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A uniformly closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t1301504.png" />-algebra of operators on a Hilbert space (a uniformly closed [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t1301505.png" />-algebra]]). Such algebras are closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables.
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In the one-dimensional case one considers the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t1301506.png" /> over the one-dimensional torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t1301507.png" /> (cf. also [[Hardy spaces|Hardy spaces]]), and defines the [[Toeplitz operator|Toeplitz operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t1301508.png" /> with "symbol" function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t1301509.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015012.png" /> is the orthogonal projection given by the [[Cauchy integral theorem|Cauchy integral theorem]]. The [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015013.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015014.png" /> generated by all operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015015.png" /> with continuous symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015016.png" /> is not commutative, but defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015017.png" />-algebra extension
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A uniformly closed $*$-algebra of operators on a Hilbert space (a uniformly closed [[C*-algebra|$C ^ { * }$-algebra]]). Such algebras are closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables.
  
of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015020.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015021.png" /> of all compact operators; in fact, this "Toeplitz extension" is the generator of the Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015022.png" />.
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In the one-dimensional case one considers the Hardy space $H ^ { 2 } ( \mathbf{T} )$ over the one-dimensional torus $\bf T$ (cf. also [[Hardy spaces|Hardy spaces]]), and defines the [[Toeplitz operator|Toeplitz operator]] $T _ { f }$ with "symbol" function $f \in L ^ { \infty } ( \mathbf{T} )$ by $T _ { f } h : = P ( f h )$ for all $h \in H ^ { 2 } ( \mathbf{T} )$, where $P : L ^ { 2 } ( \mathbf{T} ) \rightarrow H ^ { 2 } ( \mathbf{T} )$ is the orthogonal projection given by the [[Cauchy integral theorem|Cauchy integral theorem]]. The [[C*-algebra|$C ^ { * }$-algebra]] ${\cal T} ({\bf  T} ) : = C ^ { * } ( T _ { f } : f \in {\cal C} ({\bf T} ) )$ generated by all operators $T _ { f }$ with continuous symbol $f$ is not commutative, but defines a $C ^ { * }$-algebra extension
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015023.png" />-algebra extensions are the building blocks of [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015024.png" />-theory]] and [[Index theory|index theory]]; in our case a Toeplitz operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015025.png" /> is Fredholm (cf. also [[Fredholm-operator(2)|Fredholm operator]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015026.png" /> has no zeros, and then the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015027.png" /> is the (negative) [[Winding number|winding number]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015028.png" />.
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\begin{equation*} 0 \rightarrow \mathcal{K} ( H ^ { 2 } ( \mathbf{T} ) ) \triangleleft  \mathcal{T} ( \mathbf{T} ) \rightarrow \mathcal{C} ( \mathbf{T} ) \rightarrow 0 \end{equation*}
  
In the multi-variable case, Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015029.png" />-algebras have been studied in several important cases, e.g. for strictly pseudo-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015030.png" /> [[#References|[a1]]], including the unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015031.png" /> [[#References|[a2]]], [[#References|[a10]]], for tube domains and Siegel domains over convex "symmetric" cones [[#References|[a5]]], [[#References|[a8]]], and for general bounded symmetric domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015032.png" /> having a transitive semi-simple [[Lie group|Lie group]] of holomorphic automorphisms [[#References|[a7]]]. Here, the principal new feature is the fact that Toeplitz operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015033.png" /> (say, on the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015034.png" /> over the Shilov boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015035.png" /> of a pseudo-convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015036.png" />) with continuous symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015037.png" /> are not essentially commuting, i.e.
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of the $C ^ { * }$-algebra $\mathcal{K}$ of all compact operators; in fact, this "Toeplitz extension" is the generator of the Abelian group $\operatorname{Ext} ( {\cal C } ( {\bf T } ) ) \approx \bf Z$.
  
