in this paper, we aresupposed to introduce the definitions of n-fold commutative, andimplicative hyper k-ideals. these definitions are thegeneralizations of the definitions of commutative, andimplicative hyper k-ideals, respectively, which have been definedin [12]. then we obtain some related results. in particular wedetermine the relationships between n-fold implicative hyperk-ideal and n-fold...

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.

We define the angle of a bounded linear operator A along curve emanating from origin and use it to characterize range-kernel complementarity. In particular we show that if $$\sigma (A)$$ does not separate 0 $$\infty $$ , then $$X=R(A)\oplus N(A)$$ only R(A) is closed some less than $$\pi . first apply this result invertible operators have spectral set Next extend notion Banach algebras prove tw...

we shall prove an existence inequality for two maps on banach algebra, withan example and in sequel we have some results on r and rn spaces. this waycan be applied for generalization of some subjects of mathematics in teachingwhich how we can extend a math problem to higher level.

Let *3* be the algebra of polynomials in one indeterminate x over the complex field C. Suppose || • || is a norm on 9 such that the coefficient functionals c,: X a.oc' —* a, (j = 0,1,2, • • •) are all continuous with respect to || • ||, and let K CC be the set of characters on 9> which are || • ||-continuous. Then K is compact, C\K is connected, and 0 G K. Let A be the completion of 9 with resp...

For given triangular real Lie algebra $\mathfrak{g}$ we construct a certain completion $C^\infty(\mathfrak{g})$ of its universal enveloping algebra. It is Frechet-Arens-Michael algebra, which consists elements polynomial growth and satisfies to the following property: every homomorphism from Banach all whose are has an extension continuous $C^\infty(\mathfrak{g})$. Elements this can be called f...

let $a$ be a $c^*$-algebra and $e$ be a left hilbert $a$-module. in this paper we define a product on $e$ that making it into a banach algebra and show that under the certain conditions $e$  is arens regular. we also study the relationship between derivations of $a$ and $e$.

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

in the present paper an operator-differential equation of second order in complex domain isconsidered when the coefficients have singularity of pole type at the point z=0. a theorem of existence of thesolution of the equation is proved and the spectral property of the solution is separately investigated when thecoefficients are spectral operators.