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In this paper we prove an Ozguç, Yurdakadim and Taş version of the Korovkin-type approximation by operators in the sense of the power series method. That is, we try to extend the Korovkin approximation theorems, obtained by Ozguç and Taş in 2016, and Taş and Yurdakadim in 2017, for concrete classes of Banach spaces to the class of Riesz spaces. Some applications are presented.

We extend \L ukasiewicz logic obtaining the infinitary logic $\mathcal{IR}\L$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in $\sigma$-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of $\mathcal{IR}\L$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.

Within the infinitary variety of σ-complete Riesz MV-algebras RMV σ , we introduce the algebraic analogue of a random variable as a homomorphism defined on the free algebra in RMV σ . After a preliminary study of the proposed notion, we use it to define stochastic processes in the framework of non-classical logic (Łukasiewicz logic, more precisely) and we define stochastic independence.

Recently Flaminio and Montagna, [FlMo], extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis–Sikorski Theorem for monotone σ-complete effect algebras with inter- nal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order de- termining system of states is dual to the category of Bauer simplices Ω such that ∂eΩ is an F-space., peer-reviewed

Concavity and supermodularity are in general independent properties. A class of functionals defined on a lattice cone of a Riesz space has the Choquet property when it is the case that its members are concave whenever they are supermodular. We show that for some important Riesz spaces both the class of positively homogeneous functionals and the class of translation invariant functionals have the Choquet property. We extend in this way the results of Choquet [G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1953–1954) 131–295] and Konig [H. Konig, The (sub/super) additivity assertion of Choquet, Studia Math. 157 (2003) 171–197].

A Perron-type integral of order 1 and 2 for Riesz-space-valued functions is investigated. Some versions of integration by parts formula for this integral are proved for both orders.

Ph.D. (Mathematics) Our objective in this thesis is to investigate two fundamental questions concerning Riesz operators de ned on a Banach space. Recall that Riesz operators are generalizations of compact operators in the sense that Riesz operators have the same spectral properties as compact operators. However, they do not possess the same algebraic properties as compact operators. Our rst question that we investigate is: When is a Riesz operator a nite rank operator? This question is motivated from the fact that if a compact operator de ned on a Banach space has closed range, then it is a nite rank operator. Also, Ghahramani proved that a compact homomorphism de ned on a C -algebra is a nite rank operator, see . Martin Mathieu, in his paper, generalized the result of Ghahramani by proving that a weakly compact homomorphism de ned on a C -algebra is a nite rank operator...

Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004. This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. Doctoral

This study revolves around operators defined by conditional expectations and operators generated by random measures. Studies of operators in function spaces defined by conditional expectations first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26]. N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their averaging properties were studied by P.G. Dodds and C.B. Huijsmans and B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert [17] studied their relationship with multiplication operators in C*-modules. It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special cases of kernel operators that were, inter alia, studied by A.R. Schep in [25] were special cases of conditional expectation operators. On the other hand, operators generated by random measures or pseudo-integral operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30], building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late 1970's and early 1980's. In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on Multiplication Conditional Expectation-representable (MCE-representable) operators. We also generalize the result of A. Sourour [27] and show that order continuous linear maps between ideals of almost everywhere finite measurable functions on u-finite measure spaces are MCE-representable. This fact enables us to easily deduce that sums and compositions of MCE-representable operators are again MCE-representable operators. We also show that operators generated by random measures are MCE-representable. The first chapter gathers the definitions and introduces notions and concepts that are used throughout. In particular, we introduce Riesz spaces and operators therein, Riesz and Boolean homomorphisms, conditional expectation operators, kernel and absolute T-kernel operators. In Chapter 2 we look at MCE-operators where we give a definition different from that given by J.J. Grobler and B. de Pagter in [8], but which we show to be equivalent. Chapter 3 involves random measures and operators generated by random measures. We solve the problem (positively) that was posed by A. Sourour in [28] about the relationship of the lattice properties of operators generated by random measures and the lattice properties of their generating random measures. We show that the total variation of a random signed measure representing an order bounded operator T, it being the difference of two random measures, is again a random measure and represents ITI. We also show that the set of all operators generated by a random measure is a band in the Riesz space of all order bounded operators. In Chapter 4 we investigate the relationship between operators generated by random measures and MCE-representable operators. It was shown by A. Sourour in [28, 271 that every order bounded order continuous linear operator acting between ideals of almost everywhere measurable functions is generated by a random measure, provided that the measure spaces involved are standard measure spaces. We prove an analogue of this theorem for the general case where the underlying measure spaces are a-finite. We also, in this general setting, prove that every order continuous linear operator is MCE-representable. This rather surprising result enables us to easily show that sums, products and compositions of MCE-representable operator are again MCE-representable. Key words: Riesz spaces, conditional expectations, multiplication conditional expectation-representable operators, random measures. Thesis (Ph.D. (Mathematics))--North-West University, Potchefstroom Campus, 2004.

In [3] we did not give explicitly the definition of measurability for realvalued functions, with respect to finitely additive measures with values in a Dedekind complete Riesz space. We note that, in [3], all involved functions are intended to be measurable. We now report the definition of measurability, which we gave in [2] (Definition 3.2).