Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise sequential limits $f(t) = \lim f_n(t),\ t \in [0,1]$ with $f_n \in A$, $n = 1, 2, \ldots$. Note that when $A=C([0,1])$ then $B_1(A)$ is the space of bounded Baire class 1 functions on $[0,1]$, which is well known to be uniformly complete. In general, $B_1(A)$ is a linear subspace of the bounded Baire-1 functions on $[0,1]$.

The existence of such an $A$ probably follows, rather indirectly, from an old result of McWilliams (see Proc. Amer. Math. Soc. 16 (1965) 1195--1199). The point of this question is to define the functions in such an $A$ more directly, with a self-contained proof, or to provide a reference to such a description.

Here is a test case. Let $A \subset C([0,1])$ be a linearly isometric copy of $l_1$. Must $B_1(A)$, in this case, necessarily be a uniformly closed subspace of Baire-1 functions?

[edit 3/16/2016] For another, more concrete, example suppose $A$ is the disk algebra on the unit circle $\Gamma$ in the complex plane. That is, $A$ is the set of continuous complex-valued functions on $\Gamma$ which can be extended to be analytic on the open unit disk and continuous on the closed unit disk. Is $B_1(A)$ a uniformly closed subalgebra of the Baire-1 functions on $\Gamma$?

Note that such an example would contradict the (presumably erroneous) claim in the otherwise nice text "A second course on real functions" by van Rooij and Schikhof, Cambridge (1982), Exercise 11N, p. 69. The point of the Exercise is to prove all the higher Baire classes are uniformly complete, which is true, but the hint claims that $B_1(A)$ is always uniformly complete for any linear space $A$.

  • $\begingroup$ The claim of the exercise is true if the linear subspace $A$ forms a lattice (i.e. if $f, g \in A$ then $\min(f,g) \in A$ and $\max(f,g) \in A$) and contains the constant function $1$ (see R. D. Mauldin, "Baire Functions, Borel Sets, and Ordinary Function Systems", 1974, Theorem 3.1). $\endgroup$
    – yada
    Mar 15, 2016 at 7:19
  • $\begingroup$ Agreed. But what if $A$ is a uniform algebra? See example of disk algebra just added to original post. $\endgroup$ Mar 16, 2016 at 19:01


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