Vol. 56, No. 1, 1975

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On the extension of continuous and almost periodic functions

Paul Milnes

Vol. 56 (1975), No. 1, 187–193

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.)

Mathematical Subject Classification 2000
Primary: 43A60
Received: 26 November 1973
Revised: 1 February 1974
Published: 1 January 1975
Paul Milnes