Factorization of EP elements in C*-algebras

We offer some extensions to C*-algebra elements of factorization properties of EP operators on a Hilbert space.     

Projectionless C*-algebras

We give a sufficient condition for a unital C*-algebra to have no nontrivial projections, and we apply this result to known examples and to free products. We also show how questions of existence of projections relate to the norm-connectedness of certain sets of operators.

     

Ring C*-algebras

We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples. Originally, our motivation comes from algebraic number theory.     

Linear maps preserving Fredholm and Atkinson elements of C *-algebras

Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.     

AW*-algebras Which are Enveloping C*-algebras of JC-algebras

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$ , $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I₂, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$ , i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

Noncommutative Jordan C*-algebras

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.     

Inner ideals in C*-algebras

Research supported by a grant from the Schweizerische Nationalfonds/Fonds national suisse     

Separably injective C*-algebras

We show that a C*-algebra is a 1-separably injective Banach space if and only if it is linearly isometric to the Banach space \({C_0(\Omega)}\) of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space \({\Omega}\).     

Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K₀-theory), that to a ring R associates the monoid V(?R?) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor:

1. There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V °?Γ?? id.
2. There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V °?Γ?? id.
3. There is a {0,1}³-indexed commutative diagram  ${\vec{D}}$ of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0.

By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality  $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(?R?) is the positive cone of a dimension group but it is not isomorphic to V(?B?) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.     

Topologically injective C*-algebras

A criterion for the topological injectivity of an AW*-algebra as a right Banach module over itself is given. A necessary condition for a C* -algebra to be topologically injective is obtained.     

Completely positive maps and*-isomorphism ofC *-algebras II

LetA, B be unitalC ^*-algebras,D _A ¹ the set of all completely positive maps ϕ fromA toM _n (C), with Tr ϕ(I)≤1(n≥3). If Ψ is an α-invariant affine homeomorphism betweenD _A ¹ andD _B ¹ with Ψ (0)=0, thenA is^*-isomorphic toB. Obtained results can be viewed as non-commutative Kadison-Shultz theorems. This work is supported by the National Natural Science Foundation of China.