In the mathematical field of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a nuclear C*-algebra is a
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
such that the
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
and
projective C*-
cross norms on
are the same for every C*-algebra
This property was first studied by under the name "Property T", which is not related to
Kazhdan's property T.
Characterizations
Nuclearity admits the following equivalent characterizations:
* The identity map, as a
completely positive map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Definition
Let A and B be C*-algebras. A linea ...
, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a
noncommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
analogue of the existence of
partitions of unity
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0, ...
.
* The
enveloping von Neumann algebra
In operator algebras, the enveloping von Neumann algebra of a C*-algebra is a von Neumann algebra that contains all the operator-algebraic information about the given C*-algebra. This may also be called the ''universal'' enveloping von Neumann alg ...
is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
.
* It is
amenable as a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
.
* It is isomorphic to a C*-subalgebra
of the
Cuntz algebra with the property that there exists a
conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – give ...
from
to
This condition is only equivalent to the others for
separable C*-algebras.
See also
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References
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{{DEFAULTSORT:Nuclear C-algebra
C*-algebras
Functional analysis
Operator theory
it:C*-algebra#C*-algebra nucleare