## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 1967

If a B * - subalgebra X of a B * - algebra Y has the same unit e as Y , then an

show that e = e * . Since e is the unit , e * = ee * , and so e = ee * = ( ee * ) * = e * *

* = ee ...

If a B * - subalgebra X of a B * - algebra Y has the same unit e as Y , then an

**element**in X with an inverse in Y has this inverse also in X . PROOF . We firstshow that e = e * . Since e is the unit , e * = ee * , and so e = ee * = ( ee * ) * = e * *

* = ee ...

Page 2108

We shall say that an

used ... The

and the mapping g + g ( a ) = valg ) is an operational calculus for a . We say that a

...

We shall say that an

**element**a € A is scalar ( although Schaefer [ 10 ; p . 143 ]used ... The

**elements**in the range of va are , in a natural sense , “ functions of a ”and the mapping g + g ( a ) = valg ) is an operational calculus for a . We say that a

...

Page 2215

is fundamental in X . Then a bounded operator is in the weakly closed operator

algebra generated by B if and only if it commutes with every

PROOF . It is clear that every

generated ...

is fundamental in X . Then a bounded operator is in the weakly closed operator

algebra generated by B if and only if it commutes with every

**element**of B .PROOF . It is clear that every

**element**in the weakly closed operator algebragenerated ...

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