Nuclear C * algebra

The nuclear C * -algebras considered in the mathematical sub-area of functional analysis form a large class of C * -algebras that includes important sub- classes. The nuclear C * -algebras have been introduced in connection with uniqueness questions about tensor products ; hence the name nuklear , which was chosen as an allusion to the nuclear spaces from the theory of locally convex spaces .

definition

If and are two C * -algebras, then one can define a C * -norm on the algebraic tensor product in several ways, i.e. a norm such that ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\odot B}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

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

applies. A C * -algebra is called nuclear if there is exactly one such C * -norm for every C * -algebra . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\odot B}$

Since there is always a minimal C * -norm, namely the norm of the spatial tensor product , and a maximal C * -norm , the nuclearity for a C * -algebra means that for every C * -algebra the minimal and maximal C * -Normally on coincide In this context, M. Takesaki spoke of C * -algebras with the property T , the designation nuclear C * -algebra goes back to C. Lance . ${\displaystyle A\odot B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\odot B}$

Examples

• Commutative C * algebras are nuclear. In this case the uniquely determined tensor product coincides with the injective tensor product .

• More generally, all postliminal C * algebras are nuclear, as already shown in the paper by Takesaki mentioned below.

• Finite-dimensional C * -algebras are nuclear, because these are finite direct sums of matrix algebras and for each C * -algebra it is based on the norm described in the article about the spatial tensor product . ${\displaystyle M_{n}}$ ${\displaystyle M_{n}\otimes B\cong M_{n}(B)}$ ${\displaystyle B}$ ${\displaystyle M_{n}(B)}$

• The reduced group C * algebra of a related or indirect group is nuclear. For discrete groups, according to a theorem of C. Lance, the converse also applies: For a discrete group is nuclear if and only if is indirect. ${\displaystyle C_{r}^{*}(G)}$ ${\displaystyle G}$ ${\displaystyle C_{r}^{*}(G)}$ ${\displaystyle G}$

• ${\displaystyle C^{*}(F_{2}),C_{r}^{*}(F_{2})}$and are examples of C * algebras that are not nuclear, where is the free group created by 2 elements and the sequence space of the square summable sequences. ${\displaystyle L(\ell ^{2})}$ ${\displaystyle F_{2}}$ ${\displaystyle \ell ^{2}}$

properties

• Completed two-sided ideals and quotients of nuclear C * algebras are again nuclear.

• Conversely, if there is a short exact sequence of C * -algebras with nuclear and , then is also nuclear. ${\displaystyle 0\rightarrow J\rightarrow A\rightarrow B\rightarrow 0}$ ${\displaystyle J}$ ${\displaystyle B}$ ${\displaystyle A}$

• Sub-C * algebras of nuclear C * algebras are generally not again nuclear. Then all sub-C * -algebras of a nuclear C * -algebra are again nuclear if the C * -algebra is postliminal.

• Inductive limits of nuclear C * algebras are again nuclear, so all AF-C * algebras are nuclear.

• If a C * -dynamic system with a nuclear C * -algebra and an indirect group , then the entangled product is also nuclear. In particular, the irrational rotational algebras are nuclear. ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle A\ltimes _{\alpha }G}$

• A C * -algebra is nuclear if and only if the identity of pointwise norm limits is completely positive , 1-bounded operators of finite order, that is, there is a network of completely positive operators with and for all and for all . ${\displaystyle A}$ ${\displaystyle \mathrm {id} _{A}}$ ${\displaystyle (P_{i})_{i\in I}}$ ${\displaystyle \mathrm {dim} P_{i}(A)<\infty }$ ${\displaystyle \|P_{i}\|\leq 1}$ ${\displaystyle i\in I}$ ${\displaystyle \textstyle \lim _{i\in I}\|P_{i}(a)-a\|=0}$ ${\displaystyle a\in A}$

• A Von Neumann algebra is called hyperfinite if it contains an ascending sequence of finite-dimensional * -algebras, the union of which is close with respect to the weak operator topology . A C * -algebra is nuclear if and only if its enveloping Von Neumann algebra is hyperfinite. See for more equivalent characterizations.

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 0-1239-3302-1 , Lemma 11.3.5
3. ↑ C. Lance: On Nuclear C * -Algebras , Journal of Functional Analysis, Volume 12 (1973), pages 157-176, Theorem 4.2
4. ↑ Gerald. J. Murphy: C * -Algebras and Operator Theory , Academic Press Inc. (1990), ISBN 0-1251-1360-9 , Theorem 6.5.3
5. ↑ B. Blackadar: Nonnuclear subalgebras of C * -algebras , Journal of Operator Theory, Volume 14 (1985), pages 347-350
6. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , sentence 3/11/12
7. ^ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 15.8.2
8. ^ B. Blackadar: K-Theory for Operator-Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 15.8.1
9. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 15.8
10. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , 8.15.15