ne the absolute value (not to be confused with the norm ) by taking the pointwise absolute value in .
only needs to be defined on the spectrum of in order for to be well-defined. Thus if is self-adjoint then we can make sense of for any continuous . If is self-adjoint and has positive spectrum then we can make sense of, for example, and .
As a more specific application (although I do not think I need the full power of the continuous functional calculus to conclude this), recall that a one-parameter subgroup of a topological group is a continuous group homomorphism .
Theorem: The map assigning an element of a C*-algebra the function is a bijection from self-adjoint elements of to norm-continuous one-parameter subgroups of the unitary group of .
Proof. If is self-adjoint, lands in the commutative C*-subalgebra generated by , and any one-parameter subgroup of takes values in a commutative C*-subalgebra, so we may assume without loss of generality that is commutative. If is self-adjoint, then , thought of as a one-parameter family of elements of , is a one-parameter subgroup by the boundedness of , and taking adjoints shows that it is unitary. Conversely, if is a one-parameter subgroup of , then for sufficiently small , as a function on , will only take values in a ball of small radius around by norm-continuity. On a sufficiently small such ball we can define a branch of the logarithm, and then is a one-parameter family of functions such that . Since is continuous in , it follows that for some , and by unitarity is self-adjoint.
The above theorem describes the exponential map from the unitary Lie algebra
of skew-adjoint elements of (which are precisely the elements where is self-adjoint) to the unitary group . However, it is stated for self-adjoint elements so that it becomes a version of Noether’s theorem for C*-algebras, with self-adjoint elements of a C*-algebra taking the role of observables of some quantum system (the noncommutative space associated to ) and one-parameter subgroups of taking the role of one-parameter groups of symmetries of the system.
Note that already to talk about position and momentum we needed unbounded operators which can’t embed into a Banach algebra or strongly continuous (not norm-continuous) one-parameter subgroups, so this is an incomplete discussion.