# Tagged Questions

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) ...

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### Is the hermitian condition a must?

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$ In this post, I saw a comment stating that "More generally, if $a$ is normal then $∥a^n∥=∥a∥^n$ for each positive ...
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Let $A$ be Von Neumann algebra. A positive linear functional ‎$‎‎‎\varphi‎$ on $A$ ‎is ‎said ‎to ‎be ‎normal ‎if ‎for ‎any ‎self‎adjoint and increasing nets such that ‎$‎‎u_{\alpha‎}\longrightarrow u‎... 0answers 20 views ### Homotopy of bounded homomorphisms between Banach algebras Let$A$and$B$be Banach algebras. Say that two bounded homomorphisms$\phi_0$and$\phi_1$from$A$to$B$are homotopic if there is a path$(\phi_t)_{t\in[0,1]}$of bounded homomorphisms from$A$... 0answers 28 views ### The image of a continuous derivation on a Banach algebra is contained in the kernel of a character. It is known that if$D$is a continuous derivation on a commutative Banach algebra$\mathcal{A}$, then for any nonzero character$\theta$on$\mathcal{A}$we have$D(\mathcal{A})\subseteq ker \theta$. ... 0answers 16 views ### multiplication on the second dual of a Banach *-algebra I am looking for an Arense regular Banach *-algebra$A$, whose multiplication on$A^{**}$is not$w^*$-separately continuous. 1answer 117 views ### K theory of finite dimenional Banach algebras Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose$K_{0}$-group is a non trivial finite group?(I ... 0answers 39 views ### Well-definedness of the logarithm in a Banach-algebra Let$x \in \mathcal{A}$be an element of a unital Banach-algebra$\mathcal{A}$and assume$\sigma(x) \subset \{ z \in \mathbf{C} : \vert 1 - z \vert < 1 \}$, where$\sigma(x)$denotes the spectrum ... 1answer 28 views ### closed convex hull of projection$1$:I know that if ‎$‎‎F$is a ‎locally convex ‎compact ‎space ‎then ‎‎$‎‎‎\overline{co}(‎Ext (F))=F$‎ ($Ext$: means extreme point)$2$:I ‎know ‎that ‎if ‎‎$‎‎M$‎is a ‎Von ‎Neumann ‎algebra ‎then ‎‎... 1answer 30 views ### projecton and positive element in$C^*$algebras Let ‎$‎‎A$‎be ‎a‎ ‎$‎‎C^*$-algebra.‎$‎‎p\in A$‎is a ‎‎projections. ‎‎‎ Assume ‎that ‎‎$‎‎a$‎is a element in‎$‎‎ Ball(A_+)$‎such ‎that ‎‎$‎‎a‎\leq p‎$‎ Q:May I‎ ‎say ‎‎$‎‎ap=pa$?why?‎ ‎ 1answer 67 views ### If an operator have only Real eigenvalues + symmetric then it's self-adjoint? I know that if an operator is self-adjoint then has Real eigenvalues but I'm not sure about the converse i.e. if it has only Real eigenvalues and is symmetric then the operator is selfadjoint. Is that ... 1answer 34 views ### *-isomorphism and spectrum ‎‎‎$A$is a ‎‎‎‎$‎‎C^∗$-algebra and$P(A)$is a set of projection of it. Assume that$A$‎admits a‎ ‎strictly ‎positive ‎element ‎‎‎‎‎$a$‎such ‎that ‎‎‎‎‎$‎‎‎‎σ(a)‎-\{‎0\}$‎is ‎discrete‎. I want to ... 0answers 27 views ### strictly positive elments$a$when$‎‎‎\sigma(a) ‎\backslash‎ {0}‎$‎is ‎discrete If ‎$‎‎A$is a ‎‎$‎‎C^*$-algebra ‎and it ‎admits a‎ ‎strictly ‎positive ‎element ‎‎$‎‎a$‎such ‎that ‎‎$‎‎‎\sigma(a) ‎\backslash‎ {0}‎$‎is ‎discrete‎ then‎ Q1:‎$‎‎A$admits ‎an ‎approximate ‎unit ‎‎$...
Let ‎$‎‎A$ ‎be a ‎‎‎‎$‎‎C^*$-algebra‎. ‎$‎‎a\in A^+$ ‎is ‎strictly ‎positive in ‎$‎‎A$‎ ‎if ‎‎$‎‎‎\overline{aAa}=A‎$‎‎ *I know that if $A$ is unital, $a\in A^+$ is strictly positive iff $a\in Inv(A)$...
If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...