Abelian Von Neumann Algebra

Abelian von Neumann algebras are in the mathematical branch of functional analysis looked Von Neumann algebras whose multiplication is commutative is.

Examples

• The algebra of the diagonal matrices on the finite-dimensional Hilbert space is an Abelian Von Neumann algebra, which is apparently isomorphic to the algebra with component-wise multiplication . The sub -algebra of the constant multiples of the identity matrix is also an Abelian Von Neumann algebra. ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$
• The sequence space with component-wise multiplication is the infinite-dimensional generalization of the first example. This Abelian Von Neumann algebra operates on the Hilbert space . ${\displaystyle \ell ^{\infty }}$ ${\displaystyle \ell ^{2}}$
• If the Lebesgue measure is on the unit interval [0.1], then each function defines a continuous linear operator through the formula . Algebra is an Abelian Von Neumann algebra, which is simply referred to as. ${\displaystyle \lambda }$ ${\displaystyle f\in L^{\infty }([0,1],\lambda )}$ ${\displaystyle M_{f}(\xi ):=f\cdot \xi }$ ${\displaystyle M_{f}:L^{2}([0,1],\lambda )\rightarrow L^{2}([0,1],\lambda )}$ ${\displaystyle \{M_{f};f\in L^{\infty }([0,1],\lambda )\}}$ ${\displaystyle L^{\infty }[0,1]}$

Abelian Von Neumann algebras as L ^∞ algebras

The above example is the most general case except for isomorphism. The following applies: ${\displaystyle L^{\infty }}$

If an Abelian Von Neumann algebra is over a Hilbert space , then there is a locally compact Hausdorff space and a positive measure on with support , so that is isomorphic to . Isomorphism means isometric * isomorphism. If the Hilbert space is separable , you can choose a compact , metric space . ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle \mu }$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle L^{\infty }(X,\mu )}$ ${\displaystyle H}$ ${\displaystyle X}$

Conversely, if a measure space is locally compact , then every function defines a continuous linear operator through the formula . Algebra is an Abelian Von Neumann algebra that is isomorphic to . is at most under all Abelian Von Neumann algebras . ${\displaystyle (X,\mu )}$ ${\displaystyle X}$ ${\displaystyle f\in L^{\infty }(X,\mu )}$ ${\displaystyle M_{f}(\xi ):=f\cdot \xi }$ ${\displaystyle M_{f}:L^{2}(X,\mu )\rightarrow L^{2}(X,\mu )}$ ${\displaystyle A=\{M_{f};f\in L^{\infty }(X,\mu )\}}$ ${\displaystyle L^{\infty }(X,\mu )}$ ${\displaystyle A}$ ${\displaystyle L^{2}(X,\mu )}$

Abelian Von Neumann algebras on separable Hilbert spaces

The isomorphism classes of the Abelian Von Neumann algebras over a separable Hilbert space can be completely surveyed; if one restricts oneself to maximal Von Neumann algebras, one can even replace isomorphism with unitary equivalence.

Let the Von Neumann algebras from the above examples be the isomorphic and the isomorphic. Every maximal Abelian Von Neumann algebra over a separable Hilbert space is unitarily equivalent to exactly one of the algebras ${\displaystyle A_{n}\subset L(\mathbb {C} ^{n})}$ ${\displaystyle C^{n}}$ ${\displaystyle A_{\infty }}$ ${\displaystyle \ell ^{\infty }}$

• ${\displaystyle A_{n},\quad 1\leq n\leq \infty }$
• ${\displaystyle L^{\infty }[0,1]}$
• ${\displaystyle A_{n}\oplus L^{\infty }[0,1],\quad 1\leq n\leq \infty }$

Two Von Neumann algebras are called over and over unitarily equivalent if there is a unitary operator such that is an isomorphism . ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle B}$ ${\displaystyle K}$ ${\displaystyle u:H\rightarrow K}$ ${\displaystyle a\mapsto uau^{*}}$ ${\displaystyle A\rightarrow B}$

Abelian Von Neumann algebras as C * algebras

Abelian Von Neumann algebras are in particular commutative C * algebras and as such, according to Gelfand-Neumark's theorem, is isomorphic to an algebra of continuous functions on a compact Hausdorff space. is an extremely disjointed space . The converse does not apply, that is, there are extremely disconnected, compact Hausdorff spaces , so that the algebra is not isomorphic to a Von Neumann algebra. ${\displaystyle C(X)}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle C(X)}$

Spectral theorem

If a self-adjoint , bounded linear operator is on the Hilbert space , then the Von Neumann algebra generated by is Abelian and contains all spectral projections of . Abelian Von Neumann algebras are therefore a natural framework for developing spectral theory , which can also be extended to unrestricted self-adjoint operators. This program is consistently executed in. ${\displaystyle a\in L(H)}$ ${\displaystyle H}$ ${\displaystyle a}$ ${\displaystyle a}$

See also

• Abelian Von Neumann algebras are Type I Von Neumann algebras .

Individual evidence

1. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , I.7.3: Structure of abelian von Neumann algebras
2. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 9.4.1
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II Academic Press (1983), ISBN 0-12-393301-3 , 5.7.21.
4. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras I , Academic Press (1983), ISBN 0-1239-3301-3 , chapters 5.2 and 5.6