Hilbert C * module

Hilbert C * modules are considered in the mathematical sub-area of functional analysis. They play an important role in the structure of the KK theory , the elements of the groups occurring there are modules with a certain additional structure. Hilbert C * modules are defined in analogy to Hilbert spaces , where the inner product takes on values ​​in a C * algebra . They were introduced in 1953 by Irving Kaplansky for the case of commutative C * algebras and in 1973 by William Paschke for the general case.

definition

Let it be a C * -algebra . A pre-Hilbert module is a right B module together with a mapping so that ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle \langle \cdot ,\cdot \rangle :E\times E\rightarrow B}$

1. ${\displaystyle \langle \cdot ,\cdot \rangle }$is sesquilinear ( conjugate linearly in the first variable)
2. ${\displaystyle \langle x,yb\rangle =\langle x,y\rangle b}$ for all ${\displaystyle x,y\in E,b\in B}$
3. ${\displaystyle \langle x,y\rangle ^{*}=\langle y,x\rangle }$ for all ${\displaystyle x,y\in E}$
4. ${\displaystyle \langle x,x\rangle \geq 0}$for all , where the order defined by the positive elements is on . ${\displaystyle x\in E}$ ${\displaystyle \geq }$ ${\displaystyle B}$
5. ${\displaystyle \langle x,x\rangle =0\Leftrightarrow x=0}$for everyone . ${\displaystyle x\in E}$

The obvious analogy to the definition of a Hilbert space can be expanded further. One shows the Cauchy-Schwarz inequality

${\displaystyle \|\langle x,y\rangle \|\leq \|\langle x,x\rangle \|^{1/2}\|\langle y,y\rangle \|^{1/2}}$for everyone (without using the fifth condition) ${\displaystyle x,y\in E}$

and thus receives a semi-norm

${\displaystyle \|x\|:=\|\langle x,x\rangle \|^{1/2}}$

on which even a fifth because of the condition norm is. If the pre-Hilbert module with respect to this standard completely , so it is called a Hilbert module . The main difference to Hilbert spaces is that you cannot prove a projection theorem, that is, there are sub-Hilbert- modules that are not continuously projectable. ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B}$

Examples

• By definition, a C * -algebra is a Hilbert- module. Its sub-Hilbert B modules are the completed legal ideals . is generated as a Hilbert- module as countable, that is, there is a countable subset such that the smallest submodule that includes this set is if σ-unital . ${\displaystyle B}$ ${\displaystyle \langle x,y\rangle :=x^{*}y}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B}$
• A Hilbert space is a Hilbert module. ${\displaystyle \mathbb {C} }$
• For a C * -algebra be the space of all sequences for which converges. With the definition it becomes a Hilbert module. Obviously the separable sequence space is the square summing sequences. ${\displaystyle B}$ ${\displaystyle H_{B}}$ ${\displaystyle (b_{n})_{n}\in B^{\mathbb {N} }}$ ${\displaystyle \textstyle \sum _{n\in \mathbb {N} }b_{n}^{*}b_{n}\in B}$ ${\displaystyle \textstyle \langle (a_{n})_{n},(b_{n})_{n}\rangle :=\sum _{n\in \mathbb {N} }a_{n}^{*}b_{n}}$ ${\displaystyle H_{B}}$ ${\displaystyle B}$ ${\displaystyle H_{\mathbb {C} }=\ell ^{2}}$

Constructions

Direct sums

With the definition , direct sums of Hilbert- modules are obviously Hilbert- modules again . ${\displaystyle E_{1}\oplus \ldots \oplus E_{n}}$ ${\displaystyle B}$ ${\displaystyle \textstyle \langle (x_{i})_{i},(y_{i})_{i}\rangle :=\sum _{i=1}^{n}\langle x_{i},y_{i}\rangle }$ ${\displaystyle B}$

Algebras of operators

For two Hilbert- modules and let be the set of all operators for which there is an operator such that for all and . One shows that such operators are -linear and form a closed subspace of continuous, linear operators . is a C * -algebra with the operator norm and the involution . In the special case is isomorphic to the multiplier algebra of . ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle L_{B}(E,F)}$ ${\displaystyle T:E\rightarrow F}$ ${\displaystyle T^{*}:F\rightarrow E}$ ${\displaystyle \langle Tx,y\rangle _{F}=\langle x,T^{*}y\rangle _{E}}$ ${\displaystyle x\in E}$ ${\displaystyle y\in F}$ ${\displaystyle B}$ ${\displaystyle E\rightarrow F}$ ${\displaystyle L_{B}(E):=L_{B}(E,E)}$ ${\displaystyle T\mapsto T^{*}}$ ${\displaystyle E=B}$ ${\displaystyle L_{B}(B)=M(B)}$ ${\displaystyle B}$

