In the paper, we characterize Jordan $*$-derivations of a $2$-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R=Uoplus V$ such that both $U$ and $V$ are invariant under $delta$. Moreover, $*$ is the identity map of $U$ and $delta,|_U$ is a derivation, and the Jordan $*$-derivation $delta,|_V$ is inner. We also prove the theorem: Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, $operatorname{char},R e 2$, and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and $delta$ is inner.

Let $M^m$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\colon M^m \to M^m$ whose fixed point set has the form $F^n \cup F^j$, where $F^n$ and $F^j$ are submanifolds with dimensions $n$ and $j$, $F^j$ is indecomposable and $ n >j$. Write $n-j=2^pq$, where $q \ge 1$ is odd and $p \geq 0$, and set $m(n-j) = 2n+p-q+1$ if $p \leq q + 1$ and $m(n-j)= 2n + 2^{p-q}$ if $p \geq q$. In this paper we show that $m \le m(n-j) + 2j+1$. Further, we show that this bound is \emph{almost} best possible, by exhibiting examples $(M^{m(n-j) +2j},T)$ where the fixed point set of $T$ has the form $F^n \cup F^j$ described above, for every $2 \le j
Keywords:involution, projective space bundle, indecomposable manifold, splitting principle, Stiefel-Whitney class, characteristic number
Category:57R85

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group.