International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 78981, 8 pages
doi:10.1155/IJMMS/2006/78981
Decompositions of a C-algebra
Department of Mathematics, Andhra University, Visakhapatnam 530 003, India
Received 12 April 2005; Revised 12 December 2005; Accepted 18 December 2005
Copyright © 2006 G. C. Rao and P. Sundarayya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We prove that if A is a C-algebra, then for each a∈A, Aa={x∈A/x≤a} is itself a C-algebra and is isomorphic to the quotient algebra A/θa of A where θa={(x,y)∈A×A/a∧x=a∧y}. If A is C-algebra with T, we prove that for every a∈B(A), the centre of A, A is isomorphic to Aa×Aa′ and that if A is isomorphic A1×A2, then there exists a∈B(A) such that A1 is isomorphic Aa and A2 is isomorphic to Aa′. Using this decomposition theorem, we prove that if a,b∈B(A) with a∧b=F, then Aa is isomorphic to Ab if and only if there exists an isomorphism φ on A such that φ(a)=b.