Off-diagonal matrix coefficients are tangents to state space: Orientation and C*-algebras

Author:
Martin E. Walter

Journal:
Proc. Amer. Math. Soc. **137** (2009), 2311-2315

MSC (2000):
Primary 46L30, 46L05; Secondary 43A30

Published electronically:
February 18, 2009

MathSciNet review:
2495264

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Any non-commutative C*-algebra , e.g., two by two complex matrices, has at least two associative multiplications for which the collection of positive linear functionals is the same. Alfsen and Shultz have shown that by selecting an orientation for the state space of , i.e., the convex set of positive linear functionals of norm one, a unique associative multiplication for is determined. We give a simple method for describing this orientation.

**1.**Erik M. Alfsen and Frederic W. Shultz,*State spaces of operator algebras*, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. Basic theory, orientations, and 𝐶*-products. MR**1828331****2.**Erik M. Alfsen and Frederic W. Shultz,*Geometry of state spaces of operator algebras*, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2003. MR**1947002****3.**Robert A. Cohen and Martin E. Walter,*An explicit duality for finite groups*, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 87–96. MR**2277205**, 10.1090/conm/414/07801**4.**Martin E. Walter,*Algebraic structures determined by 3 by 3 matrix geometry*, Proc. Amer. Math. Soc.**131**(2003), no. 7, 2129–2131 (electronic). MR**1963763**, 10.1090/S0002-9939-02-06849-1**5.**Martin E. Walter,*Differentiation on the dual of a group: an introduction*, Rocky Mountain J. Math.**12**(1982), no. 3, 497–536. MR**672234**, 10.1216/RMJ-1982-12-3-497

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46L30,
46L05,
43A30

Retrieve articles in all journals with MSC (2000): 46L30, 46L05, 43A30

Additional Information

**Martin E. Walter**

Affiliation:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, Colorado 80309

Email:
walter@euclid.colorado.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-09868-2

Keywords:
C*-algebra,
positive linear functional,
state space,
matrix coefficient,
orientation

Received by editor(s):
May 2, 2008

Published electronically:
February 18, 2009

Communicated by:
Marius Junge

Article copyright:
© Copyright 2009
American Mathematical Society

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