Nuclear C*-algebra

The considered in the mathematical branch of Functional Analysis nuclear C *-algebras form a large class of C *-algebras, which includes important sub-classes. The nuclear C *-algebras have been introduced in the context of uniqueness questions concerning tensor products; therefore stirs the name nuclear, who was elected in allusion to the nuclear facilities from the theory of locally convex spaces.

Definition

Are and two C *-algebras, we can define the algebraic tensor product in several ways, a C * - norm, that is a standard, so that

  • Is a normed algebra
  • For all

Applies. A C * - algebra is called nuclear if there is exactly one such C * - norm is for every C * - algebra.

Because there will always be a minimum C * standard, namely, the standard of the spatial tensor, and a maximum C * standard means for the nuclearity C * algebra that for each C * algebra the minimum and maximum C * norm to coincide. M. Takesaki spoke in this context of C * - algebras with the property T, the term nuclear C * - algebra goes back to C. Lance.

Examples

  • Commutative C *-algebras are nuclear. The uniquely determined tensor falls in this case with the injective tensor product together.
  • General postliminalen all C *-algebras are nuclear, as has been shown already in the below-mentioned work of Takesaki.
  • Finite- dimensional C *-algebras are nuclear, because these are finite direct sums of matrix algebras and it is for every C * - algebra with the standard described in the article about the spatial tensor product on.
  • The reduced group C * - algebra of a coherent or indirect group is nuclear. For discrete groups is valid for a set of C. Lance the converse: For a discrete group if and only nuclear, if indirectly.
  • And are examples of C *-algebras which are non-nuclear, the free group generated by two elements and the sequence space of the consequences is quadratsummierbaren.

Properties

  • Closed two-sided ideal and quotients of nuclear C *-algebras are nuclear again.
  • Conversely, a short exact sequence of C * - algebras with nuclear and so is nuclear.
  • Sub -C *-algebras of nuclear C *-algebras are in general not back nuclear. Just then all the sub -C *-algebras of a nuclear C * - algebra are again nuclear, when the C *-algebra is postliminal.
  • Inductive limits of nuclear C *-algebras are nuclear again, so all AF C *-algebras are nuclear.
  • Is a C * - dynamical system with a nuclear C * - algebra and an indirect group, including the crossed product is nuclear. In particular, the irrational Rotationsalgebren are nuclear.
  • A C * - algebra is nuclear if and only if the identity pointwise norm limit of completely positive, 1- bounded operators is of finite rank, ie there is a network completely positive operators with and for all and for all.
  • A von Neumann algebra is called hyperfinit if it contains an increasing sequence of finite-dimensional *-algebras whose union is dense with respect to the weak operator topology. A C * - algebra is nuclear if and only if its enveloping von Neumann algebra is hyperfinit. See for further equivalent characterizations.
C*-Algebra Nuclear space Masamichi Takesaki Amenable group Quotient space (linear algebra) Richard Kadison
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