Type II Von Neumann algebra

Type II Von Neumann algebras are special algebras considered in the mathematical theory of Von Neumann algebras. It is the second of three types of type classification for Von Neumann algebras . These can be further subdivided into finite , so-called type II ₁ algebras and infinite, so-called type II _∞ algebras, whereby the latter can be constructed from the former in the σ-finite case.

Definitions

A projection in a Von Neumann algebra is a self-adjoint idempotent element , that is, it holds . Such a projection is called Abelian if it is an Abelian Von Neumann algebra ; it is called finite if it follows from and always . A Von Neumann algebra is called of type II if it does not contain any Abelian projections other than 0, but each projection different from 0 from the center of a finite projection different from 0. It is called of type II ₁ if the unity element as a projection is finite, it is called of type II _∞ if none of the projections from the center other than 0 is finite. ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle e=e^{*}=e^{2}}$ ${\displaystyle eAe}$ ${\displaystyle e=v^{*}v}$ ${\displaystyle vv^{*}\leq e}$ ${\displaystyle vv^{*}=e}$ ${\displaystyle A}$

Examples

• It's a discreet group . Each element operates as a left operator and as a right operator on the Hilbert space in which and is defined. Let and be the Von Neumann algebras generated by or . Then, and finally and mutual Kommutanten . ${\displaystyle G}$ ${\displaystyle g\in G}$ ${\displaystyle l_{g}}$ ${\displaystyle r_{g}}$ ${\displaystyle \ell ^{2}(G)}$ ${\displaystyle (l_{g}(x))(h):=x(g^{-1}h)}$ ${\displaystyle (r_{g}(x))(h):=x(hg^{-1})}$ ${\displaystyle L_{G}}$ ${\displaystyle R_{G}}$ ${\displaystyle \{l_{g};\,g\in G\}}$ ${\displaystyle \{r_{g};\,g\in G\}}$ ${\displaystyle L_{G}}$ ${\displaystyle R_{G}}$

If an ICC group , i.e. only the neutral element is in a finite conjugation class, it is a type II ₁ algebras, even so-called factors, that is, the center of the algebras consists only of the multiples of the unit element . ${\displaystyle G}$

• If a type II ₁ algebra, then the tensor product is a type II _∞ algebra. ${\displaystyle A}$ ${\displaystyle L(\ell ^{2}){\overline {\otimes }}A}$
• In the article on W * -dynamic systems , a construction is described that leads to Type II Von Neumann algebras.

structure

For every Type II Von Neumann algebra there is a projection from the center of such that ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle A}$

• ${\displaystyle A=pA\oplus (1-p)A=pAp\oplus (1-p)A(1-p)}$
• ${\displaystyle pA}$is a Type II ₁ algebra.
• ${\displaystyle (1-p)A}$is a Type II _∞ algebra.

For every σ-finite type II _∞ -algebra there is a type II ₁ -algebra with . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\cong L(\ell ^{2}){\overline {\otimes }}B}$

Tensor products of type II algebras are again type II algebras. If the algebras are of type II ₁ or type II _∞ , then the tensor product is only of type II ₁ if both factors are, otherwise of type II _∞ .

See also

• Type I Von Neumann algebra
• Type III Von Neumann Algebra

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , definitions 6.3.1 and 6.5.1
2. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , 6.7.2 - 6.7.5
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 6.5.2
4. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 6.7.10
5. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Table 11.1