By Werner Ricker
This publication provides a scientific research of the speculation of these commutative, unital subalgebras (of bounded linear operators performing in a Banach area) that are closed for a few given topology and are generated by way of a uniformly bounded Boolean algebra of projections. one of many major goals is to hire the equipment of vector measures and integration as a unifying subject all through. This yields proofs of a number of classical effects that are really assorted to the classical ones. This e-book is directed to either these wishing to benefit this subject for the 1st time and to present specialists within the box.
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Extra info for Operator Algebras Generated by Commuting Projections: A Vector Measure Approach
Sample text
7- of elements from/3 is called an ideal if (a) ~z V v E jr whenever u, v E jc and (b) u C jr whenever u E/3 and u _< v for some v E jr. (ii) Suppose that/3 is abstractly a-complete . Then an ideal jr in/3 is called a a-ideal if V~un E ~ for all countable sets {u~}~eN C_ jr. 9 E x a m p l e 16. Let f~ be a compact Hausdorff space. Let /3 := 2 a. Using the fact that the union of two sets of first category is again of first category and that a subset of a set of first category is again of first category, it is clear that the collection of all subsets of first oo 1A ~, for any category is an ideal in/3.
B) The algebra sim(Co(ft)) is conjugate closed, that is, ] E sim(Co(~2)) whenever f C sim(Co(ft)), where ] denotes the function w ~ f ( w ) , for w E ~. (c) / f f e sim(Co(f~)) has standard representation given by (2), then ~ e sim(Co(f~)) if and only ifO f[ {c~y}]=l. Then ~ = ~ 2 = l ( ~ ) X s j is the standard representation of ~. (d) /f (2) is the standard representation of f C sim(Co(f~)) , then Iifil~=max{lc~Jl: l_ Is it likely that this is an equality, with perhaps certain restrictions on ft? E x e r c i s e 24. Let ft = {0} U {88 : n E N} be the totally disconnected, compact Hausdorff space of Exercise 19(a). 5 show that Ba(f~) = Bo(ft). 100]. Unfortunately, Stone spaces are rarely metrizable. Let us show that Ba(f~) C_ Bo(f~) is strict for the example of Exercise 23. 99], for example. Let f~ be a totally disconnected, compact Hausdorff space. Then every open Baire set in f~ is the countable union of sets from Co(f~).