Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Representations of a class of real $B^ *$-algebras as algebras of quaternion-valued functions
HTML articles powered by AMS MathViewer

by S. H. Kulkarni PDF
Proc. Amer. Math. Soc. 116 (1992), 61-66 Request permission

Abstract:

For a compact Hausdorff space $X$, let $C(X,{\mathbf {H}})$ denote the set of all quaternion-valued functions on $X$. It is proved that if a real ${B^*}$-algebra $A$ satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in $A$ is normal, then $A$ is isometrically $*$-isomorphic to a closed $*$-subalgebra of $C(X,{\mathbf {H}})$ for some compact Hausdorff $X$. In particular, a real ${C^*}$-algebra in which every element is normal is isometrically $*$-isomorphic to a closed $*$-subalgebra of $C(X,{\mathbf {H}})$.
References
  • Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
  • K. R. Goodearl, Notes on real and complex ${C^*}$ algebras, Shiva, Birkhauser, Boston, MA, 1982.
  • Lars Ingelstam, Real Banach algebras, Ark. Mat. 5 (1964), 239–270 (1964). MR 172132, DOI 10.1007/BF02591126
  • S. H. Kulkarni and B. V. Limaye, Gel′fand-Naimark theorems for real Banach $\ast$-algebras, Math. Japon. 25 (1980), no. 5, 545–558. MR 602283
  • S. H. Kulkarni, A note on the Shirali-Ford theorem, Math. Student 48 (1980), no. 2-4, 343–347 (1984). MR 776755
  • S. H. Kulkarni, Topological conditions for commutativity of a real Banach algebra, Houston J. Math. 14 (1988), no. 2, 235–245. MR 978730
  • Zoltán Magyar, On commutativity and spectral radius property of real generalized $^*$-algebras, Acta Sci. Math. (Szeged) 53 (1989), no. 3-4, 339–343. MR 1033606
  • Zoltán Magyar, A characterization of (real or complex) Hermitian algebras and equivalent $C^*$-algebras, Acta Sci. Math. (Szeged) 53 (1989), no. 3-4, 345–353. MR 1033607
  • K. Viswanath, Operators on real Hilbert spaces, J. Indian Math. Soc. (N.S.) 42 (1978), no. 1-4, 1–13 (1979). MR 558979
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46K05, 46L05
  • Retrieve articles in all journals with MSC: 46K05, 46L05
Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 116 (1992), 61-66
  • MSC: Primary 46K05; Secondary 46L05
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1110546-4
  • MathSciNet review: 1110546