<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/t/t130/t130150/t13015038.png" /></td> </tr></table>
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$C ^ { * }$-algebra extensions are the building blocks of [[K-theory|$K$-theory]] and [[Index theory|index theory]]; in our case a Toeplitz operator $T _ { f }$ is Fredholm (cf. also [[Fredholm-operator(2)|Fredholm operator]]) if $f \in \mathcal{C} ( \mathbf{T} )$ has no zeros, and then the index $\operatorname {Index}( T _ { f } ) = \operatorname { dim } \operatorname { Ker } T _ { f } - \operatorname { dim } \text { Coker } T _ { f }$ is the (negative) [[Winding number|winding number]] of $f$.
  
in general. Thus, the corresponding Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015039.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015040.png" /> is not a (one-step) extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015041.png" />; instead one obtains a multi-step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015043.png" />-filtration
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In the multi-variable case, Toeplitz $C ^ { * }$-algebras have been studied in several important cases, e.g. for strictly pseudo-convex domains $D \subset \mathbf{C} ^ { x }$ [[#References|[a1]]], including the unit ball $D = \left\{ z \in \mathbf{C} ^ { n } : | z _ { 1 } | ^ { 2 } + \ldots + | z _ { n } | ^ { 2 } &lt; 1 \right\}$ [[#References|[a2]]], [[#References|[a10]]], for tube domains and Siegel domains over convex "symmetric" cones [[#References|[a5]]], [[#References|[a8]]], and for general bounded symmetric domains in $\mathbf{C} ^ { n }$ having a transitive semi-simple [[Lie group|Lie group]] of holomorphic automorphisms [[#References|[a7]]]. Here, the principal new feature is the fact that Toeplitz operators $T _ { f }$ (say, on the Hardy space $H ^ { 2 } ( S )$ over the [[Shilov boundary]] $S$ of a pseudo-convex domain $D \subset \mathbf{C} ^ { x }$) with continuous symbols $f \in \mathcal{C} _ { 0 } ( S )$ are not essentially commuting, i.e.
  
<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/t/t130/t130150/t13015044.png" /></td> </tr></table>
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\begin{equation*} [ T _ { f _ { 1 } } , T _ { f _ { 2 } } ] \notin \mathcal{K} ( H ^ { 2 } ( S ) ), \end{equation*}
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015045.png" />-ideals, with essentially commutative subquotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015046.png" />, whose maximal ideal space (its spectrum) reflects the boundary strata of the underlying domain. The length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015048.png" /> of the composition series is an important geometric invariant, called the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015049.png" />. The index theory and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015050.png" />-theory of these multi-variable Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015051.png" />-algebras is more difficult to study; on the other hand one obtains interesting classes of operators arising by geometric quantization of the underlying domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015052.png" />, regarded as a complex [[Kähler manifold|Kähler manifold]].
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in general. Thus, the corresponding Toeplitz $C ^ { * }$-algebra $\mathcal{T} ( S )$ is not a (one-step) extension of $\mathcal{K}$; instead one obtains a multi-step $C ^ { * }$-filtration
  
