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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?

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

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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 $z$:

$M_\zeta\sum_{n \in \mathbb{Z}}a_nz^n = z\sum_{n \in \mathbb{Z}}a_nz^n = \sum_{n \in \mathbb{Z}}a_nz^{n+1} = \sum_{n \in \mathbb{Z}}a_{n+1}z^n$.

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