Certain operators from can be defined as follows in analogy to the one-dimensional operators on Hilbert spaces. Are and so be . It's easy to confirm the formula and thus . The closed subspace generated by these operators is designated with and its elements are called the compact operators of after , even if they are generally not compact operators in the sense of Banach space theory . It is easy to confirm for one from which it follows, and quite similarly too . This is a two-sided ideal . Apparently the ideal of compact operators is on . ${\displaystyle L_{B}(E,F)}$ ${\displaystyle x\in F}$ ${\displaystyle y\in E}$ ${\displaystyle \theta _{x,y}:E\rightarrow F,\,\theta _{x,y}(z):=x\langle y,z\rangle }$ ${\displaystyle \theta _{x,y}^{*}=\theta _{y,x}}$ ${\displaystyle \theta _{x,y}\in L_{B}(E,F)}$ ${\displaystyle K_{B}(E,F)}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle T\theta _{x,y}=\theta _{Tx,y}}$ ${\displaystyle T\in L_{B}(F)}$ ${\displaystyle L_{B}(F)K_{B}(E,F)\subset K_{B}(E,F)}$ ${\displaystyle K_{B}(E,F)L_{B}(E)\subset K_{B}(E,F)}$ ${\displaystyle K_{B}(E)\subset L_{B}(E)}$ ${\displaystyle K_{\mathbb {C} }(H_{\mathbb {C} })}$ ${\displaystyle \ell ^{2}}$

These constructions are related as follows: For every C * -algebra and every Hilbert- module is isomorphic to the multiplier algebra . In particular, is a * -isomorphism that on maps. ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle L_{B}(E)}$ ${\displaystyle M(K_{B}(E))}$ ${\displaystyle L_{B}(H_{B})\cong M(K_{\mathbb {C} }\otimes B)}$ ${\displaystyle K_{B}(H_{B})}$ ${\displaystyle K_{\mathbb {C} }\otimes B}$

Unitary equivalence

Two Hilbert modules and are called unitary equivalent , in signs , if there is a bijective, linear mapping with for all . ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle E\cong F}$ ${\displaystyle U\in L_{B}(E,F)}$ ${\displaystyle \langle U(x),U(y)\rangle _{F}=\langle x,y\rangle _{E}}$ ${\displaystyle x,y\in E}$

Inner tensor products

Let there be a Hilbert- module, a Hilbert- module and a * -homomorphism. The formula turns into a left- module and one can therefore form the algebraic tensor product , which through the definition becomes a right- module. Through the formula ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle \varphi :B\rightarrow L_{B}(F)}$ ${\displaystyle by:=\varphi (b)(y)}$ ${\displaystyle F}$ ${\displaystyle B}$ ${\displaystyle E\otimes _{B}F}$ ${\displaystyle (x\otimes _{B}y)a:=x\otimes _{B}ya}$ ${\displaystyle A}$

${\displaystyle \langle x_{1}\otimes _{B}y_{1},x_{2}\otimes _{B}y_{2}\rangle :=\langle y_{1},\varphi (\langle x_{1},x_{2}\rangle _{E})(y_{2})\rangle _{F}}$

we obtain a sesquilinear form by means of linear expansion , which fulfills all rules from the definition of the pre-Hilbert- module up to possibly point 5, that is, vectors of length 0 can exist. By the space divided out of the vectors of length, that is the factor space to passes, and then completed to obtain a Hilbert- module, the one having called and the inner tensor product of the Hilbert C * -modules calls. ${\displaystyle E\otimes _{B}F}$ ${\displaystyle A}$ ${\displaystyle N:=\{z\in E\otimes _{B}F;\langle z,z\rangle =0\}}$ ${\displaystyle E\otimes _{B}F/N}$ ${\displaystyle A}$ ${\displaystyle E\otimes _{\varphi }F}$

External tensor product

Let there be a Hilbert module and a Hilbert module. Then the algebraic tensor product is by means of the definition ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle E\otimes _{\mathbb {C} }F}$

${\displaystyle (x\otimes y)\cdot (b\otimes a):=xb\otimes ya,\quad x\in E,y\in F,a\in A,b\in B}$

a right module and by means of linear expansion one gets from ${\displaystyle B\otimes A}$

${\displaystyle \langle x_{1}\otimes _{\mathbb {C} }y_{1},x_{2}\otimes _{C}y_{2}\rangle :=\langle x_{1},x_{2}\rangle \otimes \langle y_{1},y_{2}\rangle }$

a sesquilinear form. If the spatial tensor product of the C * -algebras, then one constructs a Hilbert module by dividing out vectors of length 0 and extending it to the completions , which is called the outer tensor product of the Hilbert C * modules. ${\displaystyle B{\hat {\otimes }}A}$ ${\displaystyle B{\hat {\otimes }}A}$ ${\displaystyle E{\hat {\otimes }}F}$

Pushout

If there is a Hilbert- module and a surjective * -homomorphism, then define . If the quotient mapping is , the definitions ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle \varphi :B\rightarrow A}$ ${\displaystyle N_{\varphi }:=\{x\in E;\varphi (\langle x,x\rangle )=0\}}$ ${\displaystyle q:E\rightarrow E_{\varphi }:=E/N_{\varphi }}$ ${\displaystyle E_{\varphi }}$

${\displaystyle q(x)a:=q(xb)}$, with ${\displaystyle b\in B}$ ${\displaystyle \varphi (b)=a}$

