and let . By Theorem 13, the map defined by , is an isometric isomorphism between Banach algebras. By Lemma 18, the map is also an isometric isomorphism between Banach algebras. Let be defined by for . By Proposition 2.3 of [15], also is an isometric isomorphism between Banach algebras, too. Now we have and this theorem follows by Theorem 19.

Theorem 21. For and , let and be related by (20). Let . Then we have the following: (1) If , then(2) If and with (or equivalently ), then(3)If there exists satisfying  for any , then is given by the right-hand sides of (60). Moreover, if (62) holds for , then for any nonzero real , is given by the right-hand sides of (60) replacing by . In addition, if , then is given by the right-hand side of (61) replacing by .

Proof. Since , we have for by the Fubini theorem and the change of variable theorem, which proves (60). Moreover, it is not difficult to prove that with if and only if since is bounded and bounded away from 0. Now if with , then we have which proves (61). If (62) holds, then we have of this theorem by the definition of the generalized analytic Feynman -integral and the dominated convergence theorem.

Remark 22. Applying of Theorem 21, we can also obtain Theorem 3.1 of [2].

Letting in (62) of Theorem 21, we now have the following corollary which is one of our main results.

Corollary 23. For and , let and be related by (20). If for all , then we have for any where . Moreover, for any nonzero real , is given by the right-hand sides of (65) replacing by . In this case, we have .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education (2017R1D1A1B03029876).