Kaplansky density theorem

The leak rate of Kaplansky ( Irving Kaplansky ) is one of the basic principles of the theory of von Neumann algebras. This is a series of statements about approximability with respect to the strong operator topology.

Wording of the sentence

Be one with respect to the involution closed subalgebra of continuous linear operators on the Hilbert space. We consider the strong operator topology, ie the topology of pointwise norm convergence: A network converges to 0 if for all. The financial statements in this topology, the so-called strong degree, will be denoted by a horizontal bar. In this situation, the leak rate of Kaplansky applies:

• Includes approximated by operators of A (with respect to the strong operator topology), we can approximate T by operators of A with norm less than or equal 1:

.

• Is self-adjoint with by operators of A can be approximated, we can approximate T by self-adjoint operators of A with norm less than or equal 1:

.

• Is positive with by operators of A can be approximated, we can approximate T by positive operators of A with norm less than or equal 1:

.

• If A is a C * - algebra and the unitary operator by operators of A can be approximated, we can approximate T by unitary operators from A:

Or

,

Since the condition is redundant.

Note that the above statement about self-adjoint operators is not trivial follows from the first statement, because the involution with respect to the strong operator topology is discontinuous: Is the shift operator, it is in the strong operator topology, but does not converge to 0

It is clear that one can generalize the conditions for each of the first three points above theorem, because the multiplication by the scalar is a homeomorphism.

Importance

The leak rate of Kaplansky provides for many sets from the theory of C * - algebras and von Neumann algebras is an important technical tool, it is a fundamental theorem in the theory of von Neumann algebras. Gert K. Pedersen writes in his book, C * - Algebras and Their automorphism groups:

( The leak rate is Kaplanskys great gift to humanity. You can use it daily, and Sundays twice. )

Typical application

• Be a separable Hilbert space and with respect to the involution closed subalgebra. Then you can each be approximated by a sequence of.

For a proof of a dense sequence in. If so you can go to the above tightness set of Kaplansky to each find one with and. If now, as is with to a. Then for all

And therefore in the strong operator topology.

One sees in this evidence very well how the argument depends on that you can choose is limited in the operator norm the approximating operators, and serves the leak rate of Kaplansky.