The most important results of
this paper are two not very closely related theorems concerning the extension of
functions. For the first theorem, let A be a subspace of a topologicaI space B and let
X and Y be subsets of C(A) and C(B). respectively. The theorem then asserts that,
if every member of χ extends to a member of Y , then every member of the
C∗-subalgebra of C(A) generated by χ extends without increase in norm to a
member of the C∗-subalgebra of C(B) generated by Y . As an application of this
theorem, new proofs of some results of J. F. Berglund on the extension of almost
periodic functions are given.
The statement of the second theorem is: every weakly almost periodic function on
an open subgroup H of a locally compact group G extends to a function weakly
almost periodic on G. (K. deLeeuw and I. Glicksberg have proved this result with the
additional assumption that H is normal.)
