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

Copyright © 2018 Dong Hyun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. LetC[0, T] denote an analogue of a generalizedWiener space, that is, the space of continuous, real-valued functions on the interval [0, T]. In this paper, we introduce a Banach algebra on C[0, T] which generalizes Cameron-Storvick’s one, the space of generalized Fourier-Stieltjes transforms of the C-valued, and finite Borel measures on L2[0, T]. We also investigate properties of the Banach algebra on C[0, T] and equivalence between the Banach algebra and the Fresnel class which plays a significant role in Feynman integration theories and quantum mechanics.


Introduction
Let  0 [0, ] denote the Wiener space, that is, the space of continuous, real-valued functions  on the interval [0, ] with (0) = 0.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][2][3][4].In particular, the PWZ integral is used in the definition of Cameron-Storvick's Banach algebra S of functions on  0 [0, ] which is the space of generalized Fourier-Stieltjes transforms of the C-valued and finite Borel measures on  2 [0, ] [1].Johnson [4] showed that S 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 S 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  , [0, ] which is a generalized Wiener space with mean function  and variance function , where  and  are appropriate functions on [0, ].We note that every Wiener path  in both  0 [0, ] and  , [0, ] starts at the origin; that is, (0) = 0.
On the other hand, let [0, ] denote an analogue of a generalized Wiener space, that is, the space of continuous, real-valued functions on the interval [0, ].On the space [0, ], Ryu [8,9] introduced a finite measure  ,; and investigated its properties, where ,  : [0, ] → R are continuous functions such that  is strictly increasing, and  is arbitrary finite measure on the Borel class B(R) of R. On this space ([0, ],  ,; ), 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 B(R) and the generalized PWZ integral exists.In this paper, we will introduce a Banach algebra S ,; on [0, ] by using  , , which generalizes Cameron-Storvick's Banach algebra S with the mean function and the variance function determined by  and , respectively.We also investigate properties of S ,; , and relationships between S ,; and the Fresnel class [5] which plays a significant role in Feynman integration theories and quantum mechanics.We note that every path in [0, ] starts at arbitrary point so that [0, ] generalizes both  0 [0, ] and  , [0, ].
When the generalized PWZ integral on [0, ] is defined, one of difficulties encountered is the existence of a complete orthonormal basis of functions in  2  , [0, ] such that these functions are of bounded variation and orthogonal in  2 0, [0, ], where  2 0, [0, ] and  2 , [0, ] are the  2 -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 S ,; regardless of the existence of the

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 R. Let [0, ] denote the space of continuous, real-valued functions on the interval [0, ].Let ,  : [0, ] → R be two continuous functions, where  is strictly increasing.Let  be a positive finite measure on B(R).
For a rectangle ] is called an interval  and let C be the set of all such intervals .Define a premeasure  ,; on C by where for  →   = ( 1 , . . .,   ) ∈ R  and  0 ∈ R, The Borel -algebra B([0, ]) of [0, ] with the supremum norm coincides with the smallest -algebra generated by C and there exists a unique positive finite measure  ,; on B([0, ]) with  ,; () =  ,; () for all  ∈ C.This measure  ,; is called an analogue of a generalized Wiener measure on ([0, ], B([0, ])) according to  [8,9].For further work, we give additional conditions for  and .Let  and  be functions defined on [0, ] 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 [0, ] by where ]  () = ∫  ||() and ]  () = ∫  () for a Lebesgue measurable subset  of [0, ].Define  2 , [0, ] to be the space of functions on [0, ] that are square integrable with respect to the measure ] , ; that is, The space  2 , [0, ] is in fact a Hilbert space (as our notation suggests) and has the obvious inner product [11] ⟨, ⟩ , = ∫ Let [0, ] denote the collection of all step functions on [0, ].For  in  2 , [0, ], let {  } be a sequence of the step functions in [0, ] with lim →∞ ‖  − ‖ , = 0. Define the Itô type integral  , () of  by the  2 ([0, ])-limit for all  ∈ [0,] for which this limit exists, where ∫  0   ()() denotes the Riemann-Stieltjes integral of   with respect to .We note that  , ()() exists for  ,; a.e. ∈ [0, ] and the limit in (7)  Throughout the remainder of this paper, unless otherwise specified, we assume that (R) = 1; that is,  is a probability measure on B(R).We now have the following generalized PWZ theorem [10].] where * = means that if either side exists, then both sides exist and they are equal.
Let  : [0, ] → C be a measurable function and suppose that the integral

A Banach Algebra with Its Applications
In this section, we introduce a Banach algebra which generalizes the Banach algebra S given by Cameron and Storvick [1].To define it, we need the following lemmas.
Lemma 3. Let H 0 be a separable, real Hilbert space with the inner product ⟨⋅, ⋅⟩ H 0 .Let A be the -algebra generated by the class of sets of the form where  and  range over all elements in H 0 and over all real numbers, respectively.Then we have A = B(H 0 ), where B(H 0 ) is the Borel -algebra of H 0 .
Proof.Since ⟨⋅, ⟩ H 0 is continuous for each  ∈ H 0 , B(H 0 ) contains all sets of the form given by ( 13) so that we have A ⊆ B(H 0 ).To prove B(H 0 ) ⊆ A, it suffices to show that for each  > 0 and  0 ∈ H 0 , the closed ball belongs to A since H 0 is separable.Let  ≡ {  :  ∈ N} be a dense subset of H 0 .For each  ∈ N, take   ∈ H 0 such that ⟨  ,   ⟩ H 0 = ‖  ‖ H 0 and ‖  ‖ H 0 = 1 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  ∈ N, let Then for all positive integer , we have which belongs to A. We will prove  , 0 = ⋂ ∞ =1  ,, 0 so that  , 0 also belongs to A. Indeed, if  ∈  , 0 , then we have for all Remark 4. (1) By the Riesz representation theorem and Lemma 3, B(H 0 ) is in fact the smallest -algebra satisfying that all bounded linear functionals on H 0 are measurable.
(2) Applying the method used in the proof of [13, Theorem 4.2, p. 74], we can also prove Lemma 3.
We now have the following lemma.
For  ∈ where   is the measure on B defined by Since  2 , [0, ] =  2 0, [0, ], B is generated by the sets of the form   1 so that we have  1 () −  2 () =  3 () = 0 for all  ∈ B, that is,  1 =  2 , 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  2 , [0, ] =  2 0, [0, ] 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  2 0, [0, ] for the functions in the Hilbert space  Let (⋅, ) ∈ S ,; for each  ∈ , where  is a measure space, and let (, ) be a measurable function of (, ) on [0, ] × .Let {  } be the family of measures corresponding to (⋅, ) so that for each  ∈  and   ∈ M, for  ,; a.e. ∈ [0, ].Then, for each  ∈ B,   () 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  ∈  2 , [0, ] with ‖‖ 0, ̸ = 0, let ()  (, )  ,; () .
It is a measurable function of  by the assumption.Then we have where Letting  → 0 + , we have by the dominated convergence theorem which is still a measurable function of .Letting  → −∞, we have by the dominated convergence theorem again lim so that if then , [0, ], then we have which is also a measurable function of  by the above argument.Since B is generated by the sets of the form   1 ,   () is a measurable function of  on  for all  ∈ B. Solving   for   , we have for all  ∈ B which is a measurable function of  on , since the integrand can be expressed by a limit of simple functions on  2 , [0, ].
Using the same process used in the proof of Theorem 2.3 in [1], we can prove the following theorem.
Theorem 13. S ,; is a normed algebra.Moreover, the correspondence  →   given by (20) for  ∈ M is an algebra isometric isomorphism between M and S ,; , so that S ,; is a Banach algebra.Remark 14. (1) One can show that S ,; is a Banach space without the isomorphism in Theorem 13.For more details, see the proof of Theorem 2.2 in [1].
(2) If () = 0 and () =  for all  ∈ [0, ], 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 S ,; generalizes the Banach algebra S introduced by Cameron and Storvick [1].

The Fresnel Class with Its Equivalent Spaces
In this section, we establish that the Fresnel class [4,5] is isometrically isomorphic to S ,; .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 H be the space of real-valued functions V on [0, ] which are absolutely continuous with V(0) = 0 and V  /  ∈  2  , [0, ].Define  : H →  2 , [0, ] by V = V  /  for V ∈ H, and an inner product ⟨⋅, ⋅⟩ H on H by These equalities tell us that V = V for all V ∈ H, which implies that  is a vector space isomorphism and Let M(H) be the collection of C-valued, countably additive measures of finite variation on B(H).Define D : M(H) → M by D =  ∘  −1 for  ∈ M(H) and ) with B as its -algebra of measurable sets and both D and I are well defined by the arguments in (2) of Remark 17.Moreover, M(H) is a Banach algebra under convolution and with the total variation norm, since H is a separable, infinite dimensional, real Hilbert space by Lemmas 15 and 16 [15].Lemma 18. D is a Banach algebra isometric isomorphism between M(H) and M with D −1 = I.
Proof.Clearly, D is linear and bijective with D −1 = I.To prove that D preserves convolutions, let  1 ,  2 ∈ M(H).For  ∈ B, we have and since  2 , [0, ] =  2 0, [0, ] as vector spaces with the same Borel -algebra B, we have which completes the proof.
Let F(H) be the Fresnel class [4,15] defined by Then the correspondence  → F() is injective and carries convolution to point-wise multiplication.Letting ‖F()‖ = ‖‖, we have that M(H) is isometrically isomorphic to F(H) and F(H) is a Banach algebra [15].The Fresnel integral F  [F()] is defined for F() ∈ F(H) by the formula Now we have the following isomorphism theorem.
by the Fubini theorem and the change of variable theorem, which proves (60).Moreover, it is not difficult to prove that   ∈  2 , [0, ] with (0) = 0 if and only if  ∈ H since   is bounded and bounded away from 0. Now if   ∈  2 , [0, ] with (0) = 0, then we have which proves (61).If (62) holds, then we have (3) of this theorem by the definition of the generalized analytic Feynman  ,; -integral and the dominated convergence theorem.
Letting  = ‖‖ in (62) of Theorem 21, we now have the following corollary which is one of our main results.