Tensor product for Von Neumann algebras

In the mathematical theory of Von Neumann algebras one can define a tensor product with which one obtains a third from two Von Neumann algebras. Since Von Neumann algebras operate on Hilbert spaces and must have certain closure properties there, the formation of the algebraic tensor product is not sufficient; the construction described in this article is therefore used.

construction

Let and two Von Neumann algebras on the Hilbert spaces and . Two operators and define a continuous linear operator on the Hilbert space tensor product , and it even holds (see article Hilbert space tensor product ). The Von Neumann algebra generated by all operators of the form with and in , that is, the closure of the set of all finite sums of such operators with respect to the weak operator topology , is called the tensor product from and and is denoted by, where the dash above the tensor product symbol leads to the To remember the final surgery. ${\displaystyle {\mathcal {A}}\subset L(H)}$ ${\displaystyle {\mathcal {B}}\subset L(K)}$ ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle B\in {\mathcal {B}}}$ ${\displaystyle A\otimes B}$ ${\displaystyle H\otimes K}$ ${\displaystyle \|A\otimes B\|=\|A\|\cdot \|B\|}$ ${\displaystyle A\otimes B}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle B\in {\mathcal {B}}}$ ${\displaystyle L(H\otimes K)}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle {\mathcal {A}}{\overline {\otimes }}{\mathcal {B}}}$

The commutant theorem

If and are two Von Neumann algebras, and as well as and from the commutants or , then it is clear that and in interchange, because . It follows immediately . The commutant law states that equality even applies here: ${\displaystyle {\mathcal {A}}\subset L(H)}$ ${\displaystyle {\mathcal {B}}\subset L(K)}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle B\in {\mathcal {B}}}$ ${\displaystyle A^{'}}$ ${\displaystyle B^{'}}$ ${\displaystyle {\mathcal {A}}^{'}\subset L(H)}$ ${\displaystyle {\mathcal {B}}^{'}\subset L(K)}$ ${\displaystyle A\otimes B}$ ${\displaystyle A^{'}\otimes B^{'}}$ ${\displaystyle L(H\otimes K)}$ ${\displaystyle (A\otimes B)(A^{'}\otimes B^{'})=AA^{'}\otimes BB^{'}=A^{'}A\otimes B^{'}B=(A^{'}\otimes B^{'})(A\otimes B)}$ ${\displaystyle {\mathcal {A}}^{'}{\overline {\otimes }}{\mathcal {B}}^{'}\subset ({\mathcal {A}}{\overline {\otimes }}{\mathcal {B}})^{'}}$

• If and are two Von Neumann algebras, then . ${\displaystyle {\mathcal {A}}\subset L(H)}$ ${\displaystyle {\mathcal {B}}\subset L(K)}$ ${\displaystyle {\mathcal {A}}^{'}{\overline {\otimes }}{\mathcal {B}}^{'}=({\mathcal {A}}{\overline {\otimes }}{\mathcal {B}})^{'}}$

A simple consequence is what can easily be proven without the commutant theorem. ${\displaystyle L(H){\overline {\otimes }}L(K)=L(H\otimes K)}$

The commutant theorem can also be used to show that the center of a tensor product of Von Neumann algebras is equal to the tensor product of the centers. In particular, the tensor product of factors is again a factor.

Type of tensor product

If the Von-Neumann algebras and a pure type , then also their tensor product, and the type can be found in the following table: ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$

${\displaystyle {\overline {\otimes }}}$	${\displaystyle I_{n},\,n}$ at last	${\displaystyle I_{n},\,n}$ infinite	${\displaystyle \quad \,II_{1}\quad }$	${\displaystyle \quad II_{\infty }\quad }$	${\displaystyle \quad III\quad }$
${\displaystyle I_{m},\,m}$ at last	${\displaystyle I_{mn}\,}$	${\displaystyle I_{mn}\,}$	${\displaystyle II_{1}\,}$	${\displaystyle II_{\infty }}$	${\displaystyle III\,}$
${\displaystyle I_{m},\,m}$ infinite	${\displaystyle I_{mn}\,}$	${\displaystyle I_{mn}\,}$	${\displaystyle II_{\infty }}$	${\displaystyle II_{\infty }}$	${\displaystyle III\,}$
${\displaystyle II_{1}}$	${\displaystyle \quad \,II_{1}\quad }$	${\displaystyle II_{\infty }}$	${\displaystyle II_{1}\,}$	${\displaystyle II_{\infty }}$	${\displaystyle III\,}$
${\displaystyle II_{\infty }}$	${\displaystyle \quad II_{\infty }\quad }$	${\displaystyle II_{\infty }}$	${\displaystyle II_{\infty }}$	${\displaystyle II_{\infty }}$	${\displaystyle III\,}$
${\displaystyle III\,}$	${\displaystyle III\,}$	${\displaystyle III\,}$	${\displaystyle III\,}$	${\displaystyle III\,}$	${\displaystyle III\,}$

In general, a Von Neumann algebra does not have a pure type, but, according to the theorem of the type decomposition, is a finite direct sum of Von Neumann algebras of the types or . The above table can thus be used to determine the type of the components of the tensor product . ${\displaystyle I_{n},II_{1},II_{\infty }}$ ${\displaystyle III\,}$ ${\displaystyle {\mathcal {A}}{\overline {\otimes }}{\mathcal {B}}}$

See also

A very similar construction leads to the so-called spatial tensor product in the theory of C * -algebras .

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , § 11.2: Tensor products of von Neumann algebras
2. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , I.2.4: Tensor products of von Neumann algebras
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Theorem 11.2.16
4. ^ Jacques Dixmier: Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , I.6.9: Tensor products of von Neumann algebras
5. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Table 11.1