Toeplitz C*-algebra

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