A converse of the maximum modulus Theorem

Question

W. Rudin in Real and Complex Analysis (262) mentioned that

Theorem Suppose $M$ is a vector space of continuous complex functions on the closed unit disc $\bar U$, with the following properties:

(a) $1 \in M$

(b) If $f \in M$,then also $jf \in M$, where $j$ denotes the identity function: $j(z)=z$.

(c) If $f \in M$, then $\lVert f\rVert_U=\lVert f\rVert_T$, where T is the boundary of $\bar U$.

Then every $f\in M$ is holomorphic in $U$.

What I interested in is the origin and extension of this theorem. Where did it come form? Are there any references about this?

Answer 1

In the notes and comments section at the end of Real and Complex Analysis (page 397 in my copy), Rudin attributes this result "in a slightly different form" to:

W. Rudin, Analyticity, and the maximum modulus principle. Duke Math. J. 20, (1953). 449–457.