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The disc algebra, as a set, consists of the functions on the unit disc $D$, which are analytic on the interior of the disc and continuous on its boundary. Its addition and multiplication is obvious. And, as a normed algebra, its norm is given by

$\| f\| = \sup\{ |f(z)| | z \in D \}$.

One page 16 of Gerard J. Murphy's C star algebras and Operator Theory, an element in this algebra is called its "canonical generator".

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I am wondering which one in this algebra is the canonical generator. Thanks a lot.

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It's not terminology I've heard before, but it probably doesn't make sense for it to mean anything but the identity function ($f(x)=x$). –  Henning Makholm Dec 1 '12 at 11:38
    
@Henning Makholm: Thank you very much for the comment. I have add some details, trying to make the question more clear. –  ShinyaSakai Dec 2 '12 at 10:25

1 Answer 1

As Henning said in a comment, from the context it must mean the function $z\mapsto z$, although it didn't quite say as much. This function "canonically" generates the polynomial functions, and $A$ is the closure of the polynomials in the sup norm on the closed disk. In this context, "canonical" seems to mean, "what you would expect; no tricks here."


For reasons I cannot fully explain, this function is often named awkwardly. It is sometimes tempting to call it "the identity function" but this is not an ideal way to think of the function, given that the algebraic operations in $A$ are pointwise operations with values in $\mathbb C$ (rather than, say, composition of self-maps of the disk). It is more like "inclusion," but that isn't quite right, either, in this context.

I would guess that at some level most people just think of this as "the function $z$," or just "$z$" when the context seems clear, just as we would speak of "(the function) $z$ squared" or "(the function) $e$ to the $z$". But particularly in the case of $z$ that has the potential to be confusing or ambiguous, and also might not emphasize enough the role as an element of an algebra, where we sometimes don't want to think of it explicitly as a function. Murphy might have been trying to concisely both avoid ambiguity (not entirely successfully) and emphasize the status of $z$ as generator, as the later plays a role in the application of Theorem 1.3.7 (I presume, although I don't have the book).


I have seen another functional analysis textbook use the phrase "the current variable" in a similar context.

When dealing with subsets of $\mathbb C^n$, the function $(z_1,z_2,\ldots,z_n)\mapsto z_k$ is sometimes called the $k^\text{th}$ "coordinate function." When $n=1$, the map $z\mapsto z$ can be called the coordinate function, too, and I have also seen this used in similar contexts.

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But isn't this a typo? The disk algebra is generated by $z$ and $1$. –  Student Jan 31 at 11:11
    
@Student: It is generated by $z$ as a unital Banach algebra. –  Jonas Meyer Jun 27 at 7:26

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