## Linear Operators: Spectral operators |

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strongly closed if and only if it is

1 DEFINITION . A Boolean algebra B of projections in a B - space X is said to be

strongly closed if and only if it is

**complete**in the sense of the following definition .1 DEFINITION . A Boolean algebra B of projections in a B - space X is said to be

**complete**( o -**complete**) as an abstract Boolean algebra if each subset ...Page 2204

This means that for every Borel set e in 4 there is a projection Ele ) in B with A ( e

) = E ( e ) * , and thus the proof is

Boolean algebra of projections in a B - space is o -

...

This means that for every Borel set e in 4 there is a projection Ele ) in B with A ( e

) = E ( e ) * , and thus the proof is

**complete**. Q . E . D . 10 COROLLARY . ABoolean algebra of projections in a B - space is o -

**complete**if and only if it is the...

Page 2217

Let B be a o -

, be its strong closure . By Lemma 3 , B is bounded and thus B , is also a bounded

Boolean algebra of projections in X . Suppose that B , is not

Let B be a o -

**complete**Boolean algebra of projections in a B - space X , and let B, be its strong closure . By Lemma 3 , B is bounded and thus B , is also a bounded

Boolean algebra of projections in X . Suppose that B , is not

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