projections as noncommutative characteristic functions

In this entry we try to exhibit the profound similarities there are between projections in Hilbert spaces and characteristic functions in a measure space. In fact, in the general framing of viewing von Neumann algebras as noncommutative measure spaces, projections are the noncommutative analog of characteristic functions.

Note - By a projection we always an orthogonal projection.

Let us some notation first:

• $H$ denotes a Hilbert space and $B(H)$ its algebra of bounded operators.

• $(X,\mathfrak{B},\mu)$ denotes a measure space.

• $\chi_{A}$ denotes the characteristic function of the measurable set $A\subset X$.

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.

• Characteristic functions take only the values $0$ and $1$. The spectrum of a projection is contained in $\{0,1\}$. Notice that the spectrum of a normal operator consists precisely of the values of the complex valued function associated with it (using the Gelfand-Naimark theorem and/or the continuous functional calculus).

• A partial ordering can be defined on the set of characteristic functions by saying

 $\chi_{A}\leq\chi_{B}\;\;\Longleftrightarrow\;\;A\subseteq B\text{, or % equivalently,}\;\;\;\chi_{A}\leq\chi_{B}\;\;\Longleftrightarrow\;\;\chi_{B}-% \chi_{A}\text{is a positive function}.$

Analogously, a partial ordering can be defined on the set of projections by saying

 $P\leq Q\;\;\Longleftrightarrow\;\;\mathrm{Ran}(P)\subseteq\mathrm{Ran}(Q)\text% {, or equivalently,}\;\;\;P\leq Q\;\;\Longleftrightarrow\;\;Q-P\text{is a % positive operator}.$

where $\mathrm{Ran}(P)$ denotes the range of the operator $P$.

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

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

• Consider the Banach algebra $L^{\infty}(X,\mu)$. Functions $f\in L^{\infty}(X,\mu)$ can be seen as multiplication operators in the Hilbert space $L^{2}(X,\mu)$. Thus, $L^{\infty}(X,\mu)$ can be seen as a closed subalgebra of $B(L^{2}(X,\mu))$ (it is in fact a von Neumann algebra).

Characteristic functions in $L^{\infty}(X,\mu)$ are exactly the projections of this subalgebra.

0.3 Measure Theory and the Spectral Theorem

The next key observation explores the similarities between some facts about measure theory and the spectral theorem of self-adjoint (or normal) operators.

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 $N$.

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