Cuntz algebra

The -algebra generated by isometries , where or , on some infinite-dimensional Hilbert space whose ranges are pairwise orthogonal:

(a1)

and, when , sum up to the identity operator on :

(a2)

has been introduced in [a2]. The linear span of all is a Hilbert space in , i.e. , . is called the generating Hilbert space. The role of , rather than that of the generating set of isometries, has been emphasized in [a7] and [a5]. In the latter an intrinsic description of the -algebraic structure of has been given (thus leading to the notation ): consider, for a fixed finite-dimensional Hilbert space and any , the algebraic inductive limit of spaces of (bounded) linear mappings between tensor powers of , with inclusion mappings that tensor on the right by the identity operator on . Then has a natural structure of a -graded -algebra, and has also a unique -norm for which the automorphic action of the circle group defining the grading is isometric [a5] (cf. also Norm). The completion is the Cuntz algebra , if . The case where is a separable infinite-dimensional Hilbert space can be similarly treated, but when forming the graded subspaces one has to take into consideration the spaces of compact operators between tensor powers of .

Important properties of are the following:

1) Universality. does not depend on the generating set of isometries satisfying relations (a1) and (a2), but only on its cardinality, or, in other words, is covariantly associated to the generating Hilbert space : every unitary extends uniquely to an isomorphism between the corresponding generated -algebras.

2) Simplicity. has no proper closed two-sided ideal.

3) Pure infiniteness. is a fundamental example of a purely infinite -algebra: every hereditary -subalgebra contains an infinite projection.

4) Toeplitz extension. Assume . Then the Toeplitz extension of is, by definition, the -algebra generated by the set of isometries satisfying (a1) but . The Toeplitz extension satisfies 1) and 3) as well but it is not simple: it has a unique proper closed ideal , generated by the projection , naturally isomorphic to the compact operators on the full Fock space of the generating Hilbert space. Therefore, there is a short exact sequence: .

5) Crossed product representation. Assume . Let denote the -inductive limit of the algebras under the inclusion mappings that tensor on the left by some fixed minimal projection of . Let be the right shift automorphism of the tensor product; then has as a full corner, so that . A similar construction goes through in the case .

6) -theory. ; if , . These results were first proved in [a4] and imply that if and only if .

7) Canonical groups of automorphisms. By virtue of the universality property 1), any unitary operator on the generating Hilbert space induces an automorphism on . Thus, to any closed subgroup of there corresponds a -dynamical system on , whose properties have been studied in [a5] for and [a1] for .

8) Quasi-free states. Let be a sequence of operators in , the set of positive trace-class operators on , with , , if and otherwise. Then there is a unique state on , called quasi-free, such that

Properties of quasi-free states have been studied in [a7]. In particular, it has been shown that the quasi-free states associated to a constant sequence , , is the unique state satisfying the -property at a finite inverse temperature for the -parameter automorphism group of implemented by a strongly continuous unitary group on (cf. [a9]) if and only if .

9) Absorbing properties under tensor products. The following recent (1998) results were shown by E. Kirchberg. Let be a separable simple unital nuclear -algebra. Then:

i) is isomorphic to ;

ii) is purely infinite if and only if is isomorphic to , where denotes the minimal (or spatial) tensor product.

Results from 1)–6) were first obtained by J. Cuntz in [a2], [a4]. Cuntz algebras, since their appearance, have been extensively used in operator algebras: results in 7) played an important role in abstract duality theory for compact groups [a6], those in 9) are part of deep results obtained in [a8] in the classification theory of nuclear, purely infinite, simple -algebras. Furthermore, the very construction of the Cuntz algebras has inspired a number of important generalizations, among them: the Cuntz–Krieger algebras associated to topological Markov chains [a3] (cf. also Markov chain); the -algebra associated to an object of a tensor -category [a6]; and the Pimsner algebras associated to a Hilbert -bimodule [a10].

References

[a1]	T. Ceccherini, C. Pinzari, "Canonical actions on " J. Funct. Anal. , 103 (1992) pp. 26–39
[a2]	J. Cuntz, "Simple -algebras generated by isometries" Comm. Math. Phys. , 57 (1977) pp. 173–185
[a3]	J. Cuntz, W. Krieger, "A class of -algebras and topological Markov chains" Invent. Math. , 56 (1980) pp. 251–268
[a4]	J. Cuntz, "-theory for certain -algebras" Ann. of Math. , 113 (1981) pp. 181–197
[a5]	S. Doplicher, J.E. Roberts, "Duals of compact Lie groups realized in the Cuntz algebras and their actions on -algebras." J. Funct. Anal. , 74 (1987) pp. 96–120
[a6]	S. Doplicher, J.E. Roberts, "A new duality theory for compact groups." Invent. Math. , 98 (1989) pp. 157–218
[a7]	D.E. Evans, "On " Publ. Res. Inst. Math. Sci. , 16 (1980) pp. 915–927
[a8]	E. Kirchberg, "Lecture on the proof of Elliott's conjecture for purely infinite separable unital nuclear -algebras which satisfy the UCT for their -theory" Talk at the Fields Inst. during the Fall semester (1994/5)
[a9]	G.K. Pedersen, "-algebras and their automorphism groups" , Acad. Press (1990)
[a10]	M. Pimsner, "A class of -algebras generalizing both Cuntz–Krieger algebras and crossed products by " D.-V. Voiculescu (ed.) , Free Probability Theory , Amer. Math. Soc. (1997)