# projections as noncommutative characteristic functions

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Note - By a projection we always an orthogonal projection.

Let us some notation first:

Recall that a projection in $B(H)$ is a bounded operator $P$ such that $P^{*}P=P$. Let us now point out what projections and characteristic functions have in common. Note that, although we have written our observations in separate points (making it easier to read), they are all closely related.

## 0.1 Basic Facts

• Just like projections, characteristic functions satisfy: $\chi_{A}^{2}=\chi_{A}$. Thus, both characteristic functions and projections are idempotents.

• Characteristic functions are functions (meaning they take only positive or zero values), just like projections are positive operators.

## 0.2 Projections of $L^{\infty}(X,\mu)$

The above observations could all be easily derived from the general fact we describe next:

## 0.3 Measure Theory and the Spectral Theorem

It is a well known fact from measure theory that a continuous function   $f:X\longrightarrow\mathbb{R}$ can be approximated by linear combinations  of characteristic functions. With some additional effort it can be seen that, in fact, each continuous function $f$ is a (vector valued) integral of characteristic functions

 $f=\int_{X}f\,d\chi$

where $\chi$ is the vector measure of characteristic functions $\chi(A):=\chi_{A}$.

An analogous phenomenon in the spectral theory of normal operators. Notice (as pointed earlier) that the $C^{*}$-algebra (http://planetmath.org/CAlgebra) theory allows one to see a normal operator as a continuous function. With this in mind, the spectral theorem of normal operators can be seen as an analog of the previous measure  theoretic construction. Recall that the spectral theorem that a normal operator $N$ can be approximated by linear combinations of projections and can, in fact, be given by a (vector valued) integral of projections:

 $N=\int_{\sigma(N)}\lambda\;dP(\lambda)$

where $\sigma(N)$ denotes the spectrum of $N$ and $P$ is the projection valued measure associated with $N$.

Title projections as noncommutative characteristic functions ProjectionsAsNoncommutativeCharacteristicFunctions 2013-03-22 17:54:43 2013-03-22 17:54:43 asteroid (17536) asteroid (17536) 11 asteroid (17536) Feature msc 46C07 msc 46L10 msc 46L51 msc 46C05