We study those aspects of continuous stochastic processes in Riesz spaces that enable us to state and prove the Doob-Meyer decomposition theorem for submartingales. We use the concepts developed for countable processes by Kuo, Labuschagne and Watson.
Abstract. We show that the existence of a trace on an ideal in a Banach algebra provides an elegant way to develop the abstract index theory of Fredholm elements in the algebra. We prove some results on the problem of the equality of the nonzero exponential spectra of elements ab and ba and use the index theory to formulate a condition guaranteeing this equality in a quotient algebra.1. Introduction. The existence of a continuous trace on an operator ideal of operators on a Banach space has long been known to provide a useful tool for developing the Fredholm theory of operators (see for instance the monograph of A. Pietsch [20] and the paper by the present authors [13]). The problem of defining traces on ideals in a Banach algebra attracted the attention of many authors (see the papers by Puhl [22] and Aupetit and Mouton [2]). The main thrust of these papers was to show that a trace and a Fredholm determinant exist on the socle of a semisimple Banach algebra ([2] and [22]) and on the question of extending the trace or the determinant to larger ideals ([22] and [3]). The aim of this paper, on the other hand, is to present an axiomatic approach by assuming a trace to exist and then to show how useful such a trace is by applying it firstly to develop the index theory for abstract Fredholm elements in a semisimple Banach algebra (see the research notes of Barnes, Murphy, Smyth and West [5] and of Caradus, Pfaffenberger and Yood [10]). Here the commutative property of the trace provides some elegant proofs for the properties of the index. In the definitions and proofs we avoid using results from operator theory as far as possible.An index function in a semisimple Banach algebra was also defined by Kraljević, Suljagić and Veselić (see [14]). We do not follow their approach, which is to define the trace of a Fredholm element a as an element of Z n(a)
The notions of stopping times and stopped processes for continuous stochastic processes are defined and studied in the framework of Riesz spaces. This leads to a formulation and proof of Doob's optional sampling theorem.
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