Hilbert algebra

Hilbert algebras are examined in the mathematical sub-area of functional analysis. They are algebras with an additional pre-Hilbert space structure, which explains the name Hilbert algebra . On the completion can be Von Neumann algebras construct, ultimately leading to a characterization of the semi-finite von Neumann algebras leads. A generalization of this construction, which applies to every Von-Neumann algebra, leads to the concept of the generalized Hilbert algebra and is the starting point of the Tomita-Takesaki theory .

Definitions

A Hilbert algebra is an associative algebra over the field of complex numbers together with an involution and a scalar product , which makes it a pre-Hilbert space, so that the following conditions are met: ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle x\mapsto x^{*}}$ ${\displaystyle \langle \cdot |\cdot \rangle }$ ${\displaystyle A}$

• ${\displaystyle \langle x|y\rangle =\langle y^{*}|x^{*}\rangle }$ for all ${\displaystyle x,y\in A}$
• ${\displaystyle \langle xy|z\rangle =\langle y|x^{*}z\rangle }$ for all ${\displaystyle x,y,z\in A}$
• For each , the mapping is continuous in the norm topology defined by the scalar product . ${\displaystyle x\in U}$ ${\displaystyle A\rightarrow A,\,y\mapsto xy}$
• Out for everyone follows . ${\displaystyle \langle xy|z\rangle =0}$ ${\displaystyle x,y\in A}$ ${\displaystyle z=0}$

In the following is the Hilbert space , which results as the completion of . From the first condition it follows that the involution continues to a continuous, conjugate linear mapping for which ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle J:H\rightarrow H}$

${\displaystyle J^{2}=\mathrm {id} _{H}}$and for everyone ${\displaystyle \langle Jx|Jy\rangle =\langle y|x\rangle }$ ${\displaystyle x,y\in H}$

holds, one calls the canonically defined involution on . ${\displaystyle J}$ ${\displaystyle A}$ ${\displaystyle H}$

The maps and continue for each to form continuous linear operators and such that: ${\displaystyle y\mapsto xy}$ ${\displaystyle y\mapsto yx}$ ${\displaystyle x\in A}$ ${\displaystyle U_{x}:H\rightarrow H}$ ${\displaystyle V_{x}:H\rightarrow H}$

${\displaystyle x\mapsto U_{x}}$is an involutive homomorphism

${\displaystyle x\mapsto V_{x}}$is an involutive antiomomorphism

${\displaystyle U_{x}V_{y}=V_{y}U_{x}}$ for all ${\displaystyle x,y\in A}$

${\displaystyle JU_{x}J=V_{x^{*}}}$and for everyone ${\displaystyle JV_{x}J=U_{x^{*}}}$ ${\displaystyle x\in A}$

The completed cases concerning. The weak operator topologies of and be with and called and called the left-associated or right-associated von Neumann algebra . To prove that these are actually Von Neumann algebras, in particular that these algebras contain the identity , one needs the fourth condition of the above definition. ${\displaystyle \{U_{x}|x\in A\}}$ ${\displaystyle \{V_{x}|x\in A\}}$ ${\displaystyle U=U(A)}$ ${\displaystyle V=V(A)}$ ${\displaystyle A}$ ${\displaystyle \mathrm {id} _{H}}$

A von Neumann algebra is called a standard von Neumann algebra if it is of the form . ${\displaystyle U(A)=}$

example

The H * -algebra algebra of the Hilbert-Schmidt operators on a Hilbert space is a Hilbert algebra. Denotes the conjugated Hilbert space , then is isomorphic to the Hilbert space tensor product . For is the one-dimensional operator if the scalar product is linear in the first component and conjugate linear in the second. Then ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle {\overline {H}}}$ ${\displaystyle A}$ ${\displaystyle {\overline {H}}\otimes H}$ ${\displaystyle \xi ,\eta \in H}$ ${\displaystyle \xi \otimes \eta }$ ${\displaystyle H\rightarrow H,(\xi \otimes \eta )\zeta :=\langle \zeta ,\eta \rangle \xi }$

${\displaystyle (\xi _{1}\otimes \eta _{1})(\xi _{2}\otimes \eta _{2})\zeta =(\xi _{1}\otimes \eta _{1})(\langle \zeta ,\xi _{2}\rangle \eta _{2})=\langle \zeta ,\xi _{2}\rangle \langle \eta _{2},\xi _{1}\rangle \eta _{1}=\langle \eta _{2},\xi _{1}\rangle (\xi _{2}\otimes \eta _{1})\zeta }$

and therefore

${\displaystyle (\xi _{1}\otimes \eta _{1})(\xi _{2}\otimes \eta _{2})=\langle \eta _{2},\xi _{1}\rangle (\xi _{2}\otimes \eta _{1})}$,

so

${\displaystyle U_{\xi _{1}\otimes \eta _{1}}(\xi _{2}\otimes \eta _{2})=\langle \eta _{2},\xi _{1}\rangle (\xi _{2}\otimes \eta _{1})=\xi _{2}\otimes (\langle \xi _{1},\eta _{2}\rangle \eta _{1})}$

${\displaystyle =\xi _{2}\otimes ((\xi _{1}\otimes \eta _{1})\eta _{2})=(\mathrm {id} _{H}{\overline {\otimes }}(\xi _{1}\otimes \eta _{1}))(\xi _{2}\otimes \eta _{2})}$,

where the tensor product denoted by the slash is the tensor product for Von Neumann algebras . One reads from it

