arXiv subjects

Ilja Gogić

Publications and source records attributed to Ilja Gogić.

22 records · Page 2Linked to original sources

arXiv2019

Centrally Stable Algebras

We define an algebra $A$ to be centrally stable if, for every epimorhism $φ$ from $A$ to another algebra $B$, the center $Z(B)$ of $B$ is equal to $φ(Z(A))$, the image of the center of $A$. After providing some examples and basic observations, we consider in somewhat greater detail central stability in tensor products of algebras, and finally establish our main result which states that a finite-dimensional unital algebra $A$ over a perfect field $F$ is centrally stable if and only if $A$ is isomorphic to a direct product of algebras of the form $C_i\otimes_{F_i}A_i$, where $F_i$ is a field extension of $F$, $C_i$ is a commutative $F_i$-algebra, and $A_i$ is a central simple $F_i$-algebra.

math.RA↗

arXiv2019

The cb-norm approximation of generalized skew derivations by elementary operators

Let $A$ be a ring and $σ: A \to A$ a ring endomorphism. A generalized skew (or $σ$-)derivation of $A$ is an additive map $d: A \to A$ for which there exists a map $δ:A \to A$ such that $d(xy)=δ(x)y+σ(x)d(y)$ for all $x,y \in A$. If $A$ is a prime $C^*$-algebra and $σ$ is surjective, we determine the structure of generalized $σ$-derivations of $A$ that belong to the cb-norm closure of elementary operators $\mathcal{E}\ell(A)$ on $A$; all such maps are of the form $d(x)=bx+axc$ for suitable elements $a,b,c$ of the multiplier algebra $M(A)$. As a consequence, if an epimorphism $σ: A \to A$ lies in the cb-norm closure of $\mathcal{E}\ell(A)$, then $σ$ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime $C^*$-algebras like $C(X,\mathbb{M}_2 )$.

math.OA↗

arXiv2016

The closure of two-sided multiplications on C*-algebras and phantom line bundles

For a C*-algebra A we consider the problem of when the set $TM_0(A)$ of all two-sided multiplications $x \mapsto axb$ ($a,b \in A$) on A is norm closed, as a subset of B(A). We first show that $TM_0(A)$ is norm closed for all prime C*-algebras A. On the other hand, if $A\cong Γ_0(E )$ is an n-homogeneous C*-algebra, where E is the canonical $\mathbb{M}_n $-bundle over the primitive spectrum X of A, we show that $TM_0(A)$ fails to be norm closed if and only if there exists a $σ$-compact open subset U of X and a phantom complex line subbundle L of E over U (i.e. L is not globally trivial, but is trivial on all compact subsets of U). This phenomenon occurs whenever $n \geq 2$ and X is a CW-complex (or a topological manifold) of dimension $3 \leq d<\infty$.

math.OA↗

arXiv2009

Derivations which are inner as completely bounded maps

We consider derivations from the image of the canonical contraction $θ_A$ from the Haagerup tensor product of a C*-algebra A with itself to the space of completely bounded maps on A. We show that such derivations are necessarily inner if A is prime or if A is quasicentral with Hausdorff primitive spectrum. We also provide an example of a C*-algebra which has outer elementary derivations.

math.OA↗