Stone's Theorem On One-parameter Unitary Groups - Formal Statement

Formal Statement

Let U be a strongly continuous 1-parameter unitary group, then there exists a unique self-adjoint operator A such that

Conversely, let A be a self-adjoint operator on a Hilbert space H. Then

is a strongly continuous one-parameter family of unitary operators.

The infinitesimal generator of {U_t}_t is the operator . This mapping is a bijective correspondence. A will be a bounded operator if and only if the operator-valued function is norm continuous.

Stone's theorem can be recast using the language of Fourier transform. The real line R is a locally compact abelian group. Nondegenerate representations of the group C*-algebra C*(R) is in one-to-one correspondence with strongly continuous unitary representations R, i.e. strongly continuous one-parameter family of unitary operators. On the other hand, the Fourier transform is a *-isomorphism between C*(R) and C₀(R), the C*-algebra of continuous functions on the line vanishing at infinity. So there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and representations of C₀(R). Since every representation of C₀(R) corresponds to a self-adjoint operator, Stone's theorem holds.

The precise definition of C*(R) is as follows. Form the convolution algebra on C_c(R), the continuous functions of compact support, where the multiplication is convolution. The completion of this algebra in the L1 norm is a *-algebra, denoted by L1(R). C*(R) is then defined to be the enveloping C*-algebra of L1(R), i.e., its completion in the largest possible C*-norm. It is not trivial that C*(R) is isomorphic to C₀(R), under the Fourier transform. A result in this direction is the Riemann–Lebesgue lemma, which says the Fourier transform maps L1(R) to C₀(R).