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The [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203001.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203002.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203003.png" /> isometries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203004.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203005.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203006.png" />, on some infinite-dimensional [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203007.png" /> whose ranges are pairwise orthogonal:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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and, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c1203009.png" />, sum up to the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030010.png" />:
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The [[C*-algebra|$C ^ { * }$-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|Hilbert space]] $H$ whose ranges are pairwise orthogonal:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a1} S _ { i } ^ { * } S _ { j } = 0 , i \neq j, \end{equation}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030012.png" /> has been introduced in [[#References|[a2]]]. The linear span <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030013.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030014.png" /> is a Hilbert space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030015.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030017.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030018.png" /> is called the generating Hilbert space. The role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030019.png" />, rather than that of the generating set of isometries, has been emphasized in [[#References|[a7]]] and [[#References|[a5]]]. In the latter an intrinsic description of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030020.png" />-algebraic structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030021.png" /> has been given (thus leading to the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030022.png" />): consider, for a fixed finite-dimensional Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030023.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030024.png" />, the algebraic [[Inductive limit|inductive limit]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030025.png" /> of spaces of (bounded) linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030026.png" /> between tensor powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030027.png" />, with inclusion mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030028.png" /> that tensor on the right by the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030029.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030030.png" /> has a natural structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030031.png" />-graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030032.png" />-algebra, and has also a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030033.png" />-norm for which the automorphic action of the circle group defining the grading is isometric [[#References|[a5]]] (cf. also [[Norm|Norm]]). The completion is the Cuntz algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030034.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030035.png" />. The case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030036.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030037.png" />.
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and, when $n &lt; \infty$, sum up to the identity operator on $H$:
  
Important properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030038.png" /> are the following:
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\begin{equation} \tag{a2} \sum _ { i = 1 } ^ { n } S _ { i } S _ { i } ^ { * } = I. \end{equation}
  
1) Universality. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030039.png" /> does not depend on the generating set of isometries satisfying relations (a1) and (a2), but only on its cardinality, or, in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030040.png" /> is covariantly associated to the generating Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030041.png" />: every unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030042.png" /> extends uniquely to an isomorphism between the corresponding generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030043.png" />-algebras.
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$\mathcal{O} _ { N }$ has been introduced in [[#References|[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 [[#References|[a7]]] and [[#References|[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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030030.png"/> 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 [[#References|[a5]]] (cf. also [[Norm|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 $\mathcal{H}$.
  
2) Simplicity. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030044.png" /> has no proper closed two-sided [[Ideal|ideal]].
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Important properties of $\mathcal{O} _ { N }$ are the following:
  
3) Pure infiniteness. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030045.png" /> is a fundamental example of a purely infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030046.png" />-algebra: every hereditary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030047.png" />-subalgebra contains an infinite projection.
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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.
  
4) Toeplitz extension. Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030048.png" />. Then the Toeplitz extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030050.png" /> is, by definition, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030051.png" />-algebra generated by the set of isometries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030052.png" /> satisfying (a1) but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030053.png" />. The Toeplitz extension satisfies 1) and 3) as well but it is not simple: it has a unique proper closed ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030054.png" />, generated by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030055.png" />, naturally isomorphic to the compact operators on the full [[Fock space|Fock space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030056.png" /> of the generating Hilbert space. Therefore, there is a short exact sequence: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030057.png" />.
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2) Simplicity. $\mathcal{O} _ { N }$ has no proper closed two-sided [[Ideal|ideal]].
  
5) Crossed product representation. Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030058.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030059.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030060.png" />-inductive limit of the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030061.png" /> under the inclusion mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030062.png" /> that tensor on the left by some fixed minimal projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030063.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030064.png" /> be the right shift automorphism of the tensor product; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030065.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030066.png" /> as a full corner, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030068.png" />. A similar construction goes through in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030069.png" />.
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3) Pure infiniteness. $\mathcal{O} _ { N }$ is a fundamental example of a purely infinite $C ^ { * }$-algebra: every hereditary $C ^ { * }$-subalgebra contains an infinite projection.
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030071.png" />-theory. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030072.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030073.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030075.png" />. These results were first proved in [[#References|[a4]]] and imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030076.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030077.png" />.
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4) Toeplitz extension. Assume $n &lt; \infty$. 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 $\sum _ { i = 1 } ^ { n } S _ { i } S _ { i } ^ { * } &lt; I$. 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|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: $0 \rightarrow {\cal K} \rightarrow {\cal T} _ { n } \rightarrow {\cal O} _ { n } \rightarrow 0$.
  
7) Canonical groups of automorphisms. By virtue of the universality property 1), any unitary operator on the generating Hilbert space induces an automorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030078.png" />. Thus, to any closed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030080.png" /> there corresponds a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030082.png" />-dynamical system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030083.png" />, whose properties have been studied in [[#References|[a5]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030084.png" /> and [[#References|[a1]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030085.png" />.
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5) Crossed product representation. Assume $n &lt; \infty$. 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 $n = \infty$.
  
8) Quasi-free states. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030086.png" /> be a sequence of operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030087.png" />, the set of positive trace-class operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030088.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030090.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030092.png" /> otherwise. Then there is a unique state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030093.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030094.png" />, called quasi-free, such that
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6) $K$-theory. $K _ { 1 } ( {\cal O} _ { n } ) = 0$; $K _ { 0 } ( \mathcal{O} _ { n } ) = \mathbf{Z} _ { n  - 1}$ if $n &lt; \infty$, $K _ { 0 } ( \mathcal{O} _ { \infty } ) = \mathbf{Z}$. These results were first proved in [[#References|[a4]]] and imply that $\mathcal{O} _ { n } \simeq \mathcal{O} _ { m }$ if and only if $n = m$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030095.png" /></td> </tr></table>
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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 [[#References|[a5]]] for $n &lt; \infty$ and [[#References|[a1]]] for $n = \infty$.
  
