## On the $C^*$-algebra of Toeplitz operators on the quarterplane

HTML articles powered by AMS MathViewer

- by R. G. Douglas and Roger Howe PDF
- Trans. Amer. Math. Soc.
**158**(1971), 203-217 Request permission

## Abstract:

Using the device of the tensor product of ${C^ \ast }$-algebras, we study the ${C^ \ast }$-algebra generated by the Toeplitz operators on the quarter-plane. We obtain necessary and sufficient conditions for such an operator to be Fredholm, but show in this case that not all such operators are invertible.## References

- Manfred Breuer,
*Fredholm theories in von Neumann algebras. I*, Math. Ann.**178**(1968), 243–254. MR**234294**, DOI 10.1007/BF01350663 - Manfred Breuer,
*Fredholm theories in von Neumann algebras. II*, Math. Ann.**180**(1969), 313–325. MR**264407**, DOI 10.1007/BF01351884 - Henri Cartan and Samuel Eilenberg,
*Homological algebra*, Princeton University Press, Princeton, N. J., 1956. MR**0077480** - L. A. Coburn,
*The $C^{\ast }$-algebra generated by an isometry*, Bull. Amer. Math. Soc.**73**(1967), 722–726. MR**213906**, DOI 10.1090/S0002-9904-1967-11845-7 - L. A. Coburn,
*The $C^{\ast }$-algebra generated by an isometry. II*, Trans. Amer. Math. Soc.**137**(1969), 211–217. MR**236720**, DOI 10.1090/S0002-9947-1969-0236720-9 - L. A. Coburn and R. G. Douglas,
*Translation operators on the half-line*, Proc. Nat. Acad. Sci. U.S.A.**62**(1969), 1010–1013. MR**275228**, DOI 10.1073/pnas.62.4.1010 - L. A. Coburn and R. G. Douglas,
*$C^{\ast }$-algebras of operators on a half-space. I*, Inst. Hautes Études Sci. Publ. Math.**40**(1971), 59–67. MR**358417**, DOI 10.1007/BF02684693 - L. A. Coburn, R. G. Douglas, D. G. Schaeffer, and I. M. Singer,
*$C^{\ast }$-algebras of operators on a half-space. II. Index theory*, Inst. Hautes Études Sci. Publ. Math.**40**(1971), 69–79. MR**358418**, DOI 10.1007/BF02684694 - Jacques Dixmier,
*Les $C^{\ast }$-algèbres et leurs représentations*, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR**0171173** - R. G. Douglas,
*On the spectrum of a class of Toeplitz operators*, J. Math. Mech.**18**(1968/1969), 433–435. MR**0239442**, DOI 10.1512/iumj.1969.18.18034 - R. G. Douglas,
*On the spectrum of Toeplitz and Wiener-Hopf operators*, Abstract Spaces and Approximation (Proc. Conf., Oberwolfach, 1968) Birkhäuser, Basel, 1969, pp. 53–66. MR**0259638** - I. C. Gohberg and M. G. Kreĭn,
*Systems of integral equations on a half line with kernels depending on the difference of arguments*, Amer. Math. Soc. Transl. (2)**14**(1960), 217–287. MR**0113114**, DOI 10.1090/trans2/014/09 - L. S. Gol′denšteĭn,
*Criteria for one-sided invertibility of functions of several isometric operators and their applications*, Dokl. Akad. Nauk SSSR**155**(1964), 28–31 (Russian). MR**0161195** - L. S. Gol′denšteĭn and I. C. Gohberg,
*On a multidimensional integral equation on a half-space whose kernel is a function of the difference of the arguments, and on a discrete analogue of this equation*, Soviet Math. Dokl.**1**(1960), 173–176. MR**0117519** - Ulf Grenander and Gabor Szegö,
*Toeplitz forms and their applications*, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR**0094840** - Paul R. Halmos,
*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368**
M. Kac, - M. G. Kreĭn,
*Integral equations on the half-line with a kernel depending on the difference of the arguments*, Uspehi Mat. Nauk**13**(1958), no. 5 (83), 3–120 (Russian). MR**0102721** - V. A. Malyšev,
*The solution of the discrete Wiener-Hopf equations in a quarter-plane.*, Dokl. Akad. Nauk SSSR**187**(1969), 1243–1246 (Russian). MR**0262858** - Stanley Osher,
*Systems of difference equations with general homogeneous boundary conditions*, Trans. Amer. Math. Soc.**137**(1969), 177–201. MR**237982**, DOI 10.1090/S0002-9947-1969-0237982-4 - Stanley J. Osher,
*On certain Toeplitz operators in two variables*, Pacific J. Math.**34**(1970), 123–129. MR**267408**, DOI 10.2140/pjm.1970.34.123 - Walter Rudin,
*Function theory in polydiscs*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0255841** - I. B. Simonenko,
*Convolution type operators in cones*, Mat. Sb. (N.S.)**74 (116)**(1967), 298–313 (Russian). MR**0222723** - Masamichi Takesaki,
*On the cross-norm of the direct product of $C^{\ast }$-algebras*, Tohoku Math. J. (2)**16**(1964), 111–122. MR**165384**, DOI 10.2748/tmj/1178243737

*Theory and applications of Toeplitz forms*, Summer Institute on Spectral Theory and Statistical Mechanics, Brookhaven National Laboratories, 1965.

## Additional Information

- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**158**(1971), 203-217 - MSC: Primary 46.65
- DOI: https://doi.org/10.1090/S0002-9947-1971-0288591-1
- MathSciNet review: 0288591