arXiv subjects
Publications and source records attributed to Ali Taghavi.
Let $\mathcal{A}$ and $\mathcal{B}$ be two prime $C^{*}$-algebras. In this paper, we investigate the additivity of map $Φ$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$Φ(AP+λPA^{*})=Φ(A)Φ(P)+λΦ(P)Φ(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$ and $λ\in\{-1,+1\}$. Then, $Φ$ is $*$-additive.
Let X and Y be Banach spaces with dim X greater than 3. Let A and B be standard operator algebras on X and Y. We characterize the form of maps from A onto B such that completely preserve involution.
Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let $ϕ:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and $λ\neq 1$ be an nth root of the unity, then for every complex valued continuous function $f$ on $S^{2}$ the function $\sum_{i=0}^{n-1} λ^{i}f(ϕ^{i}(x))$ must be vanished at some point of $S^{2}$. We give a generalization in term of action of compact groups. We also discuss about some noncommutative versions of the Borsuk- Ulam theorem
Let H and K be infinite dimensional Hilbert spaces, while B(H) and B(K) denote the algebras of all linear bounded operators on H and K, respectively. We characterize the forms of additive mappings from B(H) into B(K) that preserve the nonzero idempotency of either Jordan products of operators or usual products of operators in both directions.
Let H be a complex Hilbert space, B(H) and S(H) be the spaces of all bounded operators and all self-adjoint operators on H, respectively. We give the concrete forms of the maps on B(H) and also S(H) which preserve the spectrum of certain products of operators.
In this paper, we introduce the concept of inner product on weak hypervector spaces and prove some results about them.
Let B(X) be the algebra of all bounded linear operators on a complex Banach space X with dim X greater than 3. In this paper, we characterize the forms of surjective linear maps on B(X) which preserve the dimension of the vector space containing of all fixed points of operators, whenever X is a fifinite dimensional Banach space. Moreover, we characterize the forms of linear maps on B(X) which preserve the vector space containing of all fixed points of operators.
Let $X$ be a complex Banach space with $\dim X\geq3$ and $B(X)$ the algebra of all bounded linear operators on $X$. Suppose $ϕ:B(X)\longrightarrow B(X)$ is a surjective map satisfying the following property: $Fix(AB)=Fix(ϕ(A)ϕ(B)), (A, B\in B(X))$. Then the form of $ϕ$ is characterized, where $Fix(T)$ is the set of all fixed points of an operator $T$.
We study some properies of $Z^{*}$ algebras, thos C^* algebra which all positive elements are zero divisors. We show by means of an example that an extension of a Z* algebra by a Z* algebra is not necessarily Z* algebra. However we prove that an extension of a non Z* algebra by a non Z* algebra is again a Z^* algebra. As an application of our methods, we prove that evey compact subset of the positive cones of a C* algebra has an upper bound in the algebra.
In this paper we concern with positive zero divisors in $C^{*}$ algebras. By means of zero divisors, we introduce a hereditary invariant for $C^{*}$ algebras. Using this invariant, we give an example of a $C^{*}$ algebra $A$ and a $C^{*}$ sub algebra $B$ of $A$ such that there is no a hereditary imbedding of $B$ into $A$. We also introduce a new concept zero divisor real rank of a $C^{*}$ algebra, as a zero divisor analogy of real rank theory of $C^{*}$ algebras. We observe that this quantity is zero for $A=C(X)$ when $X$ is a separable compact Hausdorff space or $X$ is homeomorphic to the unit square with the lexicographic topology. To a $C^{*}$ algebra $A$ with $\dim A > 1$, we assign the undirected graph $Γ^{+} (A)$ of non zero positive zero divisors. For the Calkin algebra $A=B(H)/K(H)$, we show that $Γ^{+}(A)$ is a connected graph and diam $Γ^{+}(A)= 3$. We show that $Γ^{+}(A)$ is a connected graph with $\text {diam}\; Γ^{+}(A)\leq 6$, if $A$ is a factor.
A consequence of the Gauss Bonnet theorem is interpreted in term of operator theory by Alain Connes in his book, Non Commutative geometry. In this note we explain in details about his method. We also introduce an operator theoretical nature for limit cycle theory.
For a monic polynomial p(z) with coefficients in a unital complex Banach algebra, we prove that there exist a complex number z such that p(z)is not invertible
We give a proof for the fundamental theorem of algebra,using the Fredholm index phenomena
We Prove That The Uniform Upper Bound for the Number Of Limit Cycles Of The Lienard Equation of Degree 4 Can be equals to 2. Further We Suggest to Embedding Planar Lienard Equations In Higher Dimension and Present question of completly integrabiolity of Corresponding Hamiltonians. Actually this Question Suggest To Look at to Limit Cycles Problems with KAM methods
We prove that coexistence of two limit cycles with disjoint interior can not occur in certain quadratic systems
We show that the number of limit cycles of Lienard vector field L is equal to the index of first order linear PDE defined by L.