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A. Karasev

Publications and source records attributed to A. Karasev.

6 recordsLinked to original sources
arXiv2005

Root closed function algebras on compacta of large dimension

Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued functions on $X$, endowed with the supremum norm. We say that $C(X)$ is (approximately) $n$-th root closed if any function from $C(X)$ is (approximately) equal to the $n$-th power of another function. We characterize the approximate $n$-th root closedness of $C(X)$ in terms of $n$-divisibility of first $\check {\rm C}$ech cohomology groups of closed subsets of $X$. Next, for each positive integer $m$ we construct $m$-dimensional metrizable compactum $X$ such that $C(X)$ is approximately $n$-th root closed for any $n$. Also, for each positive integer $m$ we construct $m$-dimensional compact Hausdorff space $X$ such that $C(X)$ is $n$-th root closed for any $n$.

math.FA↗
arXiv2005

On commutative and non-commutative C*-algebras with the approximate n-th root property

We say that a C*-algebra X has the approximate n-th root property (n\geq 2) if for every a\in X with ||a||\leq 1 and every ε>0 there exits b\in X such that ||b||\leq 1 and ||a-b^n||<ε. Some properties of commutative and non-commutative C*-algebras having the approximate n-th root property are investigated. In particular, it is shown that there exists a non-commutative (resp., commutative) separable unital C*-algebra X such that any other (commutative) separable unital C*-algebra is a quotient of X. Also we illustrate a commutative C*-algebra, each element of which has a square root such that its maximal ideal space has infinitely generated first Cech cohomology.

math.OA↗
arXiv2002

Topological model categories generated by finite complexes

Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -- describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups -- is obtained by letting $L = \{{\rm point}\}$. The other -- describing n-homomotopy equivalences between at most $(n+1)$-dimensional CW-complexes as maps inducing isomorophisms of k-dimensional homotopy groups with $k \leq n$ -- by letting $L = S^{n+1}$, $n \geq 0$.

math.AT↗
arXiv2002

Real rank and squaring mapping for unital C*-algebras

It is proved that if X is a compact Hausdorff space of Lebesgue dimension $\dim(X)$, then the squaring mapping $α_{m} \colon (C(X)_{\mathrm{sa}})^{m} \to C(X)_{+}$, defined by $α_{m}(f_{1},..., f_{m}) = \sum_{i=1}^{m} f_{i}^{2}$, is open if and only if $m -1 \ge \dim(X)$. Hence the Lebesgue dimension of X can be detected from openness of the squaring maps $α_m$. In the case m=1 it is proved that the map $x \mapsto x^2$, from the self-adjoint elements of a unital $C^{\ast}$-algebra A into its positive elements, is open if and only if A is isomorphic to C(X) for some compact Hausdorff space X with $\dim(X)=0$.

math.FA↗
arXiv2001

Topological semigroups and universal spaces related to extension dimension

It is proved that there is no structure of left (right) cancelative semigroup on $[L]$-dimensional universal space for the class of separable compact spaces of extensional dimension $\le [L]$. Besides, we note that the homeomorphism group of $[L]$-dimensional space whose nonempty open sets are universal for the class of separable compact spaces of extensional dimension $\le [L]$ is totally disconnected.

math.GN↗