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In the late 70s, Cuntz and Behncke had a paper H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95 …

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. Cl …

Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite grou …

Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$ $$\inf \lbrace\|T …

Let $f : G \to H$ be a homomorphism of discrete groups. The homomorphism $f$ extends to a homomorphism of reduced group $C^{\ast}$-algebras if and only of $\ker(f)$ is amenable, and extends to a h …

The answer is that $G=SU(\infty)$ (with the direct limit topology of the usual Hilbert-Schmidt topologies) is extremely amenable. This means (by definition) that every continuous action of $G$ on a co …

Let $H$ be a Hilbert space and let $u_1 \in U(H)$ be a unitary operator on $H$. Consider the self-map $w: U(H) \to U(H)$ which is given by $$w(v) := v^2 u_1 v^{-1}.$$ Since $U(H)$ is connected, there …

Theorem Let $A$ be a unital ring and $I_1,\dots,I_n \subset A$ be 2-sided commutative ideals such that $A=I_1+\dots + I_n$. Then, $A$ is commutative. Proof: If $A=I_1+\dots+I_n$, then $1 = x_1+\dots+ …

The answer is yes, and much more is true. Any hyperfinite von Neumann algebra (with separable predual) has a unique embedding (up to conjugacy) into the ultra-product of the hyperfinite $II_1$-factor. …

In the theory of subfactors, it frequently happens that a subfactor of finite index $N \subset M$ satisfies $N' \cap M = {\mathbb C}$. Those subfactors are called irreducible. In this case the relativ …

Passman showed that whenever there are zero-divisors in a group ring one also has (non-zero) nilpotent elements. He shows that for any field $k$ and any torsionfree group $G$, the ring $kG$ is a prime …

This post is closely related to this one. (In fact I copied some of its content.) Let $H$ be an infinite dimensional separable complex Hilbert space. All $C^{\star}$-subalgebras of $B(H)$ are assumed …

If Kazhdan's property (T) is reflected in the structure of the Cayley graph, then not in a very geometric way. Steve Gersten (that is what I read in the book by B. Bekka, P. de la Harpe and A. Valet …

The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other …

Alex Lubotzky and Yehuda Shalom have shown in Finite representations in the unitary dual and Ramanujan groups., (Discrete geometric analysis, 173–189, Contemp. Math., 347, Amer. Math. Soc., Providence …