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ABSTRACT: Lie superautomorphisms of prime associative superalgebras are considered. A
definitive result is obtained for central simple superalgebras: their Lie
superautomorphisms are of standard forms, except when the dimension of the
superalgebra in question is 2 or 4.
04/2012;
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ABSTRACT: Let $\A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial
in noncommuting indeterminates $x_i$. We consider the problem of describing
linear maps $\phi:\A\to \A$ that preserve zeros of $f$. Under certain technical
restrictions we solve the problem for general polynomials $f$ in the case where
$\A=M_n(F)$. We also consider quite general algebras $\A$, but only for
specific polynomials $f$.
04/2012;
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ABSTRACT: We describe subalgebras of the Lie algebra $\mf{gl}(n^2)$ that contain all
inner derivations of $A=M_n(F)$ (where $n\ge 5$ and $F$ is an algebraically
closed field of characteristic 0). In a more general context where $A$ is a
prime algebra satisfying certain technical restrictions, we establish a density
theorem for the associative algebra generated by all inner derivations of $A$.
04/2012;
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ABSTRACT: Let $A$ be a Banach algebra. By $\sigma(x)$ and $r(x)$ we denote the spectrum
and the spectral radius of $x\in A$, respectively. We consider the relationship
between elements $a,b\in A$ that satisfy one of the following two conditions:
(1) $\sigma(ax) = \sigma(bx)$ for all $x\in A$, (2) $r(ax) \le r(bx)$ for all
$x\in A$. In particular we show that (1) implies $a=b$ if $A$ is a
$C^*$-algebra, and (2) implies $a\in \mathbb C b$ if $A$ is a prime
$C^*$-algebra. As an application of the results concerning the conditions (1)
and (2) we obtain some spectral characterizations of multiplicative maps.
04/2012;
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ABSTRACT: We consider the relationship between derivations $d$ and $g$ of a Banach
algebra $B$ that satisfy $\s(g(x)) \subseteq \s(d(x))$ for every $x\in B$,
where $\s(\, . \,)$ stands for the spectrum. It turns out that in some basic
situations, say if $B=B(X)$, the only possibilities are that $g=d$, $g=0$, and,
if $d$ is an inner derivation implemented by an algebraic element of degree 2,
also $g=-d$. The conclusions in more complex classes of algebras are not so
simple, but are of a similar spirit. A rather definitive result is obtained for
von Neumann algebras. In general $C^*$-algebras we have to make some
adjustments, in particular we restrict our attention to inner derivations
implemented by selfadjoint elements. We also consider a related condition
$\|[b,x]\|\leq M\|[a,x]\|$ for all selfadjoint elements $x$ from a
$C^*$-algebra $B$, where $a,b\in B$ and $a$ is normal.
04/2012;
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ABSTRACT: The main theorem states that a bounded linear operator $h$ from a unital $C^{\ast}$-algebra $A$ into a unital Banach algebra $B$ must be a homomorphism provided that $h(\bm{1})=\bm{1}$ and the following condition holds: if $x,y,z\in A$ are such that $xy=yz=0$, then $h(x)h(y)h(z)=0$. This theorem covers various known results; in particular it yields Johnson's theorem on local derivations.
Proceedings of the Royal Society of Edinburgh Section A Mathematics 01/2007; 137(01):1 - 7. · 0.68 Impact Factor
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Communications in Algebra 01/2003; No. 3(pp. 1207–1234):1207-1234. · 0.35 Impact Factor
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Studia Mathematica - STUD MATH. 01/2001; 146(2):177-200.
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ABSTRACT: The main result of the paper characterizes continuous bilinear maps ϕ from C1[0,1]×C1[0,1] into a Banach space X with the property that ϕ(f,g)=0 whenever fg=0. This is applied to the study of zero product preserving operators on C1[0,1], and operators on C1[0,1] satisfying a version of the condition of the locality of an operator.
Journal of Mathematical Analysis and Applications. 347(2):472-481.
