## Some $C^{\ast }$-alegebras with a single generator

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- by Catherine L. Olsen and William R. Zame
- Trans. Amer. Math. Soc.
**215**(1976), 205-217 - DOI: https://doi.org/10.1090/S0002-9947-1976-0388114-7
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## Abstract:

This paper grew out of the following question: If*X*is a compact subset of ${C^n}$, is $C(X) \otimes {{\mathbf {M}}_n}$ (the ${C^\ast }$-algebra of $n \times n$ matrices with entries from $C(X)$) singly generated? It is shown that the answer is affirmative; in fact, $A \otimes {{\mathbf {M}}_n}$ is singly generated whenever

*A*is a ${C^\ast }$-algebra with identity, generated by a set of $n(n + 1)/2$ elements of which $n(n - 1)/2$ are selfadjoint. If

*A*is a separable ${C^\ast }$-algebra with identity, then $A \otimes K$ and $A \otimes U$ are shown to be singly generated, where

*K*is the algebra of compact operators in a separable, infinite-dimensional Hilbert space, and

*U*is any UHF algebra. In all these cases, the generator is explicitly constructed.

## References

- J. Brenner,
*The problem of unitary equivalence*, Acta Math.**86**(1951), 297–308. MR**46334**, DOI 10.1007/BF02392670
J. Bunce and J. Deddens, - Chandler Davis,
*Generators of the ring of bounded operators*, Proc. Amer. Math. Soc.**6**(1955), 907–972. MR**73138**, DOI 10.1090/S0002-9939-1955-0073138-1 - Jacques Dixmier,
*Les $C^{\ast }$-algèbres et leurs représentations*, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR**0246136** - R. G. Douglas and Carl Pearcy,
*Von Neumann algebras with a single generator*, Michigan Math. J.**16**(1969), 21–26. MR**244775**, DOI 10.1307/mmj/1029000161 - Theodore W. Gamelin,
*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR**0410387** - James G. Glimm,
*On a certain class of operator algebras*, Trans. Amer. Math. Soc.**95**(1960), 318–340. MR**112057**, DOI 10.1090/S0002-9947-1960-0112057-5 - Carl Pearcy,
*$W^*$-algebras with a single generator*, Proc. Amer. Math. Soc.**13**(1962), 831–832. MR**152904**, DOI 10.1090/S0002-9939-1962-0152904-1 - Carl Pearcy,
*On certain von Neumann algebras which are generated by partial isometries*, Proc. Amer. Math. Soc.**15**(1964), 393–395. MR**161172**, DOI 10.1090/S0002-9939-1964-0161172-8 - Carl Pearcy and David Topping,
*Sums of small numbers of idempotents*, Michigan Math. J.**14**(1967), 453–465. MR**218922** - Shôichirô Sakai,
*$C^*$-algebras and $W^*$-algebras*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR**0442701** - Teishirô Saitô,
*On generators of von Neumann algebras*, Michigan Math. J.**15**(1968), 373–376. MR**236724** - Noboru Suzuki and Teishirô Saitô,
*On the operators which generate continuous von Neumann algebras*, Tohoku Math. J. (2)**15**(1963), 277–280. MR**154143**, DOI 10.2748/tmj/1178243811 - David M. Topping,
*$\textrm {UHF}$ algebras are singly generated*, Math. Scand.**22**(1968), 224–226 (1969). MR**244783**, DOI 10.7146/math.scand.a-10886
—, - Warren Wogen,
*On generators for von Neumann algebras*, Bull. Amer. Math. Soc.**75**(1969), 95–99. MR**236725**, DOI 10.1090/S0002-9904-1969-12157-9

*A family of simple*${C^\ast }$-

*algebras related to weighted shift operators*(preprint).

*Lectures on von Neumann algebras*, Van Nostrand, Princeton, N. J., 1971.

## Bibliographic Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**215**(1976), 205-217 - MSC: Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9947-1976-0388114-7
- MathSciNet review: 0388114