Concerning the ring of all real valued functions of bounded variation on $[a,b]$

Question

Let $B[a,b]$ the ring of all real valued functions of bounded variation on $[a,b]$ . What is the cardinality of $B[a,b]$ ? How does the maximal ideals of $B[a,b]$ look like ? How does the prime ideals of $B[a,b]$ look like ? Is this ring isomorphic with $C[a,b]$ ? What is the structure of nil-radical of this ring ? And please can someone provide some texts or online references where I can read about this ring ? Please help . Thanks in advance .

Answer 1

An elementary proof that $BV$ is a ring / Banach algebra was given by Russell:

A. M. Russell, A commutative Banach algebra of functions of bounded variation, Amer. Math. Monthly. 87 (1980), 39–40.

As already mentioned in the comments, $BV$ contains non-trivial idempotent so is not isomorphic to $C[0,1]$ as ring. Observe that $BV$ is non-separable, hence is not isomorphic to $C[0,1]$ as a Banach space either.

$BV$ was studied as a Banach algebra in this seminal paper by Newman.

S. E. Newman, Measure algebras and functions of bounded variation on idempotent semigroups, Trans. Amer. Math. Soc. 163 (1972), 189–205.