Cuntz algebra explained
In mathematics, the Cuntz algebra
, named after
Joachim Cuntz, is the
universal C*-algebra generated by
isometries of an infinite-dimensional
Hilbert space
satisfying certain relations.
[1] These algebras were introduced as the first concrete examples of a
separable infinite simple C*-algebra, meaning as a Hilbert space,
is isometric to the
sequence space
and it has no nontrivial closed ideals. These algebras are fundamental to the study of simple infinite C*-algebras since any such algebra contains, for any given
n, a subalgebra that has
as quotient.
Definitions
Let n ≥ 2 and
be a
separable Hilbert space. Consider the
C*-algebra
generated by a set
of isometries (i.e.
) acting on
satisfying
This universal C*-algebra is called the Cuntz algebra, denoted by
.
A simple C*-algebra is said to be purely infinite if every hereditary C*-subalgebra of it is infinite.
is a separable, simple, purely infinite C*-algebra. Any simple infinite C*-algebra contains a subalgebra that has
as a quotient.
Properties
Classification
The Cuntz algebras are pairwise non-isomorphic, i.e.
and
are non-isomorphic for
n ≠
m. The
K0 group of
is
, the
cyclic group of
order n − 1. Since
K0 is a
functor,
and
are non-isomorphic.
Relation between concrete C*-algebras and the universal C*-algebra
Theorem. The concrete C*-algebra
is isomorphic to the universal C*-algebra
generated by
n generators
s1...
sn subject to relations
si*si = 1 for all
i and ∑
sisi* = 1.
type
n∞. Namely
is spanned by words of the form
The *-subalgebra
, being
approximately finite-dimensional, has a unique C*-norm. The subalgebra
plays role of the space of
Fourier coefficients for elements of the algebra. A key technical lemma, due to Cuntz, is that an element in the algebra is zero
if and only if all its Fourier coefficients vanish. Using this, one can show that the quotient map from
to
is
injective, which proves the theorem.
The UHF algebra
has a non-unital subalgebra
that is canonically isomorphic to
itself: In the M
n stage of the direct system defining
, consider the rank-1 projection
e11, the
matrix that is 1 in the upper left corner and zero elsewhere. Propagate this projection through the direct system. At the M
nk stage of the direct system, one has a rank
nk − 1 projection. In the
direct limit, this gives a projection
P in
. The corner
is isomorphic to
. The *-endomorphism Φ that maps
onto
is implemented by the isometry
s1, i.e. Φ(·) =
s1(·)
s1*.
is in fact the
crossed product of
with the
endomorphism Φ.
Cuntz algebras to represent direct sums
The relations defining the Cuntz algebras align with the definition of the biproduct for preadditive categories. This similarity is made precise in the C*-category of unital *-endomorphisms over C*-algebras. The objects of this category are unital *-endomorphisms, and morphisms are the elements
, where
if
for every
. A unital *-endomorphism
is the direct sum of endomorphisms
\sigma1,\sigma2,...,\sigman
if there are isometries
satisfying the
relations and
\rho(x)=
Sk\sigmak(x)S
\forallx\inA.
In this direct sum, the inclusion morphisms are
, and the projection morphisms are
.
Generalisations
Cuntz algebras have been generalised in many ways. Notable amongst which are the Cuntz–Krieger algebras, graph C*-algebras and k-graph C*-algebras.
Applied mathematics
In signal processing, a subband filter with exact reconstruction give rise to representations of a Cuntz algebra. The same filter also comes from the multiresolution analysis construction in wavelet theory.[2]
See also
References
- Cuntz. Joachim. 1977. Simple $C^*$-algebras generated by isometries. Communications in Mathematical Physics. en. 57. 2. 173–185. 0010-3616.
- Book: Analysis and Probability: Wavelets, Signals, Fractals. 234. . Palle E. T.. Jørgensen. Brian. Treadway . Springer-Verlag. 0-387-29519-4 .