Is the radical of a homogeneous ideal homogeneous?

Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality, the answer is no (see comments). However, if $M$ is a cancellative monoid with a total order, then the usual proof for $M=\mathbb{N}$ … Read more

Slightly noncommutative Nakayama’s lemma?

Nakayama’s lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $r-1 $ belongs to the ideal generated by $s$. Is the following known (or false)? Question: … Read more

Universal characterization and explicit description of elements of the group von Neumann algebra and the crossed product

Group von Neumann algebras and crossed products for a locally compact group G can be constructed in many different ways. For example, one can take the von Neumann algebra generated by certain operators on a certain Hilbert space. However, none of these constructions give an explicit description of elements of the group algebra or the … Read more

Subalgebra of group algebra generated by idempotents

Let G be a finite group, and let A and B be two abelian subgroups of G. Let K be a number field such that all characters of A and of B take values in K. Let OK be the ring of algebraic integers in K. Write {e1,…,en} for the set of primitive idempotents of … Read more

The functoriality of group C* algebra structure

Let G and H be discrete groups and f:G→H be any homomorphism of these groups. I have three questions about it: 1) How to prove the functoriality of the construction of universal C∗-algebra of discrete group (the existence of induced homomorphism C∗(G)→C∗(H))? 2) How to prove that the construction of reduced C∗-algebra of discrete group … Read more

Is there a countable discrete infinite group GG over which the group algebra CG\mathbb{C} G is semisimple?

I am seeking for an Artin k-algebra (especially for group algebra) which is infinite-dimensional over some field k. It’s known that any complex group algebra has trivial Jacobson radical. So I have the following question: Is there a countable discrete infinite group G over which the group algebra CG is semisimple? Answer The answer is … Read more

Real rank zero of group C∗C^*-algebras

The concept of real rank zero of a C∗-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if X is a compact Hausdorff space, then RR C(X)=dim(X). Let A be a C∗-algebra. we define RR(A)=0 iff every self-adjoint element in A can be approximate … Read more

Units of group algebra of dihedral group

Question: Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite dihedral group? I’m also interested in other fields $K$ (in which case one can focus on describing the quotient by its central subgroup $K^*$). If $A=G_{\mathrm{ab}}$ is the Klein group, I’d already … Read more

Are the reduced group Von Neumann algebra/ Group C∗C^{\ast} algebra functorial in the case of LCH groups

Let G be a LCH group and μ be its left Haar measure. Call λG:G→U(L2(G,μ)) the left regular representation. We can define the reduced C∗ algebra and reduced Von Neumann algebra, C∗λG and W∗λG respectively, as the smallest C∗, resp. Von Neumann, subalgebras of B(L2(G,μ)) containing λG[Cc(G)]. It is easy to see that C∗λG⊂M(C∗λG)⊂W∗λG and … Read more

About the classification of commutative and of cocommutative, fin. dim. Hopf algebras

I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f.d. Hopf algebras (over an algebraically closed field of characteristic zero) are dual to group Hopf algebras (for some finite group). More precisely, I want … Read more