21. These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus . 22. In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. 23. The functional calculus can be defined in exactly the same way for an element in " A ". 24. Some of these properties can be established by using the continuous functional calculus or by reduction to commutative C *-algebras. 25. By the functional calculus , this C *-algebra is the continuous functions on the unit circle in the complex plane. 26. Using the bounded functional calculus , one can prove part of the Stone's theorem on one-parameter unitary groups: 27. The idea is to first establish the continuous functional calculus then pass to measurable functions via the Riesz-Markov representation theorem. 28. Applying the spectral theorem, or Borel functional calculus for infinite dimensional systems, we see that it generalizes the classical entropy. 29. Using the machinery of measure theory, this can be extended to functions which are only measurable ( see Borel functional calculus ). 30. In more modern treatments however, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus .
