114 research outputs found
Elementary approach to homogeneous C*-algebras
A C*-algebra is n-homogeneous (where n is finite) if every its nonzero
irreducible representation acts on an n-dimensional Hilbert space. An
elementary proof of Fell's characterization of n-homogeneous C*-algebras (by
means of their spectra) is presented. A spectral theorem and a functional
calculus for finite systems of elements which generate n-homogeneous
C*-algebras are proposed.Comment: 22 page
Spaces of measurable functions
For a metrizable space and a finite measure space
let and be the spaces
of all equivalence classes (under the relation of equality almost everywhere
mod ) of -measurable functions from to whose
images are separable and finite, respectively, equipped with the topology of
convergence in measure. The main aim of the paper is to prove the following
result: if is (nonzero and) nonatomic and has more than one point,
then the space is a noncompact absolute retract and
is homotopy dense in for each dense subset of . In
particular, if is completely metrizable, then is homeomorphic
to an infinite-dimensional Hilbert space.Comment: 22 page
Isometry groups of proper metric spaces
Given a locally compact Polish space X, a necessary and sufficient condition
for a group G of homeomorphisms of X to be the full isometry group of (X,d) for
some proper metric d on X is given. It is shown that every locally compact
Polish group G acts freely on GxY as the full isometry group of GxY with
respect to a certain proper metric on GxY, where Y is an arbitrary locally
compact Polish space with (card(G),card(Y)) different from (1,2). Locally
compact Polish groups which act effectively and almost transitively on complete
metric spaces as full isometry groups are characterized. Locally compact Polish
non-Abelian groups on which every left invariant metric is automatically right
invariant are characterized and fully classified. It is demonstrated that for
every locally compact Polish space X having more than two points the set of
proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper
metrics on X.Comment: 24 page
Universal valued Abelian groups
The counterparts of the Urysohn universal space in category of metric spaces
and the Gurarii space in category of Banach spaces are constructed for
separable valued Abelian groups of fixed (finite) exponents (and for valued
groups of similar type) and their uniqueness is established. Geometry of these
groups, denoted by G_r(N), is investigated and it is shown that each of
G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are
Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are
studied and it is proved that every separable metrizable topological vector
space may be enlarged to G_r(0) with a `linear-like' structure which extends
the linear structure of the given space.Comment: 60 page
Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces
An \textit{ideal} of -tuples of operators is a class invariant with
respect to unitary equivalence which contains direct sums of arbitrary
collections of its members as well as their (reduced) parts. New decomposition
theorems (with respect to ideals) for -tuples of closed densely defined
linear operators acting in a common (arbitrary) Hilbert space are presented.
Algebraic and order (with respect to containment) properties of the class
of all unitary equivalence classes of such -tuples are established
and certain ideals in are distinguished. It is proved that infinite
operations in may be reconstructed from the direct sum operation of a
pair. \textit{Prime decomposition} in is proposed and its (in a sense)
uniqueness is established. The issue of classification of ideals in (up
to isomorphism) is discussed. A model for is described and its concrete
realization is presented. A new partial order of -tuples of operators is
introduced and its fundamental properties are established. Extremal importance
of unitary disjointness of -tuples and the way how it `tidies up' the
structure of are emphasized.Comment: 115 page
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