Vol. 12 (2006), no. 4

A Dirichlet form associated with the infinite dimensional symmetrized Levy-Laplace operator is constructed. It is shown that there exists a natural connection between this form and a Markov process. This correspondence is similar to that studied in a previous paper by the same authors for the non-symmetric Levy Laplacian.

We consider the complexity of the representation theory of free products of $C^*$-algebras. Necessary and sufficient conditions for the free product of finite-dimensional $C^*$-algebras to be $*$-wild is presented. As a corollary we get criteria for $*$-wildness of free products of finite groups. It is proved that the free product of a non-commutative nuclear $C^*$-algebra and the algebra of continuous functions on the one-dimensional sphere is $*$-wild. This result is applied to estimate the complexity of the representation theory of certain $C^*$-algebras generated by isometries and partial isometries.

We investigate spectral points of type $\pi_{+}$ and type $\pi_{-}$ for self-adjoint operators in Krein spaces. In particular a sharp lower bound for the codimension of the linear manifold $H_0$ occuring in the definition of spectral points of type $\pi_+$ and type $\pi_-$ is determined. Furthermore, we describe the structure of the spectrum in a small neighbourhood of such points and we construct a finite dimensional perturbation which turns a real spectral point of type $\pi_{+}$ (type $\pi_{-}$) into a point of positive (resp.\ negative) type. As an application we study a singular Sturm-Liouville operator with an indefinite weight.

We prove that the natural action of permutations in a tensor product of type $\mathrm{II}$ factors is free, and compute the von Neumann trace of the projection onto the space of symmetric and antisymmetric elements respectively. We apply this result to computation of von Neumann dimensions of the spaces of square-integrable harmonic forms ($L^{2}$-Betti numbers) of $N$-point configurations in Riemannian manifolds with infinite discrete groups of isometries.

For a closed linear operator $A$ in a Banach space, the notion of a vector accessible in the resolvent sense at infinity is introduced. It is shown that the set of such vectors coincides with the space of exponential type entire vectors of this operator and the linear span of root vectors if, in addition, the resolvent of $A$ is meromorphic. In the latter case, the completeness criteria for the set of root vectors are given in terms of behavior of the resolvent at infinity.

We introduce and study a generalized stochastic derivative on the Kondratiev-type space of regular generalized functions of Gamma white noise. Properties of this derivative are quite analogous to the properties of the stochastic derivative in the Gaussian analysis. As an example we calculate the generalized stochastic derivative of the solution of some stochastic equation with Wick-type nonlinearity.

We study properties of operators which are left-inverses to the operator of multiplication by an independent variable in the space $\mathcal H (G)$ of functions that are analytic in an arbitrary domain $G$. This space is endowed with topology of compact convergence. A description of cyclic elements for such operators is obtained. The obtained statements generalize known results in this direction.

We study Kronrod-Reeb graphs of functions with isolated critical points on smooth manifolds. We prove that any finite graph, which satisfies the condition $\Im$ is a Kronrod-Reeb graph for some such function on some manifold. In this connection, monotone functions on graphs are investigated.