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#

Spectral Algebras

**Definition 3.8**

*: A*

**spectral algebra**of an operator A is the Boolean algebra^{3.2}associated with those projectors that form the spectral decomposition of A, projectors that project onto the eigenspaces associated with the eigenvectors of A.If we consider two physical quantities A and B, whose representative operators have the following spectral decomposition:

such that the operators and have a common projector , the lattice of properties would then be [7] where

Now, let us consider a Borel function such that .

According to diagram the spectral projectors of are with respective eigenvalues . Therefore, the Boolean sublattice of , which represents the spectral algebra of is composed by , , , , , and . If we choose f such that the eigenvalues of are then, the spectral projectors of are and . This entails that the spectral algebra of the operator is a subalgebra of , i.e. .

Since in Quantum Mechanics propositions are represented by spectral projectors the question that arises is the following: how do the relations between different spectral projectors determine the relations between the propositions that they represent? To answer this question let us consider, again, the above example where, the proposition is represented by the projection operator which is equivalent to that, in turn, is equivalent to since f is one to one with respect to . Therefore, in this case, the two propositions and are equivalent. If, instead, we consider propositions and then, the situation changes since f is many to one with respect to . In fact, we would have the following: . Since represents the join

^{3.3}of and it follows, by definition of join, that with respect to the partial ordering of the lattice. Therefore the proposition is weaker than proposition and so we say that is the coarse graining of .

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**Previous:**Presheaves Cecilia Flori 2007-01-02