## 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

element in X with 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 2065

Since , for a in A , , the function â is continuous on the compact space S , it follows

that an operator a in A , has an

celebrated theorem of N . Wiener gives more by asserting that the

in ...

Since , for a in A , , the function â is continuous on the compact space S , it follows

that an operator a in A , has an

**inverse**in A if à ( s ) does not vanish on S . Acelebrated theorem of N . Wiener gives more by asserting that the

**inverse**a - 1 isin ...

Page 2069

Let the operator a in A have an

subalgebra of A which contains all

determinant S = det ( a , j ) has an

- 1 is in A . .

Let the operator a in A have an

**inverse**in B ( H ) . If a is of type ... Let A . be asubalgebra of A which contains all

**inverses**. ... 6 , A - 1 is in AP and thedeterminant S = det ( a , j ) has an

**inverse**in A . Since A , contains all**inverses**, 8- 1 is in A . .

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