Spatial tensor product

The spatial tensor product considered in the mathematical branch of functional analysis offers the possibility of constructing new ones from C * -algebras . In general, there are several ways the algebraic tensor product of two C * -algebras to a C * algebra complete ; the C * -norm treated here on the tensor product turns out to be minimal among these possibilities, which is why one speaks of the minimal tensor product . The construction presented here goes back to M. Takesaki .

Definitions

Let and two C * -algebras. A C * -norm on the algebraic tensor product is a norm such that ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\odot B}$ ${\displaystyle \alpha }$

• ${\displaystyle (A\odot B,\alpha )}$is a normalized algebra
• ${\displaystyle \alpha (s*s)=\alpha (s)^{2}}$ for all ${\displaystyle s\in A\odot B}$

If there is such a C * -norm, then the completion marked with is a C * -algebra. If there is a C * -norm that can be defined for every pair of C * -algebras and , then one speaks of a -tensor product. ${\displaystyle \alpha }$ ${\displaystyle A\otimes _{\alpha }B}$ ${\displaystyle \alpha }$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \alpha }$

One can show that C * norms automatically have the cross norm property , that is, it applies to all . ${\displaystyle \alpha (a\otimes b)=\|a\|\cdot \|b\|}$ ${\displaystyle a\in A,b\in B}$

This article by means of Hilbert spaces on which the C * -algebras operate with defined designated C * norms, which because of Hilbert spaces used in spatial (German space should) remember. ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

construction

Let and two C * -algebras. According to Gelfand-Neumark's theorem, there are Hilbert spaces and and isometric * - homomorphisms and , that is, we can assume that the C * -algebras are subalgebras of the full operator algebra over suitable Hilbert spaces. One can take the universal representations , for example . The Hilbert space tensor product is now formed and an element of the algebraic tensor product is considered as an operator , which by ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle H}$ ${\displaystyle K}$ ${\displaystyle A\rightarrow L(H)}$ ${\displaystyle B\rightarrow L(K)}$ ${\displaystyle H\otimes K}$ ${\displaystyle \sum _{i=1}^{n}a_{i}\otimes b_{i}}$ ${\displaystyle A\odot B}$ ${\displaystyle H\otimes K}$

${\displaystyle (\sum _{i=1}^{n}a_{i}\otimes b_{i})(x\otimes y):=\sum _{i=1}^{n}a_{i}x\otimes b_{i}y}$

is defined, whereby well-definedness is to be shown. Then it is clear that the restriction of the operator norm of on a C norm is *. ${\displaystyle \sigma }$ ${\displaystyle L(H\otimes K)}$ ${\displaystyle A\odot B}$

Independence from the Hilbert rooms

The above construction initially depends on the choice of Hilbert spaces. Here a formula is set up for the spatial norm that is independent of the Hilbert spaces. If and are states on or , then there is exactly one state labeled open with for all and , the so-called product state from and . For an element of the algebraic tensor product we now have ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f\otimes g}$ ${\displaystyle A\otimes _{\sigma }B}$ ${\displaystyle (f\otimes g)(a\otimes b)=f(a)g(b)}$ ${\displaystyle a\in A}$ ${\displaystyle b\in B}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle c=\sum _{i=1}^{n}a_{i}\otimes b_{i}}$ ${\displaystyle A\odot B}$

${\displaystyle \sigma (c)^{2}=\sup {\frac {(f\otimes g)(s^{*}c^{*}cs)}{(f\otimes g)(s^{*}s)}}}$

whereby the supremum is formed over all states of , of and with . This formula shows the independence of the choice of Hilbert spaces, because on the right side only data of the abstract C * -algebras and their algebraic tensor product can be found. ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle g}$ ${\displaystyle B}$ ${\displaystyle s\in A\odot B}$ ${\displaystyle (f\otimes g)(s^{*}s)>0}$