${\displaystyle U(A)=\mathbb {C} \cdot 1_{\overline {H}}\,{\overline {\otimes }}\,L(H)}$

because the linear combinations from the one-dimensional operators lie close together in the respective algebras. ${\displaystyle \xi _{1}\otimes \eta _{1}}$

Semi-infinite Von Neumann algebras

The Von Neumann algebras, which appear as the left-associated Von Neumann algebras of Hilbert algebras, are precisely the semi-finite Von Neumann algebras . If a Hilbert algebra is given, then there is a semi-finite, normal, faithful trace that makes a semi-finite Von Neumann algebra. Conversely, if a semi-finite Von Neumann algebra with such a trace is a Hilbert algebra with the scalar product defined by , whose left-associated Von Neumann algebra is too isomorphic. ${\displaystyle A}$ ${\displaystyle \phi (U_{a}^{*}U_{a}):=\langle a|a\rangle }$ ${\displaystyle U(A)}$ ${\displaystyle U}$ ${\displaystyle A:=\{a\in U|\phi (a^{*}a)<\infty ,\phi (aa^{*})<\infty \}}$ ${\displaystyle \langle a|b\rangle :=\phi (b^{*}a)}$ ${\displaystyle U}$

Generalized Hilbert Algebra

A generalized Hilbert algebra is an associative algebra over the field of complex numbers together with an involution and a scalar product , which makes it a pre-Hilbert space, so that the following conditions are met: ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle x\mapsto x^{*}}$ ${\displaystyle \langle \cdot |\cdot \rangle }$ ${\displaystyle A}$

• The mapping is a closable , conjugate-linear operator in the completion . ${\displaystyle x\mapsto x^{*}}$ ${\displaystyle H}$
• ${\displaystyle \langle xy|z\rangle =\langle y|x^{*}z\rangle }$ for all ${\displaystyle x,y,z\in A}$
• For each , the mapping is continuous in the norm topology defined by the scalar product . ${\displaystyle x\in U}$ ${\displaystyle A\rightarrow A,\,y\mapsto xy}$
• Out for everyone follows . ${\displaystyle \langle xy|z\rangle =0}$ ${\displaystyle x,y\in A}$ ${\displaystyle z=0}$

Generalized Hilbert algebras are also called left Hilbert algebras.

Hilbert algebras are generalized Hilbert algebras. For this purpose it is necessary to show that the picture , is lockable, ie from and already follows. For each one follows a Hilbert algebra using the first defining property ${\displaystyle A\rightarrow A}$ ${\displaystyle x\mapsto x^{*}}$ ${\displaystyle x_{n}\rightarrow 0}$ ${\displaystyle x_{n}^{*}\rightarrow z}$ ${\displaystyle z=0}$ ${\displaystyle y\in A}$

${\displaystyle \langle z,y\rangle =\lim _{n\to \infty }\langle x_{n}^{*},y\rangle =\lim _{n\to \infty }\langle y^{*},x_{n}\rangle =\langle y^{*},0\rangle =0}$

and therefore , because was arbitrary, that is, is perpendicular to a dense subset of the completion. ${\displaystyle z=0}$ ${\displaystyle y\in A}$ ${\displaystyle z}$

As above, the displays put on operators to continue their weakly closed hull forms the left-associated von Neumann algebra . The end of the figure is called the Sharp operator, which is why many authors write the involution with the Sharp character #. Its polar decomposition leads to the formulas that are described in the article on Tomita-Takesaki theory . ${\displaystyle y\mapsto xy}$ ${\displaystyle U_{x}}$ ${\displaystyle H}$ ${\displaystyle A}$ ${\displaystyle S}$ ${\displaystyle x\mapsto x^{*}}$ ${\displaystyle \textstyle S=J\Delta }$

Any Von Neumann algebra has a faithful, normal, semi-infinite weight . Then there is a generalized Hilbert algebra whose left-associated Von Neumann algebra is too isomorphic. ${\displaystyle U}$ ${\displaystyle \omega }$ ${\displaystyle A:=\{a\in U|\,\omega (a^{*}a)<\infty ,\omega (aa^{*})<\infty \}}$ ${\displaystyle U}$

Individual evidence

1. ^ Jacques Dixmier : Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter I.5.1: Definition of Hilbert algebras.
2. ^ Jacques Dixmier: Von Neumann algebras. North-Holland, Amsterdam 1981, ISBN 0-444-86308-7 , Chapter I, §5, Paragraph 5, Definition 7.
3. ^ Jacques Dixmier: Von Neumann algebras. North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Volume 27), ISBN 0-444-86308-7 , Part I, Chap. 6, para. 5: Normal traces on L (H).
4. ^ Jacques Dixmier: Von Neumann algebras. North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Volume 27), ISBN 0-444-86308-7 , Part I, Chap. 6, para. 2, Theorem 1 and Theorem 2.
5. ^ M. Takesaki : Tomita's theory of modular Hilbert-algebras and its applications. Lecture Notes in Mathematics, Volume 128, Springer-Verlag 1970, §2.
6. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II. Academic Press, 1983, ISBN 0-12-393302-1 , definition 9.2.41.
7. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-12-393302-1 , sentence 9.2.40.