Kaplansky tightness theorem

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

Formulation of the sentence

Let be a subalgebra of continuous linear operators on the Hilbert space that is closed with respect to the involution . We look at the strong operator topology , i. H. the topology of point-wise norm convergence: A network converges to 0 if and only if for all . The termination in this topology, the so-called strong termination, is denoted by a slash. In this situation, Kaplansky’s tightness theorem applies: ${\displaystyle A\subset L(H)}$ ${\displaystyle H}$ ${\displaystyle L(H)}$ ${\displaystyle (T_{i})_{i}}$ ${\displaystyle \|T_{i}x\|\rightarrow 0}$ ${\displaystyle x\in H}$

• If with can be approximated by operators from A (with regard to the strong operator topology), then T can also be approximated by operators from A with norm less than or equal to 1: ${\displaystyle T\in L(H)}$ ${\displaystyle \|T\|\leq 1}$

${\displaystyle \{T\in {\overline {A}};\,\|T\|\leq 1\}={\overline {\{T\in A;\,\|T\|\leq 1\}}}}$.

• If self-adjoint with can be approximated by operators from A , then T can also be approximated by self-adjoint operators from A with norm less than or equal to 1: ${\displaystyle T\in L(H)}$ ${\displaystyle \|T\|\leq 1}$

${\displaystyle \{T\in {\overline {A}};\,\|T\|\leq 1,T^{*}=T\}={\overline {\{T\in A;\,\|T\|\leq 1,T^{*}=T\}}}}$.

• If positive can be approximated with by operators from A , then T can also be approximated by positive operators from A with norm less than or equal to 1: ${\displaystyle T\in L(H)}$ ${\displaystyle \|T\|\leq 1}$

${\displaystyle \{T\in {\overline {A}};\,\|T\|\leq 1,T\geq 0\}={\overline {\{T\in A;\,\|T\|\leq 1,T\geq 0\}}}}$.

• If A is a C * -algebra with 1 and the unitary operator can be approximated by operators from A , then T can also be approximated by unitary operators from A : ${\displaystyle T\in L(H)}$

${\displaystyle \{T\in {\overline {A}};\,T^{*}=T^{-1}\}\subset {\overline {\{T\in A;\,T^{*}=T^{-1}\}}}}$,

the addition is not necessary here, because it even follows for all elements . ${\displaystyle \|T\|\leq 1}$ ${\displaystyle \|T\|=1}$ ${\displaystyle T^{*}=T^{-1}}$

Note that the above statement about self-adjoint operators does not follow trivially from the first statement, because the involution is discontinuous with respect to the strong operator topology: If the shift operator , then in the strong operator topology, but does not converge to 0. It is clear that one set above in the first three points, the conditions for each can be generalized, because the multiplication by the scalar is a homeomorphism . ${\displaystyle S\in L(\ell ^{2})}$ ${\displaystyle (S^{n})^{*}\to 0}$ ${\displaystyle ((S^{n})^{*})^{*}\,=\,S^{n}}$ ${\displaystyle \|\cdot \|\leq 1}$ ${\displaystyle \|\cdot \|\leq r}$ ${\displaystyle r>0}$ ${\displaystyle r}$

In Kaplansky's original work the sentence is:

Are and * -algebras of operators on a Hilbert space, and be strongly dense in . Then the unit sphere of is strongly dense in the unit sphere of . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {N}}}$ ${\displaystyle {\mathcal {M}}\subset {\mathcal {N}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {N}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {N}}}$

meaning

Kaplansky's density theorem is an important technical aid for many theorems from the theory of C * algebras and Von Neumann algebras; 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 density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.

(The density kit is Kaplansky's great gift to mankind. It can be used daily and twice on Sundays.)

Typical application

• Let be a separable Hilbert space and a subalgebra that is closed with respect to the involution. Then each can be approximated by a sequence from . ${\displaystyle H}$ ${\displaystyle A\subset L(H)}$ ${\displaystyle T\in {\overline {A}}}$ ${\displaystyle A}$

To prove it, consider a dense sequence in . If , according to Kaplansky’s impermeability principle above, one can find a with and for everyone . Is now so admits one with . Then applies to everyone ${\displaystyle (x_{n})_{n}}$ ${\displaystyle H}$ ${\displaystyle r=\|T\|}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle T_{n}\in A}$ ${\displaystyle \|T_{n}\|\leq r}$ ${\displaystyle \textstyle \|T_{n}x_{k}-Tx_{k}\|<{\frac {1}{n}},\,k=1,\ldots ,n}$ ${\displaystyle x\in H}$ ${\displaystyle \epsilon >0}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle \textstyle \|x_{N}-x\|<{\frac {\epsilon }{3r}}}$ ${\displaystyle \textstyle n\geq \max\{{\frac {3}{\epsilon }},N\}}$

${\displaystyle \|T_{n}x-Tx\|\leq \|T_{n}\|\|x-x_{N}\|+\|T_{n}x_{N}-Tx_{N}\|+\|T\|\|x_{N}-x\|\leq r{\frac {\epsilon }{3r}}+{\frac {1}{n}}+r{\frac {\epsilon }{3r}}\leq \epsilon }$

and therefore in the strong operator topology. ${\displaystyle T_{n}\rightarrow T}$

This proof shows very clearly how the argument depends on the fact that one can choose the approximating operators in the operator norm with restrictions, and Kaplansky's density theorem is used for this.

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras I , 1983, ISBN 0-12-393301-3 , Theorem 5.3.5 and corollaries
2. ↑ I. Kaplansky: A theroem on rings of operators , Pacific Journal of Mathematics (1951), Volume 43, pages 227-232, available online here
3. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups ISBN 0125494505 , 2.3.4
4. ^ W. Arveson : Invitation to C * -algebras , ISBN 0387901760 , corollary to Theorem 1.2.2