Cuntz algebra

The $C ^ { * }$-algebra $\mathcal{O} _ { N }$ generated by $n$ isometries $\{ S _ { i } \} _ { i = 1 } ^ { n }$, where $n \geq 2$ or $n = \infty$, on some infinite-dimensional Hilbert space $H$ whose ranges are pairwise orthogonal:

\begin{equation} \tag{a1} S _ { i } ^ { * } S _ { j } = 0 , i \neq j, \end{equation}

and, when $n < \infty$, sum up to the identity operator on $H$:

\begin{equation} \tag{a2} \sum _ { i = 1 } ^ { n } S _ { i } S _ { i } ^ { * } = I. \end{equation}

$\mathcal{O} _ { N }$ has been introduced in [a2]. The linear span $\mathcal{H}$ of all $S _ { i }$ is a Hilbert space in $\mathcal{O} _ { N }$, i.e. $S ^ { * } S ^ { \prime } \in \mathbf{C}I$, $S , S ^ { \prime } \in \mathcal{H}$. $\mathcal{H}$ is called the generating Hilbert space. The role of $\mathcal{H}$, rather than that of the generating set of isometries, has been emphasized in [a7] and [a5]. In the latter an intrinsic description of the $C ^ { * }$-algebraic structure of $\mathcal{O} _ { N }$ has been given (thus leading to the notation $\mathcal{O} _ { \mathcal{H} }$): consider, for a fixed finite-dimensional Hilbert space $\mathcal{H}$ and any $k \in \mathbf{Z}$, the algebraic inductive limit $\square ^ { 0 } \mathcal{O} _ { \mathcal{H} } ^ { ( k ) }$ of spaces of (bounded) linear mappings $( \mathcal{H} ^ { \otimes r } , \mathcal{H} ^ { \otimes r + k } )$ between tensor powers of $\mathcal{H}$, with inclusion mappings $( \mathcal{H} ^ { \otimes r } , \mathcal{H} ^ { \otimes r + k } ) \rightarrow ( \mathcal{H} ^ { \otimes r + 1 } , \mathcal{H} ^ { \otimes r + 1 + k } )$ that tensor on the right by the identity operator on $( \mathcal{H} , \mathcal{H} )$. Then has a natural structure of a $\bf Z$-graded $\square ^ { * }$-algebra, and has also a unique $C ^ { * }$-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 $\mathcal{O} _ { N }$, if $n = \operatorname { dim } ( \mathcal{H} ) \geq 2$. The case where $\mathcal{H}$ 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 $\mathcal{H}$.

Important properties of $\mathcal{O} _ { N }$ are the following:

1) Universality. $\mathcal{O} _ { N }$ does not depend on the generating set of isometries satisfying relations (a1) and (a2), but only on its cardinality, or, in other words, $\mathcal{O} _ { N }$ is covariantly associated to the generating Hilbert space $\mathcal{H}$: every unitary $u : \mathcal{H} \rightarrow \mathcal{H} ^ { \prime }$ extends uniquely to an isomorphism between the corresponding generated $C ^ { * }$-algebras.

2) Simplicity. $\mathcal{O} _ { N }$ has no proper closed two-sided ideal.

3) Pure infiniteness. $\mathcal{O} _ { N }$ is a fundamental example of a purely infinite $C ^ { * }$-algebra: every hereditary $C ^ { * }$-subalgebra contains an infinite projection.

4) Toeplitz extension. Assume $n < \infty$. Then the Toeplitz extension ${\cal T} _ { n }$ of $\mathcal{O} _ { N }$ is, by definition, the $C ^ { * }$-algebra generated by the set of isometries $\{ S _ { i } \} _ { i = 1 } ^ { n }$ satisfying (a1) but $\sum _ { i = 1 } ^ { n } S _ { i } S _ { i } ^ { * } < I$. The Toeplitz extension satisfies 1) and 3) as well but it is not simple: it has a unique proper closed ideal $\mathcal{K}$, generated by the projection $P = I - \sum _ { i = 1 } ^ { n } S _ { i } S _ { i } ^ { * }$, naturally isomorphic to the compact operators on the full Fock space $F ( \mathcal{H} ) = \mathbf{C} \oplus \oplus _ { n = 1 } ^ { \infty } \mathcal{H} ^ { \otimes n }$ of the generating Hilbert space. Therefore, there is a short exact sequence: $0 \rightarrow {\cal K} \rightarrow {\cal T} _ { n } \rightarrow {\cal O} _ { n } \rightarrow 0$.