For the designation: In the textbook by Kadison and Ringrose given below, instead of is written, Murphy uses the notation . ${\displaystyle A\otimes B}$ ${\displaystyle A\otimes _{\sigma }B}$ ${\displaystyle A\otimes _{*}B}$

properties

• If and * -homomorphisms are between C * -algebras, there is exactly one * -homomorphism marked with , so that for all . If both and are isometric or * -isomophisms, then has the same property. ${\displaystyle \pi _{1}:A_{1}\rightarrow B_{1}}$ ${\displaystyle \pi _{2}:A_{2}\rightarrow B_{2}}$ ${\displaystyle \pi _{1}\otimes \pi _{2}}$ ${\displaystyle A_{1}\otimes A_{2}\rightarrow B_{1}\otimes B_{2}}$ ${\displaystyle \pi _{1}\otimes \pi _{2}(a_{1}\otimes a_{2})=\pi _{1}(a_{1})\otimes \pi _{2}(a_{2})}$ ${\displaystyle a_{i}\in A_{i}}$ ${\displaystyle \pi _{1}}$ ${\displaystyle \pi _{2}}$ ${\displaystyle \pi _{1}\otimes \pi _{2}}$

• If there is a C * -norm on the algebraic tensor product , then . For this reason the spatial tensor product is also called the minimal tensor product, and the notation is sometimes found . ${\displaystyle \alpha }$ ${\displaystyle A\odot B}$ ${\displaystyle \sigma \leq \alpha }$ ${\displaystyle A\otimes _{\mathrm {min} }B}$

Examples

Let be a C * -algebra and a compact Hausdorff space . be the set of all continuous functions . For , and define: ${\displaystyle A}$ ${\displaystyle X}$ ${\displaystyle C(X,A)}$ ${\displaystyle X\rightarrow A}$ ${\displaystyle f,g\in C(X,A)}$ ${\displaystyle \lambda \in \mathbb {C} }$ ${\displaystyle x\in X}$

${\displaystyle {\begin{array}{rcl}(\lambda f)(x)&:=&\lambda \cdot f(x)\\(f+g)(x)&:=&f(x)+g(x)\\(f\cdot g)(x)&:=&f(x)\cdot g(x)\\(f^{*})(x)&:=&f(x)^{*}\\\|f\|&:=&\sup\{\|f(x)\|;\,x\in X\}\\\end{array}}}$.

This becomes a C * -algebra and you have an isometric isomorphism . ${\displaystyle C(X,A)}$ ${\displaystyle C(X)\otimes _{\sigma }A\rightarrow C(X,A),f\otimes a\mapsto f(\cdot )a}$

Let be the C * -algebra of the complex matrices and a C * -algebra operating on a Hilbert space . Let the algebra of the matrices with entries be off ; this operates in the usual way on , that is ${\displaystyle M_{n}}$ ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle M_{n}(A)}$ ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle H^{n}}$

${\displaystyle {\begin{pmatrix}a_{1,1}&\ldots &a_{1,n}\\\vdots &\ddots &\vdots \\a_{n,1}&\ldots &a_{n,n}\end{pmatrix}}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1,j}x_{j}\\\vdots \\\sum _{j=1}^{n}a_{n,j}x_{j}\end{pmatrix}}}$

This carries the norm of and shows that , where is mapped to. ${\displaystyle M_{n}(A)}$ ${\displaystyle L(H^{n})}$ ${\displaystyle M_{n}\otimes A\cong M_{n}(A)}$ ${\displaystyle (c_{i,j})_{i,j}\otimes a}$ ${\displaystyle (c_{i,j}\cdot a)_{i,j}}$

See also

• Maximum tensor product , another tensor product norm for C * algebras
• Nuclear C * -algebra , C * -algebras with a unique tensor product norm
• A very similar construction leads to a tensor product for Von Neumann algebras .

Individual evidence

1. ↑ M. Takesaki: On the cross-norm of the direct product of C * -algebras , Tohoku Mathematical Journal, Volume 10 (1958), pages 111-122
2. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , §11.3
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Lemma 11.3.3
4. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , sentence 11.1.2 and §11.3.1
5. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Theorem 11.1.3
6. ↑ Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-12-511360-9 , Theorem 6.4.18
7. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , Theorem 11.3.9
8. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1 , example 11.1.7

literature

• Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-12-511360-9
• RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-12-393302-1