Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Non-commutative disc algebras and their representations

Author: Gelu Popescu
Journal: Proc. Amer. Math. Soc. 124 (1996), 2137-2148
MSC (1991): Primary 47D25; Secondary 47A67
MathSciNet review: 1343719
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the smallest closed subalgebra

\begin{equation*}Alg(I_{ \mathcal {K}} ,V_{1},\dots ,V_{n})\subset \mathcal {B} (\mathcal {K}) \qquad (n=2,3,\dots ,\infty )\end{equation*}

generated by any sequence $V_{1},\dots , V_{n}$ of isometries on a Hilbert space $\mathcal {K}$ such that $V_{1}V_{1}^{*}+\cdots +V_{n}V_{n}^{*}\le I_{\mathcal {K}} $ is completely isometrically isomorphic to the non-commutative ``disc'' algebra $\mathcal {A} _{n}$ introduced in Math. Scand. 68 (1991), 292--304. We also prove that for $n\ne m$ the Banach algebras $\mathcal {A} _{n}$ and $\mathcal {A} _{m}$ are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the ``disc'' algebras $\mathcal {A} _{n} \ (n=2,3,\dots ,\infty )$ on a Hilbert space are characterized. In particular, we prove that a sequence of operators $A_{1},A_{2},\dots $ is simultaneously similar to a contractive sequence $T_{1},T_{2},\dots $ (i.e., $T_{1}T_{1}^{*}+\cdots +T_{n}T_{n}^{*} \le I$ ) if and only if it is completely polynomially bounded. The first cohomology group of $\mathcal {A} _{n}$ with coefficients in $\mathbb {C}$ is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras $F_{n}^{\infty }$ introduced in Math. Scand. 68 (1991), 292--304. The right joint spectrum of the left creation operators on the full Fock space is also determined.

References [Enhancements On Off] (What's this?)

  • 1. A. Arias, Completely bounded isomorphisms of operator algebras, Proc.Amer.Math.Soc. 124 (1966), 1091--1101.
  • 2. W.B. Arveson, Subalgebras of $C^{*}$-algebras, Acta.Math. 123 (1969), 141--224. MR 40:6274
  • 3. F.F. Bonsall and J. Duncan, Complete normed algebras, Springer--Verlag, New York Heidelberg Berlin (1973). MR 54:11013
  • 4. J.W. Bunce, The joint spectrum of commuting nonnormal operators, Proc. Amer. Math.
    Soc. 29 (1971), 449--505. MR 44:832
  • 5. J.W. Bunce, Models for n-tuples of noncommuting operators, J.Funct.Anal. 57 (1984), 21--30. MR 85k:47019
  • 6. L.A. Coburn, The $C^{*}$--algebra generated by an isometry, Bull.AMS 73 (1967), 722--726. MR 35:4760
  • 7. J. Cuntz, Simple $C^{*}$--algebras generated by isometries, Commun.Math.Phys. 57 (1977), 173--185. MR 57:7189
  • 8. J. Dixmier, $C^{*}$-algebras, North-Holland, Amsterdam (1977). MR 56:16388
  • 9. D.E. Evans, On $O_{n}$, Publ.Res.Int.Math.Sci. 16 (1980), 915--927. MR 82g:46099
  • 10. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press (1941). MR 3:312b
  • 11. W.L. Paschke and N. Salinas,, Matrix algebras over $\mathcal O _{n}$, Michigan Math.J. 26 (1979), 3--12. MR 81a:46075
  • 12. V.I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J.Funct.Anal. 55 (1984), 1--17. MR 86c:47021
  • 13. V.I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics,Vol.146, New York, 1986. MR 88h:46111
  • 14. M. Pimsner and S. Popa, The Ext-groups of some $C^{*}$-algebras considered by J. Cuntz, Rev.Roumaine Math.Pures Appl. 23 (1978), 1069--1076. MR 81j:46094
  • 15. G. Popescu, Models for infinite sequences of noncommuting operators, Acta.Sci. Math. 53 (1989), 355--368. MR 91b:47025
  • 16. G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer.Math.Soc. 316 (1989), 523--536. MR 90c:47006
  • 17. G. Popescu, Von Neumann inequality for $(B(H)^{n})_{1}$, Math.Scand. 68 (1991), 292--304. MR 92k:47073
  • 18. G. Popescu, Multi--analytic operators on Fock spaces, Math.Ann. 303 (1995), 31--46. CMP 95:17
  • 19. G. Popescu, Functional calculus for noncommuting operators, Michigan Math.J. 42
    (1995), 345--356. CMP 95:15
  • 20. W.F. Stinespring, Positive functions on $C^{*}$-algebras, Proc.Amer.Math.Soc. 6 (1955). MR 16:1033b
  • 21. B.Sz.-Nagy, C. Foias, Harmonic analysis on operators on Hilbert space, North--Holland, Amsterdam (1970). MR 43:947
  • 22. J. von Neumann, Eine Spectraltheorie für allgemeine Operatoren eines unitären Raumes, Math.Nachr. 4 (1951), 258--281. MR 13:254a

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47D25, 47A67

Retrieve articles in all journals with MSC (1991): 47D25, 47A67

Additional Information

Gelu Popescu
Affiliation: Division of Mathematics, Computer Science and Statistics, The University of Texas at San Antonio, San Antonio, Texas 78249

Received by editor(s): January 30, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society