Hilbert bundles with ends

Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.

Partial C*-dynamics and Rokhlin dimension

We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$ -algebra. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

An example regarding Kalton's paper ‘isomorphisms between spaces of vector-valued continuous functions’

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which is isomorphic to a closed subspace of a space with a basis and $C(I,\,X)$ is linearly homeomorphic to $C(\Delta ,\, X)$ , then $X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces $X$ with a basis for which $C(I,\,X)$ and $C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, $X$ is isomorphic to $C(K,\,X)$ if $K$ is a metric compactum of finite covering dimension.

Regularity of simple nuclear real C*-algebras under tracial conditions

The Toms–Winter conjecture is verified for those separable, unital, nuclear, infinite-dimensional real C*-algebras for which the complexification has a tracial state space with compact extreme boundary of finite covering dimension.

The Birman-Hilden property of covering spaces of nonorientable surfaces

UDC 517.5 Let p : N ˜ → N be a finite covering space of nonorientable surfaces, where χ ( N ˜ ) < 0 . We search whether or notphasthe Birman – Hilden property.

Characterizations of an MW-topological rough set structure

Regarding the study of digital topological rough set structures, the present paper explores some mathematical and systemical structures of the Marcus-Wyse (MW-, for brevity) topological rough set structures induced by the locally finite covering approximation (LFC-, for brevity) space (R2,C) (see Proposition 3.4 in this paper), where R2 is the 2-dimensional Euclidean space. More precisely, given the LFC-space (R2,C), based on the set of adhesions of points in R2 inducing certain LFC-rough concept approximations, we systematically investigate various properties of the MW-topological rough concept approximations (D -M, D+M) derived from this LFC-space (R2,C). These approaches can facilitate the study of an estimation of roughness in terms of an MW-topological rough set. In the present paper each of a universe U and a target set X(? U) need not be finite and further, a covering C is locally finite. In addition, when regarding both an M-rough set and an MW-topological rough set in Sections 3, 4, and 5, the universe U(? R2) is assumed to be the set R2 or a compact subset of R2 or a certain set containing the union of all adhesions of x ? X (see Remark 3.6).

Conformally related Riemannian metrics with non-generic holonomy

In this paper we show that if a compact connected n-dimensional manifold M has a conformal class containing two non-homothetic metrics g and {\tilde{g}=e^{2\varphi}g} with non-generic holonomy, then after passing to a finite covering, either {n=4} and {(M,g,\tilde{g})} is an ambikähler manifold, or {n\geq 6} is even and {(M,g,\tilde{g})} is obtained by the Calabi Ansatz from a polarized Hodge manifold of dimension {n-2}, or both g and {\tilde{g}} have reducible holonomy, M is locally diffeomorphic to a product {M_{1}\times M_{2}\times M_{3}}, the metrics g and {\tilde{g}} can be written as{g=g_{1}+g_{2}+e^{-2\varphi}g_{3}}\quad\text{and}\quad{\tilde{g}=e^{2\varphi}(% g_{1}+g_{2})+g_{3}}for some Riemannian metrics {g_{i}} on {M_{i}}, and φ is the pull-back of a non-constant function on {M_{2}}.