Let A be a unital complex semisimple Banach algebra, and MA denote its maximal ideal space. For matrix M∈An×n, Mˆ denotes the obtained by taking entry-wise Gelfand transforms. M∈Cn×n, σ(M)⊂C set of eigenvalues M. It is shown that if A∈An×n B∈Am×m are such for all φ∈MA, σ(Aˆ(φ))∩σ(Bˆ(φ))=∅, then C∈An×m, Sylvester equation AX−XB=C has unique solution X∈An×m. As an application, Roth's removal rule...

Let R be a commutative ring with non-zero identity, S multiplicatively closed subset of and M unital R-module. In this paper, we define submodule N (N :R M) ? = to weakly S-primary if there exists s such that whenever m 0 , am N, then either sa ??(N or sm N. We present various properties characterizations concept (especially in faithful multiplication modules). Moreover, the behavior structure ...

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 . Homogenous ideals of R(M) have the form I (+)N , where I is an ideal of R and N a submodule of M such that IM ⊆ N . A ring R (M) is called a homogeneous ring if every ideal of R (M) is homogeneous. In this paper we continue our recent work on the idealization of m...

Guided by the construction of the unital *-algebra of closed operators which are ‘affiliated’ with a given unital *-algebra, we introduce the notion of the ‘monotone closure’ for certain increasing sequences of unital *-algebras. Then we study an example of a unital *-algebra F0(A) constructed from a countable number of copies of a unital *-algebra A. We endow F0(A) with a quantum semigroup str...

We study the range of a classifiable class A of unital separable simple amenable C∗algebras which satisfy the Universal Coefficient Theorem. The class A contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle C∗-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable C∗-alg...

We show that morphisms from n homotopy unital A∞-algebras to a single one are maps over an operad module with n + 1 commuting actions of the operad A ∞, whose algebras are homotopy unital A∞-algebras. The operad A∞ and modules over it have two useful gradings related by isomorphisms which change the degree. The composition of A ∞-morphisms with several entries is presented as a convolution of a...

Let G be the group of projectivities stabilizing a unital U in PG(2, q2). In this paper, we prove that U is a classical unital if and only if there are two points in U such that the stabilizer of these two points in G has order q2 − 1.

Let A be a simple separable unital C*-algebra satisfying the UCT, and assume that A has finite decomposition rank. Let Q denote the UHF algebra with K0(Q) = Q. Then A⊗Q can be tracially approximated by unital Elliott-Thomsen algebras, and therefore A ⊗ Z is an ASH algebra (hence classifiable), where Z is the Jiang-Su algebra.