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nuclear C*-algebra
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(Definition)
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Definition 0.1 A C*-algebra $A$ is called a nuclear C*-algebra if all C*-norms on every algebraic tensor product $A \otimes X$ , of $A$ with any other C*-algebra $X$ , agree with, and also equal the spatial C*-norm ( viz Lance, 1981). Therefore, there is a unique completion of $A \otimes X$ to a C*-algebra , for any other C*-algebra $X$ .
In general terms, a $C^*$ -algebra is exact if it is isomorphic with a $C^*$ -subalgebra of some nuclear $C^*$ -algebra. The precise definition of an exact $C^*$ -algebra follows.
Definition 0.2 Let $M_n$ be a matrix space, let $\mathcal{A}$ be a general operator space, and also let $\mathbb{C}$ be a C*-algebra. A $C^*$ -algebra $\mathbb{C}$ is exact if it is `finitely representable' in $M_n$ , that is, if for every finite dimensional subspace $E$ in $\mathcal{A}$ and quantity $epsilon > 0$ , there exists a subspace $F$ of some $M_n$ , and also a linear isomorphism $T:E \to F$ such that the $cb$ -norm $$|T|_{cb}|T^{-1}|_{cb} < 1 + epsilon.$$
A $C^*$ -subalgebra of a nuclear C*-algebra need not be nuclear.
- 1
- E. C. Lance. 1981. Tensor Products and nuclear C*-algebras., in Operator Algebras and Applications, R.V. Kadison, ed., Proceed. Symp. Pure Maths., 38: 379-399, part 1.
- 2
- N. P. Landsman. 1998. ``Lecture notes on $C^*$ -algebras, Hilbert $C^*$ -Modules and Quantum Mechanics", pp. 89 a graduate level preprint discussing general C*-algebras in Postscript format.
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"nuclear C*-algebra" is owned by bci1.
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See Also: quantum operator algebras in quantum field theories, noncommutative geometry, groupoid C*-convolution algebras, groupoid C*-dynamical system, compact quantum groupoids related to C*-algebras,  -algebra, quotients in -algebras
| Other names: |
quantum operator algebra, C*-algebra,  -algebra |
| Also defines: |
generated C*-algebra, exact C^*-algebra |
| Keywords: |
quantum operator algebra, nuclear C*-algebra, generators |
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Cross-references: linear isomorphism, subspace, finite dimensional, representable, operator, matrix, isomorphic, terms, type, discrete, amenable, products, amenable groups, group, finite-dimensional, commutative, completion, viz, tensor product, algebraic, C*-norms, nuclear
There are 13 references to this entry.
This is version 58 of nuclear C*-algebra, born on 2008-07-14, modified 2009-02-02.
Object id is 10787, canonical name is NuclearCAlgebra.
Accessed 2618 times total.
Classification:
| AMS MSC: | 81R15 (Quantum theory :: Groups and algebras in quantum theory :: Operator algebra methods) | | | 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods) | | | 81T05 (Quantum theory :: Quantum field theory; related classical field theories :: Axiomatic quantum field theory; operator algebras) |
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Pending Errata and Addenda
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