A general method for studying the structure and representations of Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015053.png" />-algebras, at least for Shilov boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015054.png" /> arising as a symmetric space (not necessarily Riemannian), is the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015056.png" />-duality [[#References|[a11]]], [[#References|[a9]]]. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015057.png" /> is a [[Lie group|Lie group]] with (reduced) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015058.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015059.png" />, then the so-called co-crossed product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015061.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015062.png" /> induced by a natural co-action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015063.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015064.png" />. Now the Cauchy–Szegö orthogonal projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015065.png" /> (cf. also [[Cauchy operator|Cauchy operator]]) defines a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015066.png" />-completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015067.png" />, and the corresponding Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015068.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015069.png" /> can be realized as (a corner of) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015070.png" />. In this way the well-developed representation theory of (co-) crossed product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015072.png" />-algebras [[#References|[a4]]] can be applied to obtain Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015073.png" />-representations related to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015074.png" />. For example, the two-dimensional torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015075.png" /> gives rise to non-type-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015078.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015079.png" />-algebras (for cones with irrational slopes), and the underlying "Reinhardt" domains (cf. also [[Reinhardt domain|Reinhardt domain]]) have interesting complex-analytic properties, such as a non-compact solution operator of the [[Neumann d-bar problem|Neumann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015080.png" />-problem]] [[#References|[a6]]].
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\begin{equation*} \mathcal{K} =\mathcal{ I} _ { 1 } \lhd \ldots \lhd  \mathcal{ I}_ { r } \lhd  \mathcal{T} ( S ) \end{equation*}
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of $C ^ { * }$-ideals, with essentially commutative subquotients $\mathcal{ I}_ { k + 1 } / \mathcal{I} _ { k }$, whose maximal ideal space (its spectrum) reflects the boundary strata of the underlying domain. The length $r$ of the composition series is an important geometric invariant, called the rank of $D$. The index theory and $K$-theory of these multi-variable Toeplitz $C ^ { * }$-algebras is more difficult to study; on the other hand one obtains interesting classes of operators arising by geometric quantization of the underlying domain $D$, regarded as a complex [[Kähler manifold|Kähler manifold]].
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A general method for studying the structure and representations of Toeplitz $C ^ { * }$-algebras, at least for Shilov boundaries $S$ arising as a symmetric space (not necessarily Riemannian), is the so-called $C ^ { * }$-duality [[#References|[a11]]], [[#References|[a9]]]. For example, if $S$ is a [[Lie group|Lie group]] with (reduced) group $C ^ { * }$-algebra $C ^ { * } ( S )$, then the so-called co-crossed product $C ^ { * }$-algebra $C ^ { * } ( S ) \otimes _ { \delta } \mathcal{C} _ { 0 } ( S )$ induced by a natural co-action $\delta$ can be identified with $\mathcal{K} ( L ^ { 2 } ( S ) )$. Now the Cauchy–Szegö orthogonal projection $E : L ^ { 2 } ( S ) \rightarrow H ^ { 2 } ( S )$ (cf. also [[Cauchy operator|Cauchy operator]]) defines a certain $C ^ { * }$-completion $C ^ { *_ E } ( S ) \supset C ^ { * } ( S )$, and the corresponding Toeplitz $C ^ { * }$-algebra $\mathcal{T} ( S )$ can be realized as (a corner of) $C ^ { * _ E} ( S ) \otimes _ { \delta } \mathcal{C} _ { 0 } ( S )$. In this way the well-developed representation theory of (co-) crossed product $C ^ { * }$-algebras [[#References|[a4]]] can be applied to obtain Toeplitz $C ^ { * }$-representations related to the boundary $\partial D$. For example, the two-dimensional torus $S = \mathbf{T} ^ { 2 }$ gives rise to non-type-$I$ $C ^ { * }$-algebras (for cones with irrational slopes), and the underlying "Reinhardt" domains (cf. also [[Reinhardt domain|Reinhardt