Let A be a unital C∗-algebra. Denote by P the space of selfadjoint projections ofA. We study the relationship between P and the spaces of projections Pa determined by the different involutions #a induced by positive invertible elements a ∈ A. The maps φp : P → Pa sending p to the unique q ∈ Pa with the same range as p and Ωa : Pa → P sending q to the unitary part of the polar decomposition of t...

and call it the essential weight of the partition μ. For our purpose it is convenient to ignore the parts equal to 1 in the partition because an element like (1, 2, 3) ∈ S3 is also an element of bigger symmetric groups. So we write μ = 22 , ..., nn for a partition and the corresponding class Cμ is a class of an arbitrary symmetric group Sn with n ≥ W (μ) depending on the context, i.e. C2 denote...

Let [Formula: see text] and be two alternative ∗-algebras with identities text], respectively, nontrivial symmetric idempotents in text]. In this paper, we study the characterization of multiplicative ∗-Lie-type maps. As application, get a result on text]-algebras.

Let IK be a complete ultrametric field and let A unital commutative Banach IK-algebra. Suppose that the multiplicative spectrum admits partition in two open closed subsets. Then there exist unique idempotents u, v ϵ such ϕ (u) = 1, (v) 0 ∀ U, 1 V . is algebraically closed. If an element x has empty annulus r < |ξ − a| s its sp(x), then 1; whenever (x a) ≤ 0; ≥ s.

Proof. The idempotents correspond to the Borel sets modulo sets of first category. Since, in addition, the idempotents generate B(X), B(X) is an AW* and hence an injective algebra. The natural map U of C(X) into B(X) induced by the inclusion map is clearly a homomorphism. It is one-one since continuous functions which are not identically equal must differ on a set of second category. To complet...

In this paper, we characterize the idempotency, cleanness, and unit-regularity of commutator $[E_1, E_2]=E_1E_2-E_2E_1$ involving idempotents $E_1,E_2$ in certain subrings $M_{2}(\mathbb{Z})$.