arXiv subjects
Publications and source records attributed to Mohammad Sal Moslehian.
Let $\mathscr{A}$ be a unital $C^*$-algebra and let $Φ: \mathscr{A} \to {\mathbb B}({\mathscr H})$ be a unital $n$-positive linear map between $C^*$-algebras for some $n \geq 3$. We show that $$\|Φ(AB)-Φ(A)Φ(B)\| \leq Δ(A,||\cdot||)\,Δ(B,||\cdot||)$$ for all operators $A, B \in \mathscr{A}$, where $Δ(C,\|\cdot\|)$ denotes the operator norm distance of $C$ from the scalar operators.
We prove two new reverse Cauchy--Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C*-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy--Schwarz inequality due to G Polya and G. Szego.
We give an extension of Hua's inequality in pre-Hilbert $C^*$-modules without using convexity or the classical Hua's inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert $C^*$-modules, another extension of Hua's inequality is obtained. We also present an operator Hua's inequality, which is equivalent to operator convexity of given continuous real function.
We define the notion of $φ$-perturbation of a densely defined adjointable mapping and prove that any such mapping $f$ between Hilbert ${\mathcal A}$-modules over a fixed $C^*$-algebra ${\mathcal A}$ with densely defined corresponding mapping $g$ is ${\mathcal A}$-linear and adjointable in the classical sense with adjoint $g$. If both $f$ and $g$ are everywhere defined then they are bounded. Our work concerns with the concept of Hyers--Ulam--Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper [On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297--300]. We also indicate interesting complementary results in the case where the Hilbert $C^*$-modules admit non-adjointable $C^*$-linear mappings.
Let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a probability measure $P$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$ is norm continuous on $T$ and the function $t \mapsto \|A_t\|$ is integrable. Then the following equality including Bouchner integrals holds \begin{eqnarray}\label{oi} \int_T|A_t - \int_TA_s{\rm d}P|^2 {\rm d}P=\int_T|A_t|^2{\rm d}P - |\int_TA_t{\rm d}P|^2 . \end{eqnarray} This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten $p$-norm inequalities.
An operator $T$ acting on a Hilbert space is called $(α,β)$-normal ($0\leq α\leq 1\leq β$) if \begin{equation*} α^{2}T^{\ast }T\leq TT^{\ast}\leq β^{2}T^{\ast}T. \end{equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(α,β)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.
The classical Mazur--Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur--Ulam theorem in the non-Archimedean strictly convex normed spaces.
In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1, x\in {\mathbb C}^n\}$$ for all $A \in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.
A linear map $ϕ:{\mathcal A}\to {\mathcal B} $ between (Banach) algebras is called 3-homomorphism if $ϕ(abc)=ϕ(a)ϕ(b)ϕ(c)$ for each $a, b, c \in {\mathcal A}$. We investigate 3-homomorphisms on Banach algebras with bounded approximate identities and establish in two ways that every involution preserving homomorphism between $C^*$-algebras is norm decreasing.
In this paper we view some fundamentals of the theory of Hilbert C*-modules and examine some ways in which Hilbert C*-modules differ from Hilbert spaces.
In this paper, we establish the generalized Hyers--Ulam--Rassias stability of $C^*$-ternary ring homomorphisms associated to the Trif functional equation \begin{eqnarray*} d \cdot C_{d-2}^{l-2} f(\frac{x_1+... +x_d}{d})+ C_{d-2}^{l-1}\sum_{j=1}^d f(x_j) = l \cdot \sum_{1\leq j_1< ... < j_l\leq d} f(\frac{x_{j_1} + ... + x_{j_l}}{l}). \end{eqnarray*}
Let $A$ be a $C^*$-algebra acting on a Hilbert space $H$, $σ:A\to B(H)$ be a linear mapping and $d:A\to B(H)$ be a $σ$-derivation. Generalizing the celebrated theorem of Sakai, we prove that if $σ$ is a continuous $*$-mapping then $d$ is automatically continuous. In addition, we show the converse is true in the sense that if $d$ is a continuous $*-σ$-derivation then there exists a continuous linear mapping $Σ:A\to B(H)$ such that $d$ is $*-Σ$-derivation. The continuity of the so-called $*-(σ,τ)$-derivations is also discussed.
The generalized Hyers--Ulam--Rassias stability of generalized derivations on unital Banach algebras into Banach bimodules is established.
A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \| \leq ϕ(x, y),\] for an appropriate control function $ϕ(x,y)$ and all $x, y \in {\mathcal M}$. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers--Ulam--Rassias stability of the orthogonality equation.
In this paper, we establish the generalized Hyers--Ulam--Rassias stability of homomorphisms between ternary algebras associted to the generalized Jensen functional equation $r f(\frac{sx+ty}{r}) = s f(x) + t f(y)$.
In this paper, we establish the conditional Hyers-Ulam-Rassias stability of the generalized Jensen functional equation $r f \left (\frac{sx+ty}{r}) = s g(x) + t h(y)$ on various restricted domains such as inside balls, outside balls, and punctured spaces. In addition, we prove the orthogonal stability of this equation and study orthogonally generalized Jensen mappings on Balls in inner product spaces.
The main purpose of this paper is to prove the generalized Hyers-Ulam-Rassias stability of J*-homomorphisms between J*-algebras.
In this paper, we establish the Pexiderized stability of coboundaries and cocycles and use them to investigate the Hyers--Ulam stability of some functional equations. We prove that for each Banach algebra $A$, Banach $A$-bimodule $X$ and positive integer $n, H^n(A,X)=0$ if and only if the $n$-th cohomology group approximately vanishes.