domain]]) have interesting complex-analytic properties, such as a non-compact solution operator of the [[Neumann d-bar problem|Neumann $\overline { \partial }$-problem]] [[#References|[a6]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Boutet de Monvel, "On the index of Toeplitz operators of several complex variables" ''Invent. Math.'' , '''50''' (1979) pp. 249–272 {{MR|}} {{ZBL|0398.47018}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L. Coburn, "Singular integral operators and Toeplitz operators on odd spheres" ''Indiana Univ. Math. J.'' , '''23''' (1973) pp. 433–439 {{MR|0322595}} {{ZBL|0271.46052}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Douglas, R. Howe, "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015081.png" />-algebra of Toeplitz operators on the quarter-plane" ''Trans. Amer. Math. Soc.'' , '''158''' (1971) pp. 203–217 {{MR|288591}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Landstad, J. Phillips, I. Raeburn, C. Sutherland, "Representations of crossed products by coactions and principal bundles" ''Trans. Amer. Math. Soc.'' , '''299''' (1987) pp. 747–784 {{MR|0869232}} {{ZBL|0722.46031}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Muhly, J. Renault, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015082.png" />-algebras of multivariable Wiener–Hopf operators" ''Trans. Amer. Math. Soc.'' , '''274''' (1982) pp. 1–44 {{MR|0670916}} {{ZBL|0509.46050}} {{ZBL|0509.46049}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation algebras" ''Ann. Math.'' , '''130''' (1989) pp. 531–565 {{MR|1025166}} {{ZBL|0708.47021}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> H. Upmeier, "Toeplitz <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015083.png" />-algebras on bounded symmetric domains" ''Ann. Math.'' , '''119''' (1984) pp. 549–576 {{MR|744863}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> H. Upmeier, "Toeplitz operators on symmetric Siegel domains" ''Math. Ann.'' , '''271''' (1985) pp. 401–414 {{MR|0787189}} {{ZBL|0565.47016}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> H. Upmeier, "Toeplitz operators and index theory in several complex variables" , Birkhäuser (1996) {{MR|1384981}} {{ZBL|0957.47023}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130150/t13015084.png" />" ''J. Funct. Anal.'' , '''9''' (1972) pp. 349–373 {{MR|0315502}} {{ZBL|0241.47023}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Wassermann, "Algèbres d'opérateurs de Toeplitz sur les groupes unitaires" ''C.R. Acad. Sci. Paris'' , '''299''' (1984) pp. 871–874 {{MR|0777751}} {{ZBL|}} </TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> L. Boutet de Monvel, "On the index of Toeplitz operators of several complex variables" ''Invent. Math.'' , '''50''' (1979) pp. 249–272 {{MR|}} {{ZBL|0398.47018}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> L. Coburn, "Singular integral operators and Toeplitz operators on odd spheres" ''Indiana Univ. Math. J.'' , '''23''' (1973) pp. 433–439 {{MR|0322595}} {{ZBL|0271.46052}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> R. Douglas, R. Howe, "On the $C ^ { * }$-algebra of Toeplitz operators on the quarter-plane" ''Trans. Amer. Math. Soc.'' , '''158''' (1971) pp. 203–217 {{MR|288591}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> M. Landstad, J. Phillips, I. Raeburn, C. Sutherland, "Representations of crossed products by coactions and principal bundles" ''Trans. Amer. Math. Soc.'' , '''299''' (1987) pp. 747–784 {{MR|0869232}} {{ZBL|0722.46031}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> P. Muhly, J. Renault, "$C ^ { * }$-algebras of multivariable Wiener–Hopf operators" ''Trans. Amer. Math. Soc.'' , '''274''' (1982) pp. 1–44 {{MR|0670916}} {{ZBL|0509.46050}} {{ZBL|0509.46049}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation algebras" ''Ann. Math.'' , '''130''' (1989) pp. 531–565 {{MR|1025166}} {{ZBL|0708.47021}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> H. Upmeier, "Toeplitz $C ^ { * }$-algebras on bounded symmetric domains" ''Ann. Math.'' , '''119''' (1984) pp. 549–576 {{MR|744863}} {{ZBL|}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> H. Upmeier, "Toeplitz operators on symmetric Siegel domains" ''Math. Ann.'' , '''271''' (1985) pp. 401–414 {{MR|0787189}} {{ZBL|0565.47016}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> H. Upmeier, "Toeplitz operators and index theory in several complex variables" , Birkhäuser (1996) {{MR|1384981}} {{ZBL|0957.47023}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in $\mathbf{C} ^ { n }$" ''J. Funct. Anal.'' , '''9''' (1972) pp. 349–373 {{MR|0315502}} {{ZBL|0241.47023}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> A. Wassermann, "Algèbres d'opérateurs de Toeplitz sur les groupes unitaires" ''C.R. Acad. Sci. Paris'' , '''299''' (1984) pp. 871–874 {{MR|0777751}} {{ZBL|}} </td></tr></table>

