TY - JOUR
T1 - Non-isomorphic C*-algebras with isomorphic unitary groups
AU - Al-Rawashdeh, Ahmed
N1 - Funding Information:
The author would like to thank Professor Thierry Giordano from the University of Ottawa-Canada, for his guidance and help. Also, the author would like to thank the referee for the valuable comments and suggestions.
Publisher Copyright:
© 2016 by the Tusi Mathematical Research Group.
PY - 2016/9/1
Y1 - 2016/9/1
N2 - Dye, [Ann. of Math. (2) 61 (1955), 73-89] proved that the discrete unitary group in a factor determines the algebraic type of the factor. Afterwards, for a large class of simple unital C*-algebras, Al-Rawashdeh, Booth and Giordano [J. Funct. Anal. 262 (2012), 4711-4730] proved that the algebras are *-isomorphic if and only if their unitary groups are isomorphic as abstract groups. In this paper, we give a counterexample in the non-simple case. Indeed, we give two C*-algebras with isomorphic unitary groups but the algebras themselves are not *-isomorphic.
AB - Dye, [Ann. of Math. (2) 61 (1955), 73-89] proved that the discrete unitary group in a factor determines the algebraic type of the factor. Afterwards, for a large class of simple unital C*-algebras, Al-Rawashdeh, Booth and Giordano [J. Funct. Anal. 262 (2012), 4711-4730] proved that the algebras are *-isomorphic if and only if their unitary groups are isomorphic as abstract groups. In this paper, we give a counterexample in the non-simple case. Indeed, we give two C*-algebras with isomorphic unitary groups but the algebras themselves are not *-isomorphic.
KW - -isomorphism
KW - C-algebra
KW - Unitary group
UR - http://www.scopus.com/inward/record.url?scp=85052651953&partnerID=8YFLogxK
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U2 - 10.22034/aot.1609.1004
DO - 10.22034/aot.1609.1004
M3 - Article
AN - SCOPUS:85052651953
SN - 2538-225X
VL - 1
SP - 160
EP - 163
JO - Advances in Operator Theory
JF - Advances in Operator Theory
IS - 2
ER -