5) Crossed product representation. Assume $n < \infty$. Let $\mathcal{B}$ denote the $C ^ { * }$-inductive limit of the algebras $\mathcal{B} _ { i } = \otimes _ { k \geq - i} M _ { n } ( \mathbf{C} )$ under the inclusion mappings $\mathcal{B} _ { i } \rightarrow \mathcal{B} _ { i +1} $ that tensor on the left by some fixed minimal projection of $M _ { n } ( \mathbf C )$. Let $\alpha$ be the right shift automorphism of the tensor product; then $\mathcal{B} \rtimes _ { \alpha } \bf Z$ has $\mathcal{O} _ { N }$ as a full corner, so that ${\cal B} \rtimes _ { \alpha } {\bf Z} \simeq {\cal O} _ { n } \otimes \cal K$. A similar construction goes through in the case $n = \infty$.

6) $K$-theory. $K _ { 1 } ( {\cal O} _ { n } ) = 0$; $K _ { 0 } ( \mathcal{O} _ { n } ) = \mathbf{Z} _ { n - 1}$ if $n < \infty$, $K _ { 0 } ( \mathcal{O} _ { \infty } ) = \mathbf{Z}$. These results were first proved in [a4] and imply that $\mathcal{O} _ { n } \simeq \mathcal{O} _ { m }$ if and only if $n = m$.

7) Canonical groups of automorphisms. By virtue of the universality property 1), any unitary operator on the generating Hilbert space induces an automorphism on $\mathcal{O} _ { N }$. Thus, to any closed subgroup $G$ of $U ( \mathcal{H} )$ there corresponds a $C ^ { * }$-dynamical system on $\mathcal{O} _ { N }$, whose properties have been studied in [a5] for $n < \infty$ and [a1] for $n = \infty$.

8) Quasi-free states. Let $\{ K _ { i } \}$ be a sequence of operators in $T _ { 1 } ( \mathcal{H} )$, the set of positive trace-class operators on $\mathcal{H}$, with $\operatorname { tr } ( K _ { i } ) = 1$, $i \in \mathbf{N}$, if $\dim ( \mathcal{H} ) < \infty$ and $\operatorname { tr } ( K _ { i } ) \leq 1$ otherwise. Then there is a unique state $\omega \{ K _ { i } \}$ on $\mathcal{O} _ { N }$, called quasi-free, such that

\begin{equation*} \omega \{ K _ { i } \} ( S _ { i_1 } \ldots S _ { i_r } S _ { j_{s} } ^ { * } \ldots S _ { j_{ 1} } ^ { * } ) = \prod _ { h = 1 } ^ { r } \{ S _ { j _ { h } } , K _ { h } S _ { i_h } \} \delta _ { r , s }. \end{equation*}

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 $K _ { i } = K$, $i \in \mathbf{N}$, is the unique state satisfying the $KMS$-property at a finite inverse temperature $\beta$ for the $1$-parameter automorphism group of $\mathcal{O} _ { N }$ implemented by a strongly continuous unitary group $u ( t ) = e ^ { i h t }$ on $\mathcal{H}$ (cf. [a9]) if and only if $K = e ^ { - \beta h } \in T _ { 1 } ( H )$.

9) Absorbing properties under tensor products. The following recent (1998) results were shown by E. Kirchberg. Let $\mathcal{A}$ be a separable simple unital nuclear $C ^ { * }$-algebra. Then:

i) $\mathcal{A} \otimes \mathcal{O} _ { 2 }$ is isomorphic to $\mathcal{O} _ { 2 }$;

ii) $\mathcal{A}$ is purely infinite if and only if $\mathcal{A}$ is isomorphic to $\mathcal{A} \otimes \mathcal{O}_\infty$, where $\otimes$ 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 $C ^ { * }$-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 $C ^ { * }$-algebra associated to an object of a tensor $C ^ { * }$-category [a6]; and the Pimsner algebras associated to a Hilbert $C ^ { * }$-bimodule [a10].

References

[a1]	T. Ceccherini, C. Pinzari, "Canonical actions on $O _ { \infty }$" J. Funct. Anal. , 103 (1992) pp. 26–39
[a2]	J. Cuntz, "Simple $C ^ { * }$-algebras generated by isometries" Comm. Math. Phys. , 57 (1977) pp. 173–185
[a3]	J. Cuntz, W. Krieger, "A class of $C ^ { * }$-algebras and topological Markov chains" Invent. Math. , 56 (1980) pp. 251–268
[a4]	J. Cuntz, "$K$-theory for certain $C ^ { * }$-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 $C ^ { * }$-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 $O _ { n }$" 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 $C ^ { * }$-algebras which satisfy the UCT for their $K K$-theory" Talk at the Fields Inst. during the Fall semester (1994/5)
[a9]	G.K. Pedersen, "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press (1990)
[a10]	M. Pimsner, "A class of $C ^ { * }$-algebras generalizing both Cuntz–Krieger algebras and crossed products by $\bf Z$" D.-V. Voiculescu (ed.) , Free Probability Theory , Amer. Math. Soc. (1997)