## Date of Defense

16-4-2024 10:00 AM

## Location

F3-036

## Document Type

Dissertation Defense

## College

College of Science

## Department

Mathematical Sciences

## First Advisor

Prof. Ahmed Al Rawashdeh

## Abstract

H. Dye proved that the unitary group in a factor determines the algebraic type of that factor. Al-Rawashdeh, Booth and Giordano established that, for a large class of simple unital C*- algebras, an isomorphism between the unitary groups, induces an isomorphism between their K0-ordered group and K1-groups. Then using the results of Dadarlat-Elliot-Gong and KirchbergPhillips, the C*-algebras are isomorphic. Dye introduced special projections P{i,j}(a) of the matrix algebra Mn(A), and he used it as a main tool to establish his results in the case of von Neumann factors. Precisely, in case of von Neumann algebra, he proved that if is an orthoisomprhism which fixes all the P{i,j}(a), then it is the identity mapping on the projections. We discuss these projections and we give more properties in the case of C*-matrix algebras. Using Dye's approach, we prove that for a unital C*-algebra A, if is an orthoisomorphism on P(Mn(A)) which fixes the P{i,j}(a), then fixes all the projections on class D, consisting of some decomposition of P{i,j}(a). We introduce the invariant unitary groups property (IUG-P), the orthogonal IUG-P and the topological IUG-P. We investigate that some properties are IUG-P, orthogonal IUG-P or topological IUG-P, for certain C*-algebras. If the general linear groups (GL(A)) are isomorphic, we prove that the induced mapping between idempotents preserves the orthogonality, for a large class of unital C*-algebras, including certain type of UHFalgebras, 2-divisible K0-groups, Cuntz algebras On, 2 ≤ n ≤ ∞, and for simple unital purely infinite C*-algebras having 2-divisible K0- groups. We prove that if N is a normal subgroup of GL(On), then N contains all the symmetries of On. Also, we show that if N is any normal subgroup of unitary groups of compact operator K, which contains some certain type of involution, then N contains all the involutions of K.

#### Included in

ON THE PROJECTIONS AND UNITARY GROUPS OF UNITAL C*-ALGEBRAS

F3-036

H. Dye proved that the unitary group in a factor determines the algebraic type of that factor. Al-Rawashdeh, Booth and Giordano established that, for a large class of simple unital C*- algebras, an isomorphism between the unitary groups, induces an isomorphism between their K0-ordered group and K1-groups. Then using the results of Dadarlat-Elliot-Gong and KirchbergPhillips, the C*-algebras are isomorphic. Dye introduced special projections P{i,j}(a) of the matrix algebra Mn(A), and he used it as a main tool to establish his results in the case of von Neumann factors. Precisely, in case of von Neumann algebra, he proved that if is an orthoisomprhism which fixes all the P{i,j}(a), then it is the identity mapping on the projections. We discuss these projections and we give more properties in the case of C*-matrix algebras. Using Dye's approach, we prove that for a unital C*-algebra A, if is an orthoisomorphism on P(Mn(A)) which fixes the P{i,j}(a), then fixes all the projections on class D, consisting of some decomposition of P{i,j}(a). We introduce the invariant unitary groups property (IUG-P), the orthogonal IUG-P and the topological IUG-P. We investigate that some properties are IUG-P, orthogonal IUG-P or topological IUG-P, for certain C*-algebras. If the general linear groups (GL(A)) are isomorphic, we prove that the induced mapping between idempotents preserves the orthogonality, for a large class of unital C*-algebras, including certain type of UHFalgebras, 2-divisible K0-groups, Cuntz algebras On, 2 ≤ n ≤ ∞, and for simple unital purely infinite C*-algebras having 2-divisible K0- groups. We prove that if N is a normal subgroup of GL(On), then N contains all the symmetries of On. Also, we show that if N is any normal subgroup of unitary groups of compact operator K, which contains some certain type of involution, then N contains all the involutions of K.