Unitary Groups As A Complete Invariant
by
Ahmed Al-Rawashdeh
Jordan University of Science and Technology, Department of Mathematics and Statistic, Irbid 22110, Jordan.s
Coauthors: Thierry Giordano

In 1954, Dye proved that the unitary groups of von Neumann factors not of type I_2n determine the algebraic type of factors. Using Dye's result, Broise showed that any isomorphism between the unitary groups of two von Neumann factors not of type I_n is implemented by a linear or a conjugate linear *-isomorphism between the factors. Using Dye's approach, we prove that the unitary groups determine the algebraic types for a large class of simple, unital C^*-algebras such as the tracial topological rank zero (TAF-algebras) whose K₁ groups are isomorphic and a large class of simple, unital purely infinite nuclear C^*-algebras. Indeed, If j is an isomorphism between the unitary groups of such C^*-algebras (as above, including the irrational rotation algebras and the simple unital AF-algebras, the Cuntz algebras), then it induces a bijection between the sets of projections which preserves the orthogonality and the unitarily equivalence of projections, afterwards this mapping induces an isomorphism between their ordered K₀-groups.

Date received: March 9, 2008