The groupoid algebra of an eigenvalue pattern

Author:
Kin-Wai Tsang

Journal:
Proc. Amer. Math. Soc. **134** (2006), 1899-1908

MSC (2000):
Primary 46L05; Secondary 46L35

Published electronically:
January 17, 2006

MathSciNet review:
2215117

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The eigenvalue pattern of a *-homomorphism between two matrix algebras over commutative C-algebras is a generalization of the Gelfand map in the commutative case. We give a systematic formulation of abstract eigenvalue pattern and extend the classical results by using a technique involving the groupoid algebras of eigenvalue patterns. In the case with matrix algebras over the one-dimensional circle, we characterize all the *-homomorphisms up to unitary equivalence by their eigenvalue patterns. Moreover, this technique has an application to recent classification theorems of C-algebras proved by the present author.

**[Br]**Ola Bratteli,*Inductive limits of finite dimensional 𝐶*-algebras*, Trans. Amer. Math. Soc.**171**(1972), 195–234. MR**0312282**, 10.1090/S0002-9947-1972-0312282-2**[EHS]**Edward G. Effros, David E. Handelman, and Chao Liang Shen,*Dimension groups and their affine representations*, Amer. J. Math.**102**(1980), no. 2, 385–407. MR**564479**, 10.2307/2374244**[Ell1]**George A. Elliott,*On the classification of 𝐶*-algebras of real rank zero*, J. Reine Angew. Math.**443**(1993), 179–219. MR**1241132**, 10.1515/crll.1993.443.179**[Ell2]**George A. Elliott,*An invariant for simple 𝐶*-algebras*, Canadian Mathematical Society. 1945–1995, Vol. 3, Canadian Math. Soc., Ottawa, ON, 1996, pp. 61–90 (English, with English and French summaries). MR**1661611****[EV]**George A. Elliott and Jesper Villadsen,*Perforated ordered 𝐾₀-groups*, Canad. J. Math.**52**(2000), no. 6, 1164–1191. MR**1794301**, 10.4153/CJM-2000-049-9**[Eva]**David E. Evans,*Quasiproduct states on 𝐶*-algebras*, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 129–151. MR**799567**, 10.1007/BFb0074883**[Go]**Guihua Gong,*On the classification of simple inductive limit 𝐶*-algebras. I. The reduction theorem*, Doc. Math.**7**(2002), 255–461 (electronic). MR**2014489****[Ra]**Shaloub Razak,*On the classification of simple stably projectionless 𝐶*-algebras*, Canad. J. Math.**54**(2002), no. 1, 138–224. MR**1880962**, 10.4153/CJM-2002-006-7**[Re]**Jean Renault,*A groupoid approach to 𝐶*-algebras*, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR**584266****[Ts1]**K. W. Tsang, A classification of certian simple stably projectionless C-algebras, submitted.**[Ts2]**K. W. Tsang, On the positive tracial cones of simple stably projectionless C-algebras, to appear in*J. Funct. Anal.*.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L05,
46L35

Retrieve articles in all journals with MSC (2000): 46L05, 46L35

Additional Information

**Kin-Wai Tsang**

Affiliation:
Department of Mathematics, University of Toronto, 100 St. George Street, Tor- onto, Ontario, Canada M5S 3G3

Address at time of publication:
Department of Mathematics, D3-2/F-09, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, Hong Kong

Email:
tsang@math.toronto.edu, kwtsang@ied.edu.hk

DOI:
https://doi.org/10.1090/S0002-9939-06-08215-3

Keywords:
Groupoid algebra,
path space,
singular eigenvalue pattern,
Gelfand map

Received by editor(s):
June 1, 2003

Received by editor(s) in revised form:
February 1, 2005

Published electronically:
January 17, 2006

Communicated by:
David R. Larson

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.