Latest revision as of 16:46, 1 July 2020

A uniformly closed $*$-algebra of operators on a Hilbert space (a uniformly closed $C ^ { * }$-algebra). Such algebras are closely connected to important fields of geometric analysis, e.g., index theory, geometric quantization and several complex variables.

In the one-dimensional case one considers the Hardy space $H ^ { 2 } ( \mathbf{T} )$ over the one-dimensional torus $\bf T$ (cf. also Hardy spaces), and defines the Toeplitz operator $T _ { f }$ with "symbol" function $f \in L ^ { \infty } ( \mathbf{T} )$ by $T _ { f } h : = P ( f h )$ for all $h \in H ^ { 2 } ( \mathbf{T} )$, where $P : L ^ { 2 } ( \mathbf{T} ) \rightarrow H ^ { 2 } ( \mathbf{T} )$ is the orthogonal projection given by the Cauchy integral theorem. The $C ^ { * }$-algebra ${\cal T} ({\bf T} ) : = C ^ { * } ( T _ { f } : f \in {\cal C} ({\bf T} ) )$ generated by all operators $T _ { f }$ with continuous symbol $f$ is not commutative, but defines a $C ^ { * }$-algebra extension

\begin{equation*} 0 \rightarrow \mathcal{K} ( H ^ { 2 } ( \mathbf{T} ) ) \triangleleft \mathcal{T} ( \mathbf{T} ) \rightarrow \mathcal{C} ( \mathbf{T} ) \rightarrow 0 \end{equation*}

of the $C ^ { * }$-algebra $\mathcal{K}$ of all compact operators; in fact, this "Toeplitz extension" is the generator of the Abelian group $\operatorname{Ext} ( {\cal C } ( {\bf T } ) ) \approx \bf Z$.

$C ^ { * }$-algebra extensions are the building blocks of $K$-theory and index theory; in our case a Toeplitz operator $T _ { f }$ is Fredholm (cf. also Fredholm operator) if $f \in \mathcal{C} ( \mathbf{T} )$ has no zeros, and then the index $\operatorname {Index}( T _ { f } ) = \operatorname { dim } \operatorname { Ker } T _ { f } - \operatorname { dim } \text { Coker } T _ { f }$ is the (negative) winding number of $f$.

In the multi-variable case, Toeplitz $C ^ { * }$-algebras have been studied in several important cases, e.g. for strictly pseudo-convex domains $D \subset \mathbf{C} ^ { x }$ [a1], including the unit ball $D = \left\{ z \in \mathbf{C} ^ { n } : | z _ { 1 } | ^ { 2 } + \ldots + | z _ { n } | ^ { 2 } < 1 \right\}$ [a2], [a10], for tube domains and Siegel domains over convex "symmetric" cones [a5], [a8], and for general bounded symmetric domains in $\mathbf{C} ^ { n }$ having a transitive semi-simple Lie group of holomorphic automorphisms [a7]. Here, the principal new feature is the fact that Toeplitz operators $T _ { f }$ (say, on the Hardy space $H ^ { 2 } ( S )$ over the Shilov boundary $S$ of a pseudo-convex domain $D \subset \mathbf{C} ^ { x }$) with continuous symbols $f \in \mathcal{C} _ { 0 } ( S )$ are not essentially commuting, i.e.

\begin{equation*} [ T _ { f _ { 1 } } , T _ { f _ { 2 } } ] \notin \mathcal{K} ( H ^ { 2 } ( S ) ), \end{equation*}

in general. Thus, the corresponding Toeplitz $C ^ { * }$-algebra $\mathcal{T} ( S )$ is not a (one-step) extension of $\mathcal{K}$; instead one obtains a multi-step $C ^ { * }$-filtration

\begin{equation*} \mathcal{K} =\mathcal{ I} _ { 1 } \lhd \ldots \lhd \mathcal{ I}_ { r } \lhd \mathcal{T} ( S ) \end{equation*}

of $C ^ { * }$-ideals, with essentially commutative subquotients $\mathcal{ I}_ { k + 1 } / \mathcal{I} _ { k }$, whose maximal ideal space (its spectrum) reflects the boundary strata of the underlying domain. The length $r$ of the composition series is an important geometric invariant, called the rank of $D$. The index theory and $K$-theory of these multi-variable Toeplitz $C ^ { * }$-algebras is more difficult to study; on the other hand one obtains interesting classes of operators arising by geometric quantization of the underlying domain $D$, regarded as a complex Kähler manifold.

