How is Stone's Theorem Related to Lie Algebras and Unitary Groups?

[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

 

[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?

 

[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.

 

[mma]

What do you mean exactly?

 

[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>.

 

[mma]

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