# "current variable function" in Arveson's Short Course on Spectral Theory

I am reading A Short Course on Spectral Theory by William Arveson, and there is a function used throughout named the 'current variable'. The first time it is mentioned in the book (according to the index) is in the following example on page 31 (emphasis mine):

Consider the Banach algebra $$A = C(\mathbb{T})$$ of continuous functions on the unit circle, and let $$B$$ be the Banach subalgebra generated by the current variable $$\zeta(z) = z, z \in \mathbb{T}$$[...]

From the description given, $$\zeta$$ just looks like the identity function. It is used in other places with other sets, e.g. on page 54 it is "$$\zeta(z) = z, z \in X$$" (for some set $$X$$), on page 69 "$$\zeta(z) = z, z \in \sigma(N)$$" (the spectrum of some operator $$N$$).

Searching for "current variable function" online or in a few other functional analysis books turns up nothing. Is this kind of function a convention in functional analysis that I've missed, or is Arveson just describing the identity function in a strange way?

• I have not encountered the term before. Not sure why Averson decided to name it that way. (It's a nice book btw!) Commented Oct 2, 2023 at 11:37

After further study of other spectral theory texts, I have come to the conclusion that Arveson is using $$\zeta$$ for the identity function:
The smoking gun is the study of Toeplitz operators, where we have the shift operator $$Su(n) = u(n+1)$$ on $$\ell^2(\mathbb{Z})$$ associated with the operator $$M_\zeta$$ on Hardy space (where an $$\ell^2$$ sequence $$(a_n)$$ is transformed to the series $$\sum_{n \in \mathbb{Z}}a_nz^n$$ on the unit circle). It is easy to see (but not made explicit in the book) that the shift operator would be multiplying every element of the series by $$z$$:
$$M_\zeta\sum_{n \in \mathbb{Z}}a_nz^n = z\sum_{n \in \mathbb{Z}}a_nz^n = \sum_{n \in \mathbb{Z}}a_nz^{n+1} = \sum_{n \in \mathbb{Z}}a_{n+1}z^n$$.