projections as noncommutative characteristic functions


similarities \PMlinkescapephrasepoint \PMlinkescapephrasepoints \PMlinkescapephrasepositivePlanetmathPlanetmath \PMlinkescapephraseobservations \PMlinkescapephraseobservation

In this entry we try to exhibit the profound similarities there are between projections in Hilbert spaces and characteristic functionsMathworldPlanetmathPlanetmathPlanetmath in a measure spaceMathworldPlanetmath. In fact, in the general framing of viewing von Neumann algebrasMathworldPlanetmathPlanetmathPlanetmath 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:

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: χA2=χA. 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.

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

    χAχBAB, or equivalently,χAχBχB-χAis a positive function.

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

    PQRan(P)Ran(Q), or equivalently,PQQ-Pis a positive operator.

    where Ran(P) denotes the range of the operatorMathworldPlanetmath P.

0.2 Projections of L(X,μ)

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

  • Consider the Banach algebraMathworldPlanetmath L(X,μ). Functions fL(X,μ) can be seen as multiplication operators in the Hilbert space L2(X,μ). Thus, L(X,μ) can be seen as a closed subalgebra of B(L2(X,μ)) (it is in fact a von Neumann algebra).

    Characteristic functions in L(X,μ) 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 theoremMathworldPlanetmath of self-adjointPlanetmathPlanetmathPlanetmath (or normal) operators.

It is a well known fact from measure theory that a continuous functionMathworldPlanetmathPlanetmath f:X can be approximated by linear combinationsMathworldPlanetmath 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


where χ is the vector measure of characteristic functions χ(A):=χA.

An analogous phenomenon in the spectral theory of normal operators. Notice (as pointed earlier) that the C*-algebra ( 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 measureMathworldPlanetmath 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:


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

Title projections as noncommutative characteristic functions
Canonical name ProjectionsAsNoncommutativeCharacteristicFunctions
Date of creation 2013-03-22 17:54:43
Last modified on 2013-03-22 17:54:43
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Feature
Classification msc 46C07
Classification msc 46L10
Classification msc 46L51
Classification msc 46C05