The essence of Stone's theorem

1. Jun 14, 2013

mma

Could someone shortly summarise the essence of Stone's theorem ? What is the difference between Stone's theorem and the statement "the Lie-algebra of the group of orthogonal matrices consists of skew-symmetric matrices"? How Stone's theorem is related to the general notion of the exponential map between Lie-algebras and Lie-groups? What is the essential difference between Stone's theorem and its corresponding version for the finite dimensional orthogonal group? What is the significance of the strongly continuity of the one-parameter unitary subgroup? What can we say about the one-parameter subgroups that are not strongly continuous?

I would greatly appreciate if somebody could enlighten me.

*Edit

Last edited: Jun 14, 2013
2. Jun 16, 2013

mma

Perhaps I can formulate my question more specifically.

What is wrong in the following? How can it be made precise, and what fails in it, if the Hilbert space is infinite-dimensional?

Any one-parameter subgroup of the isometry-group of a finite or infinite dimensional, real or complex Hilbert space is a curve running in the group across the unit element. The tangent vector $v$ of this curve at the unit element is a skew-symmetric transformation of the Hilbert-space, and $t \mapsto \exp(tv)$ is the one-parameter subgroup itself. We say that $v$ is the infinitesimal generator of the one-parameter subgroup $t \mapsto \exp(tv)$. So every one-parameter subgroup determines a skew-symmetric transformation as its infinitesimal generator. Conversely, for every skew adjoint vector $v$, $t \mapsto \exp(tv)$ is the one-parameter subgroup of the isometry-group.

How comes here the strongly continuity?

Last edited: Jun 16, 2013
3. Jun 16, 2013

dextercioby

Stone's theorem enters the picture in connecting the strongly continuous unitary (irreducible) representations of a Lie group on a (complex separable) Hilbert space to the strongly continuous representations of the Lie algebra of the group on the same Hilbert space.

4. Jun 16, 2013

mma

What do you mean exactly?

5. Jun 16, 2013

dextercioby

Since the exponential mapping sends vectors in the Lie algebra into group elements in a neighborhood of identity, one uses this fact when trying to represent group and algebra elements as operators on a Hilbert space. See the theorem of Nelson as formulated in B. Thaller's book <The Dirac equation>.

6. Jun 16, 2013

mma

Thanks, I'll try to rake something from this book.