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Assume That $A$ is a commutative complex Banach algebra. Let $G$ be the connected component of invertible elements containing the identity. Is there an smooth embedded curve $c:(-\epsilon, \epsil …

Let $A$ be a Banach algebra. Is there a Banach algebra $B$ which contains $A$ but the spectrum of each elements of $B$ has empty interior(as a subset of $\mathbb{C}$)? The motivation comes from the …

Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a $C^{* …

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$. For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms on …

Apply holomorphic functional calculus to the following functions which are defined on a disconnected open set in the plane containing $\Gamma_0 , \Gamma_1$ $f(z)=\begin{cases} 1& \text{Aro …

A Banach algebra $A$ is called a rigid Banach algebra if for every injective Banach algebra morphism $J:A\to A$ we have either $\overline{J(A)}$ is ismorphic to $A$ or it does not contain an …

Let $A$ be a Banach algebra with the following property: For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges. …

Not a direct answer to the question but somehow related to the question: Apart from Benjamin Passer's paper, Here are two other papers about non commutative Borsuk Ulam theorem: http://arxiv.org/a …

A revision: According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$. Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal spac …

Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the $ …

Let $F_{2}$ be the free group with two generators. Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way. …

I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as $1_{ …

It is well known that the equation $$(*)\;\;\;\;ab-ba=1$$ is unsolvable in a Banach algebra. I search for some reasonable generalization of this equation in higher variable for investigation o …

Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$. Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$. Assume that for every $\phi\in …

Perhaps the first example in Teichmuller theory is the following proposition: Proposition: Let $1<r<R$. Then two annular region $U_r=\{z\in \mathbb{C}\bigm|1<|z|<r\}$ and $U_R=\{z\in \mathbb{C}\bi …