${\displaystyle \langle q(x),q(y)\rangle :=\varphi (\langle x,y\rangle )}$,

whose well-definedness is to be shown, a Hilbert module, which is called the pushout of regarding . One can show that by considering as a subalgebra of . ${\displaystyle A}$ ${\displaystyle E}$ ${\displaystyle \varphi }$ ${\displaystyle E_{\varphi }\cong E\otimes _{\varphi }A}$ ${\displaystyle A}$ ${\displaystyle M(A)\cong L_{B}(A)}$

Graduated Hilbert C * modules

Hilbert C * algebras with an additional structure, a so-called graduation, more precisely a graduation, are used especially for the KK theory . Let it be a graduated C * algebra with graduation automorphism , that is, it is ${\displaystyle \mathbb {Z} /2}$ ${\displaystyle B}$ ${\displaystyle \alpha :B\rightarrow B}$

${\displaystyle \alpha \circ \alpha =\mathrm {id} _{B}}$

${\displaystyle B_{0}:=\{b\in B;\alpha (b)=b\}}$

${\displaystyle B_{1}:=\{b\in B;\alpha (b)=-b\}}$

Then the direct sum decomposition is for -graduation. A Hilbert graduate module is a Hilbert module together with a linear bijection so that ${\displaystyle B=B_{0}\oplus B_{1}}$ ${\displaystyle \mathbb {Z} /2}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle S_{E}:E\rightarrow E}$

${\displaystyle S_{E}\circ S_{E}=\mathrm {id} _{E}}$

${\displaystyle S_{E}(xb)=S_{E}(x)\alpha (b)}$ for all ${\displaystyle x\in E,b\in B}$

${\displaystyle \langle S_{E}(x),S_{E}(y)\rangle =\alpha (\langle x,y\rangle )}$ for all ${\displaystyle x,y\in E}$

Again a direct sum breakdown is obtained , where ${\displaystyle E=E_{0}\oplus E_{1}}$

${\displaystyle E_{0}:=\{x\in E;S_{E}(x)=x\}}$

${\displaystyle E_{1}:=\{x\in E;S_{E}(x)=-x\}}$

and it follows

${\displaystyle E_{i}B_{j}\subset E_{i+j}}$and for everyone . ${\displaystyle \langle E_{i},E_{j}\rangle \subset B_{i+j}}$ ${\displaystyle i,j\in \mathbb {Z} /2=\{0,1\}}$

By the automorphism will also get and graduation. ${\displaystyle T\mapsto S_{E}TS_{E}^{-1}}$ ${\displaystyle L_{B}(E)}$ ${\displaystyle K_{B}(E)}$

Graduated Hilbert C * modules are called unitary equivalent if they are unitarily equivalent as Hilbert C * modules with a unitary operator that receives the graduation.

This generalizes the terms introduced above without graduation, because every C * -algebra can be trivially graded using and also every Hilbert- module using . ${\displaystyle \alpha =\mathrm {id} _{A}}$ ${\displaystyle B}$ ${\displaystyle S_{E}=\mathrm {id} _{E}}$

To also graduate, you have two options, namely and . We therefore define ${\displaystyle H_{B}}$ ${\displaystyle S(b_{1},b_{2},b_{3},\ldots ):=(\alpha (b_{1}),\alpha (b_{2}),\alpha (b_{3}),\ldots )}$ ${\displaystyle -S(b_{1},b_{2},b_{3},\ldots )=(-\alpha (b_{1}),-\alpha (b_{2}),-\alpha (b_{3}),\ldots )}$

${\displaystyle {\hat {H}}_{B}:=H_{B}\oplus H_{B}}$with graduation . ${\displaystyle S\oplus -S}$

The above constructions can also be defined for graduated Hilbert C * modules, whereby the graduated tensor product must be taken and all morphisms that occur must be compatible with the graduations. The details related to this are very technical and are ignored here.

Kasparov's stabilization theorem

For the KK theory, Kasparov's so-called stabilization theorem proves to be important. This theorem applies to graduate and non-graduate Hilbert C * modules, it states that all countably generated Hilbert C * modules already contain as direct summands, and analogously to graduated modules. There is even a little more: ${\displaystyle H_{B}}$

• If a countably generated Hilbert module is over a C * algebra , then is . ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle E\oplus H_{B}\cong H_{B}}$
• If a countably generated, graduated Hilbert- module over a graduated C * -algebra , then . ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle E\oplus {\hat {H}}_{B}\cong {\hat {H}}_{B}}$

Individual evidence

1. ^ I. Kaplansky: Modules over operator algebras , Amer. J. of Math. (1953) Volume 75, Pages 838-858
2. ↑ WL Paschke: Inner product modules over B * -algebras , Transactions Amer. Math. Soc. (1973) Vol. 182, pp. 443-468
3. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , definition 13.1.1
4. ^ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , Lemma 1.1.7
5. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 13.4.1
6. ^ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , Lemma 1.2.7
7. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , section 1.2.3
8. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , section 1.2.4
9. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , Lemma 1.2.5
10. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , definition 1.2.10
11. ^ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , theorem 1.1.24 and theorem 1.2.12