A general method for studying the structure and representations of Toeplitz $C ^ { * }$-algebras, at least for Shilov boundaries $S$ arising as a symmetric space (not necessarily Riemannian), is the so-called $C ^ { * }$-duality [a11], [a9]. For example, if $S$ is a Lie group with (reduced) group $C ^ { * }$-algebra $C ^ { * } ( S )$, then the so-called co-crossed product $C ^ { * }$-algebra $C ^ { * } ( S ) \otimes _ { \delta } \mathcal{C} _ { 0 } ( S )$ induced by a natural co-action $\delta$ can be identified with $\mathcal{K} ( L ^ { 2 } ( S ) )$. Now the Cauchy–Szegö orthogonal projection $E : L ^ { 2 } ( S ) \rightarrow H ^ { 2 } ( S )$ (cf. also Cauchy operator) defines a certain $C ^ { * }$-completion $C ^ { *_ E } ( S ) \supset C ^ { * } ( S )$, and the corresponding Toeplitz $C ^ { * }$-algebra $\mathcal{T} ( S )$ can be realized as (a corner of) $C ^ { * _ E} ( S ) \otimes _ { \delta } \mathcal{C} _ { 0 } ( S )$. In this way the well-developed representation theory of (co-) crossed product $C ^ { * }$-algebras [a4] can be applied to obtain Toeplitz $C ^ { * }$-representations related to the boundary $\partial D$. For example, the two-dimensional torus $S = \mathbf{T} ^ { 2 }$ gives rise to non-type-$I$ $C ^ { * }$-algebras (for cones with irrational slopes), and the underlying "Reinhardt" domains (cf. also Reinhardt domain) have interesting complex-analytic properties, such as a non-compact solution operator of the Neumann $\overline { \partial }$-problem [a6].

References

[a1] L. Boutet de Monvel, "On the index of Toeplitz operators of several complex variables" Invent. Math. , 50 (1979) pp. 249–272 Zbl 0398.47018
[a2] L. Coburn, "Singular integral operators and Toeplitz operators on odd spheres" Indiana Univ. Math. J. , 23 (1973) pp. 433–439 MR0322595 Zbl 0271.46052
[a3] R. Douglas, R. Howe, "On the $C ^ { * }$-algebra of Toeplitz operators on the quarter-plane" Trans. Amer. Math. Soc. , 158 (1971) pp. 203–217 MR288591
[a4] M. Landstad, J. Phillips, I. Raeburn, C. Sutherland, "Representations of crossed products by coactions and principal bundles" Trans. Amer. Math. Soc. , 299 (1987) pp. 747–784 MR0869232 Zbl 0722.46031
[a5] P. Muhly, J. Renault, "$C ^ { * }$-algebras of multivariable Wiener–Hopf operators" Trans. Amer. Math. Soc. , 274 (1982) pp. 1–44 MR0670916 Zbl 0509.46050 Zbl 0509.46049
[a6] N. Salinas, A. Sheu, H. Upmeier, "Toeplitz operators on pseudoconvex domains and foliation algebras" Ann. Math. , 130 (1989) pp. 531–565 MR1025166 Zbl 0708.47021
[a7] H. Upmeier, "Toeplitz $C ^ { * }$-algebras on bounded symmetric domains" Ann. Math. , 119 (1984) pp. 549–576 MR744863
[a8] H. Upmeier, "Toeplitz operators on symmetric Siegel domains" Math. Ann. , 271 (1985) pp. 401–414 MR0787189 Zbl 0565.47016
[a9] H. Upmeier, "Toeplitz operators and index theory in several complex variables" , Birkhäuser (1996) MR1384981 Zbl 0957.47023
[a10] U. Venugopalkrishna, "Fredholm operators associated with strongly pseudoconvex domains in $\mathbf{C} ^ { n }$" J. Funct. Anal. , 9 (1972) pp. 349–373 MR0315502 Zbl 0241.47023
[a11] A. Wassermann, "Algèbres d'opérateurs de Toeplitz sur les groupes unitaires" C.R. Acad. Sci. Paris , 299 (1984) pp. 871–874 MR0777751
How to Cite This Entry:
Toeplitz C*-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_C*-algebra&oldid=24133
This article was adapted from an original article by H. Upmeier (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article