Maximum tensor product

In mathematical branch of functional analysis is the maximum tensor product of C * -algebras a construction * -algebras with which one of two C and a new one gets designated C * algebra. It is a matter of completing the algebraic tensor product of and with a suitable norm . The construction presented below goes back to A. Guichardet . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\otimes _{\mathrm {max} }B}$ ${\displaystyle A}$ ${\displaystyle B}$

construction

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

• ${\displaystyle \alpha (st)\leq \alpha (s)\alpha (t)}$ for all ${\displaystyle s,t\in A\odot B}$
• ${\displaystyle \alpha (s^{*}s)\,=\,\alpha (s)^{2}}$ for all ${\displaystyle s\in A\odot B}$

One can show that for everyone and . For an element it therefore follows for every C * half-norm. Therefore , where it goes through all the C * half-norms is finite, and it is easy to confirm that a C * half-norm is and by construction the largest on . It is even a norm, because the C * half-norms include the spatial C * norm . ${\displaystyle \alpha (a\otimes b)\leq \|a\|\|b\|}$ ${\displaystyle a\in A}$ ${\displaystyle b\in B}$ ${\displaystyle s=\sum _{i=1}^{n}a_{i}\otimes b_{i}\in A\odot B}$ ${\displaystyle \alpha (s)\leq \sum _{i=1}^{n}\|a_{i}\|\|b_{i}\|}$ ${\displaystyle \mu (s):=\sup _{\alpha }\alpha (s)}$ ${\displaystyle \alpha }$ ${\displaystyle \mu }$ ${\displaystyle A\odot B}$

The completion of with respect to this maximum C * -norm is called the maximum tensor product from and and is denoted by, other authors write for it . ${\displaystyle A\odot B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\otimes _{\mu }B}$ ${\displaystyle A\otimes _{\mathrm {max} }B}$

properties

The maximum tensor product has the following useful property:

Let , and C * -algebras and as well as two * - homomorphisms with interchanging images, that is, for all and . Then there is exactly one * homomorphism with for all and . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle \varphi :A\rightarrow C}$ ${\displaystyle \psi :B\rightarrow C}$ ${\displaystyle \varphi (a)\psi (b)=\psi (b)\varphi (a)}$ ${\displaystyle a\in A}$ ${\displaystyle b\in B}$ ${\displaystyle \pi :A\otimes _{\mathrm {max} }B\rightarrow C}$ ${\displaystyle \pi (a\otimes b)=\varphi (a)\psi (b)}$ ${\displaystyle a\in A}$ ${\displaystyle b\in B}$

If and are C * -algebras, then a pair is called an exchanging pair of representations of , if and Hilbert space representations are on the same Hilbert space and for all and holds. With this concept formation the following formula can be set up for the maximum C * norm: ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle (\varphi ,\psi )}$ ${\displaystyle (A,B)}$ ${\displaystyle \varphi :A\rightarrow L(H)}$ ${\displaystyle \psi :B\rightarrow L(H)}$ ${\displaystyle H}$ ${\displaystyle \varphi (a)\psi (b)=\psi (b)\varphi (a)}$ ${\displaystyle a\in A}$ ${\displaystyle b\in B}$

For two C * -algebras and and from the algebraic tensor product holds ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle s=\sum _{j=1}^{n}a_{j}\otimes b_{j}}$ ${\displaystyle A\odot B}$

${\displaystyle \mu (s)=\sup\{\|\sum _{j=1}^{n}\varphi (a_{j})\psi (b_{j})\|;\,(\varphi ,\psi ){\mbox{ vertauschendes Paar von Darstellungen von }}(A,B)\}.}$

See also

• Spatial tensor product
• Nuclear C * algebra

Individual evidence

1. ^ A. Guichardet: Tensor products of C * -algebras , Aarhus University Lecture Notes, Volume 12 (1969)
2. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , §11.3
3. ↑ Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-1251-1360-9 , chapter 6
4. ↑ Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-1251-1360-9 , Theorem 6.3.7
5. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 11.3.4

literature

• Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-1251-1360-9
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1