Properties of quasi-free states have been studied in [[#References|[a7]]]. In particular, it has been shown that the quasi-free states associated to a constant sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030097.png" />, is the unique state satisfying the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c12030099.png" />-property at a finite inverse temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300100.png" /> for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300101.png" />-parameter automorphism group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300102.png" /> implemented by a strongly continuous unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300103.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300104.png" /> (cf. [[#References|[a9]]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300105.png" />.
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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 $\dim ( \mathcal{H} ) &lt; \infty$ 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
  
9) Absorbing properties under tensor products. The following recent (1998) results were shown by E. Kirchberg. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300106.png" /> be a separable simple unital nuclear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300108.png" />-algebra. Then:
+
\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*}
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300109.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300110.png" />;
+
Properties of quasi-free states have been studied in [[#References|[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. [[#References|[a9]]]) if and only if $K = e ^ { - \beta h } \in T _ { 1 } ( H )$.
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300111.png" /> is purely infinite if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300112.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300113.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300114.png" /> denotes the minimal (or spatial) tensor product.
+
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:
  
Results from 1)–6) were first obtained by J. Cuntz in [[#References|[a2]]], [[#References|[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 [[#References|[a6]]], those in 9) are part of deep results obtained in [[#References|[a8]]] in the classification theory of nuclear, purely infinite, simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300115.png" />-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 [[#References|[a3]]] (cf. also [[Markov chain|Markov chain]]); the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300116.png" />-algebra associated to an object of a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300117.png" />-category [[#References|[a6]]]; and the Pimsner algebras associated to a Hilbert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300118.png" />-bimodule [[#References|[a10]]].
+
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 [[#References|[a2]]], [[#References|[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 [[#References|[a6]]], those in 9) are part of deep results obtained in [[#References|[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 [[#References|[a3]]] (cf. also [[Markov chain|Markov chain]]); the $C ^ { * }$-algebra associated to an object of a tensor $C ^ { * }$-category [[#References|[a6]]]; and the Pimsner algebras associated to a Hilbert $C ^ { * }$-bimodule [[#References|[a10]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Ceccherini,  C. Pinzari,  "Canonical actions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300119.png" />"  ''J. Funct. Anal.'' , '''103'''  (1992)  pp. 26–39</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Cuntz,  "Simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300120.png" />-algebras generated by isometries"  ''Comm. Math. Phys.'' , '''57'''  (1977)  pp. 173–185</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Cuntz,  W. Krieger,  "A class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300121.png" />-algebras and topological Markov chains"  ''Invent. Math.'' , '''56'''  (1980)  pp. 251–268</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Cuntz,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300122.png" />-theory for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300123.png" />-algebras"  ''Ann. of Math.'' , '''113'''  (1981)  pp. 181–197</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Doplicher,  J.E. Roberts,  "Duals of compact Lie groups realized in the Cuntz algebras and their actions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300124.png" />-algebras."  ''J. Funct. Anal.'' , '''74'''  (1987)  pp. 96–120</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Doplicher,  J.E. Roberts,  "A new duality theory for compact groups."  ''Invent. Math.'' , '''98'''  (1989)  pp. 157–218</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.E. Evans,  "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300125.png" />"  ''Publ. Res. Inst. Math. Sci.'' , '''16'''  (1980)  pp. 915–927</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Kirchberg,  "Lecture on the proof of Elliott's conjecture for purely infinite separable unital nuclear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300126.png" />-algebras which satisfy the UCT for their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300127.png" />-theory"  ''Talk at the Fields Inst. during the Fall semester''  (1994/5)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G.K. Pedersen,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300128.png" />-algebras and their automorphism groups" , Acad. Press  (1990)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Pimsner,  "A class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300129.png" />-algebras generalizing both Cuntz–Krieger algebras and crossed products by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120300/c120300130.png" />"  D.-V. Voiculescu (ed.) , ''Free Probability Theory'' , Amer. Math. Soc.  (1997)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  T. Ceccherini,  C. Pinzari,  "Canonical actions on $O _ { \infty }$"  ''J. Funct. Anal.'' , '''103'''  (1992)  pp. 26–39</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Cuntz,  "Simple $C ^ { * }$-algebras generated by isometries"  ''Comm. Math. Phys.'' , '''57'''  (1977)  pp. 173–185</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Cuntz,  W. Krieger,  "A class of $C ^ { * }$-algebras and topological Markov chains"  ''Invent. Math.'' , '''56'''  (1980)  pp. 251–268</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J. Cuntz,  "$K$-theory for certain $C ^ { * }$-algebras"  ''Ann. of Math.'' , '''113'''  (1981)  pp. 181–197</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  S. Doplicher,  J.E. Roberts,  "A new duality theory for compact groups."  ''Invent. Math.'' , '''98'''  (1989)  pp. 157–218</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D.E. Evans,  "On $O _ { n }$"  ''Publ. Res. Inst. Math. Sci.'' , '''16'''  (1980)  pp. 915–927</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  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)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  G.K. Pedersen,  "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press  (1990)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  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)</td></tr></table>

Revision as of 15:19, 1 July 2020

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)
How to Cite This Entry:
Cuntz algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cuntz_algebra&oldid=49878
This article was adapted from an original article by Claudia Pinzari (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article