Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...

We say that a C∗-algebra X has the approximate n-th root property (n ≥ 2) if for every a ∈ X with ‖a‖ ≤ 1 and every ε > 0 there exits b ∈ X such that ‖b‖ ≤ 1 and ‖a − bn‖ < ε. Some properties of commutative and non-commutative C∗-algebras having the approximate nth root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unita...

We prove that the Cuntz-Pimsner algebra OE of a vector bundle E of rank ≥ 2 over a compact metrizable space X is determined up to an isomorphism of C(X)algebras by the ideal (1 − [E])K(X) of the K-theory ring K(X). Moreover, if E and F are vector bundles of rank ≥ 2, then a unital embedding of C(X)-algebras OE ⊂ OF exists if and only if 1 − [E] is divisible by 1 − [F ] in the ring K(X). We intr...

Let K be a Banach space, B a unital C∗-algebra, and π : B → L(K) an injective, unital homomorphism. Suppose that there exists a function γ : K×K → R+ such that, for all k, k1, k2 ∈ K, and all b ∈ B, (a) γ(k, k) = ‖k‖2, (b) γ(k1, k2) ≤ ‖k1‖ ‖k2‖, (c) γ(πbk1, k2) = γ(k1, πb∗k2). Then for all b ∈ B, the spectrum of b in B equals the spectrum of πb as a bounded linear operator on K. If γ satisfies ...

Let φ be a linear-fractional self-map of the open unit disk D, not an automorphism, such that φ(ζ) = η for two distinct points ζ, η in the unit circle ∂D. We consider the question of which composition operators lie in C∗(Cφ,K), the unital C∗-algebra generated by the composition operator Cφ and the ideal K of compact operators, acting on the Hardy space H.

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R (M), the idealization of M . Homogeneous ideals of R (M) have the form I (+)N where I is an ideal of R, N a submodule of M and IM ⊆ N . The purpose of this paper is to investigate how properties of a homogeneous ideal I (+)N of R (M) are related to those of I and N . We show that if M is a m...

Abstract. We consider three lifting questions: Given a C∗-algebra I, if there is a unital C∗-algebra A contains I as an ideal, is every unitary from A/I lifted to a unitary in A? is every unitary from A/I lifted to an extremal partial isometry? is every extremal partial isometry from A/I lifted to an extremal partial isometry? We show several constructions of I which serve as working examples o...

A unital C∗–algebra A is said to have the (APD)–property if every nonzero element in A has the approximate polar decomposition. Let J be a closed ideal of A. Suppose that J̃ and A/J have (APD). In this paper, we give a necessary and sufficient condition that makes A have (APD). Furthermore, we show that if RR(J ) = 0 and tsr(A/J ) = 1 or A/J is a simple purely infinite C∗–algebra, then A has (APD).

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for...