Cuntz algebra

Cuntz algebra

In C*-algebras, the Cuntz algebra \mathcal{O}_n (after Joachim Cuntz) is the universal C*-algebra generated by n isometries satisfying certain relations. It is the first concrete example of a separable infinite simple C*-algebra.

Every simple infinite C*-algebra contains, for any given n, a subalgebra that has \mathcal{O}_n as quotient.

Definition and basic properties

Let n ≥ 2 and H be a separable Hilbert space. Consider the C*-algebra \mathcal{A} generated by a set

\{ S_i \}_{i=1}^{n}

of isometries acting on H satisfying

\sum_{i=1}^n S_i S_i^* = I.

Note that, in particular, the Si's have the property that

S_i^* S_j = \delta_{ij} I.

Theorem. The concrete C*-algebra \mathcal{A} is isomorphic to the universal C*-algebra \mathcal{L} generated by n generators s1... sn subject to relations si*si = 1 for all i and ∑ sisi* = 1.

The proof of the theorem hinges on the following fact: any C*-algebra generated by n isometries s1... sn with orthogonal ranges contains a copy of the UHF algebra \mathcal{F} type n∞. Namely \mathcal{F} is spanned by words of the form

s_{i_1}...s_{i_k}s_{j_1}^*...s_{j_k}^*, k \geq 0.

The *-subalgebra \mathcal{F}, being approximately finite dimensional, has a unique C*-norm. The subalgebra \mathcal{F} 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 \mathcal{L} to \mathcal{A} is injective, which proves the theorem.

This universal C*-algebra is called the Cuntz algebra, denoted by \mathcal{O}_n .

A C*-algebra is said to be purely infinite if every hereditary C*-subalgebra of it is infinite. \mathcal{O}_n is a separable, simple, purely infinite C*-algebra.

Any simple infinite C*-algebra contains a subalgebra that has \mathcal{O}_n as a quotient.

The UHF algebra \mathcal{F} has a subalgebra \mathcal{F}' that is canonically isomorphic to a non-unital subalgebra of itself: In the Mn stage of the direct system defining \mathcal{F}, consider the rank-1 projection e11, the matrix that is 1 in the upper left corner and elsewhere. Propagate this projection through the direct system. At the Mnk stage of the direct system, one has a rank nk - 1 projection. In the direct limit, this gives a projection P in \mathcal{F}. The corner

P \mathcal{F} P = \mathcal{F'}

is isomorphic to \mathcal{F}. The *-endomorphism Φ that maps \mathcal{F} onto \mathcal{F}' is implemented by an isometry V, i.e. Φ(·) = V(·)V*. One can take V to be one of the generators of \; \mathcal{O}_n . \;\mathcal{O}_n is in fact the crossed product of \mathcal{F} with the endomorphism Φ.

Classification

The Cuntz algebras are pairwise non-isomorphic, i.e. \mathcal{O}_n and \mathcal{O}_m are non-isomorphic for n ≠ m. The K0 group of \mathcal{O}_n is Zn - 1, the abelian cyclic group of order n-1. Since K0 is a (functorial) invariant, \mathcal{O}_n and \mathcal{O}_m are non-isomorphic.

References

  • Cuntz, J. (1977), "Simple C*-algebras generated by isometries", Comm. Math. Phys. 57: 173-185 
  • Jørgensen, Palle E. T.; Treadway, Brian, Analysis and probability: wavelets, signals, fractals, Graduate texts in mathematics, 234, Springer-Verlag isbn=0387295194 


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