Uniform boundedness of the functional calculus

[kevinm@uwm.edu]

In deriving the functional calculus for a (possibly unbounded) self-
adjoint operator A on a Hilbert space, one typically proceeds by
defining f(A) for a nice algebra of functions, and showing that the
map f -> f(A) is bounded when the function algebra is given the
supremum (L^\infinity) norm so that it can be extended to a larger
algebra by continuity. I am interested in knowing whether there are
simple yet general criteria for determining the boundedness of f ->
f(A).

To be more specific, I would like to choose the original function
algebra to be the Schwartz space of rapidly-decreasing functions,
which is uniformly dense in the space of continuous functions
vanishing at infinity, and define f(A) via the Fourier transform. I
do have a quick proof of boundedness in this case (using an idea from
E.B. Davies, "Spectral Theory and Differential Operators", proof of
Lemma 2.2.8), but it feels as though it relies more on luck than on
general principles. The problem is really one of extending a *-
algebra homomorphism from a dense subalgebra of a C*-algebra to the
whole algebra, or of finding criteria for the boundedness of a
homomorphism on such a dense subalgebra, and I assume such criteria
are known. My immediate interest in the problem is that I will be
teaching an introductory graduate course in functional analysis this
Fall--hence the request for "simple" criteria.

Thank in advance for any help,

Kevin.