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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For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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Norm of the resolvent in the quotient algebra

Let $A$ be a Banach algebra with identity and $J$ be a two-sided ideal of $A$. It is known that $\sigma(a')\subseteq\sigma(a) \, \forall a\in A$, where $a'$ is the coset $a+J$. Suppose $\lambda-a'$ ...
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Banach Algebra convergence

I am taking a courses on Differential Calculus and I would like to know the details in the steps of the proof (1) & (2). Also I would like some details on how to solve the exercise below. (3) ...
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10 views

Spectrum of a C* subalgebra generated by a normal element

Let $A$ be a unital C*-algebra and $a$ be non-invertible normal element in $A$. Suppose $B=C^*(a)$, i.e., the commutative C*-subalgebra that is generated by $a$. Will ...
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1answer
25 views

Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
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19 views

isometrically isomorphism [duplicate]

How can embed separable Banach to Cb(X)(family of all bounded continuous functions on topological space X) with non metrizable X ? If X is locally compact or Tychonof is very well. note : We know ...
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3answers
42 views

Semi-direct decompositions of Banach algebras

Let $A$ be a Banach algebra and I an ideal of it. Is there always a subalgebra $B$ of $A$ such that A can be written as $A=B\oplus I$ where $\oplus$ denotes direct sum? If not, in what conditions we ...
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17 views

Diffeomorphism in Banach algebra

Let $X$ be an algebra $C([0,1],\mathbb{R})$. Let define $F:X\ni f \mapsto f(0)f \in X$ What is the biggest $r$ such that $F$ is $C^r$-diffeomorphism ?
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For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$

Let $\mathcal{A}$ be a $C^*$-algebra. Suppose that $a \in \mathcal{A}$ with the property that $a^* = a$ (that is, suppose that $a$ is hermitian). I would like to show that $\|a^{2n}\| = ...
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35 views

States: Extension

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit ...
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39 views

Norm of rotation matrix as element in $M_2(A)$

Let $A$ be a complex unital Banach algebra and let $R_t=\begin{pmatrix} \cos\frac{\pi t}{2} & -\sin\frac{\pi t}{2} \\ \sin\frac{\pi t}{2} & \cos\frac{\pi t}{2} \end{pmatrix}$. If I consider ...
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1answer
58 views

Nonzero projection in an irredicible C*-algebra of minimal finite rank must have rank one

The following is a part of a theorem in Murphy's C*-algebras and operator theory: Let $A$ be a C*-algebra acting irreducibly on a Hilbert space $H$ and $q$ be a nonzero projection in $A$ of minimal ...
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1answer
34 views

$*-$ isomorphism between two compact spaces $K(H)$ and $K(H')$

The following is a theorem of Murphy's C*-algebras and operator theory: I think we can write the proof more easily than Murphy's. After show that $E'$ is an orthonormal basis for $H'$, define ...
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1answer
29 views

Characterspace of algebra of rational functions on compact subset of $\mathbb C$

Let $K\subset \mathbb C$ be compact and denote with $R$ the closure of rational functions on $K$ w.r.t. $\|\cdot \|_\infty$. Show that the character space of the Banach algebra $R$ is ...
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1answer
33 views

What's difference between spectrum and eigenvectors of an operator

Let $x$ be an operator in $B(H)$. By definition $\sigma(x)=\{\lambda \in \Bbb C ~; \lambda - x \neq inv \}$. Also to find eigenvalue of an operator we should find $\lambda$ such that $x\xi = \lambda ...
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53 views

Does the equality $(T^{\ast})^{\alpha}=(T^{\alpha})^{\ast}$ hold?

Let $H$ be a complex Hilbert space and $B(H)$ be all bounded linear operator on $H$. Let $T\in B(H)$. Can I say $(T^{\ast})^{\alpha}=(T^{\alpha})^{\ast}$ for $\alpha\geq1$ where $\ast$ means ...
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101 views

$p^2=p$ in closure of ideal $I$ of Banach algebra implies $p\in I$.

Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$. Show: $p\in I$. Can someone give me a little hint how to solve this, please?
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1answer
37 views

Nonunital C*-Algebras: Closed Image

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$. Then its image is closed: $\mathrm{im}\pi\subsetneq\overline{\mathrm{im}\pi}$ The proof I ...
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1answer
103 views

Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
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1answer
18 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
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1answer
29 views

Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that ...
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1answer
37 views

If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
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1answer
20 views

$\phi(A^+) \subset B^+$ when $\phi: A\to B$ is an isometric linear map

Let $\phi: A\to B$ be an isometric linear map between unital C*-algebras $A$ and $B$ such that $\phi(a^*)=\phi(a)^* (a\in A)$ and $\phi(1)=1$. Show that $\phi(A^+) \subset B^+$. Clearly $A^+ = \{a^*a ...
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49 views

Subalgebras of $C_b(X)$ whose elements do not vanish simultaneously at any point

Let $X$ be a completely regular space. How can I find all Banach subalgebras of $C_b(X)$ (all complex-valued bounded continuous functions on $X$) with the property that for every $x\in X$ there ...
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1answer
21 views

Show that hermitian element $h=\sum p_n/3^n$ generates $ C_0(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the C*-algebra $C_0(\Omega)$ is generated by a sequence of projections $(p_n)_{n=1}^{\infty}$. Show that the hermitian element ...
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1answer
43 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
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19 views

Does this define a norm s.t. set of matrices is Banach algebra?

As I asked in this comment here: If $a$ is an $n \times n$ matrix and we define $$ \|a\|_p = \left ( \max_j \sum_i |a_{ij}|^p \right)^{1 \over p}$$ or $$ \|a\|_p = \left ( \max_i \sum_j ...
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15 views

Prove that a complex-valued homomorphism on a Banach algebra which is not identically 0, is a bounded linear functional of norm $1$

I want to prove that a complex-valued homomorphism $h$ on a Banach algebra $X$ which is not identically 0, is a bounded linear functional of norm $1$. This is a statement in the appendix D of the ...
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1answer
23 views

Joint spectrum of $\{a_1,…,a_n\}$

Let $\{a_1,...,a_n\}$ be commuting normal operators on a Hilbert space. Put $A:= C^*(1,a_1,...,a_n)$. By Gelfand theorem ,abelian C*-algebra $A$ is identified with the algebra $C(\Omega)$ of all ...
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48 views

Semigroups & Generators: Nonexample

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of automorphisms is a contraction group: ...
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1answer
24 views

CAR-Algebra: Nontriviality?

Given a Hilbert space $\mathcal{h}$. Consider the abstract CAR-algebra $a:\mathcal{h}\to\mathcal{A}_\text{CAR}$. Then their actually isometries: $$a:=a(f):\quad ...
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1answer
61 views

Hahn–Banach theorem?

In mathematics, the Hahn–Banach Theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, ...
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2answers
43 views

construction of an injective representation of $C_0(X)$

Let X be a locally compact noncompact Hausdorff space and consider the C$^*$-Algebra $C_0(X)$ of continuous functions vanishing at infinity. I want to construct an injective *-represenatation of ...
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Comput Spectrum of Idempotent

Let A be a unital banach algebra and a in A if a is idmepotent and a do not equal to 0 and 1 then the spectrum of a = {0,1}??
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1answer
33 views

an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...
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1answer
76 views

Is this equality true or it is not necessarily true?

Let $A$ and $B$ are two factor von neumann algebras that act on two infinite dimensional Hilbert spaces H and K respectively. Let $\Phi:A\longrightarrow B$ is an additive bijective map with some other ...
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41 views

positive elements in c*algebras and states

I have problems to prove that an element $a $ is a $C^*$-algebra is positive if and only if $f(a) \geq 0$ for all states $f$. The definitions I use: -$f:A\to\mathbb{C}$ linear functional on a ...
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1answer
24 views

limit of state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
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1answer
40 views

Proof that the spectrum of an element of a Banach Algebra is non-empty

I don't see why the line indicated with ***** in the following proof is true in the proof that spectrum of an element of a Banach Algebra is non-empty (Arveson, p.27) : For every $\lambda_0 \not\in ...
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1answer
26 views

strictly positive element iff A contains a countable approximative unit

I search a proof of: Let A be a c$^*$-algebra and let $(u_n)_{n\in\mathbb{N}}$ an approximative unit in A. Then $a=\sum\limits_{n=1}^{\infty}\frac{u_n}{2^n}$ is strictly positive. Could anybody tell ...
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61 views

Positive Elements in a C*algebra

Let A be a C$^*$-Algebra, $a\in A$. Why is $a\ge 0$ (a is called "positive") iff $\forall \varphi\in S(A): \varphi\ge0$? S(A) is the set of linear positive functional $\eta:A\to\mathbb{C}$ with ...
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2answers
48 views

Polar decomposition

Every $x\in B(H)$ has a representation such as $x= u|x|$ (by polar decomposition) where $u$ is a partial isometry. In a paper, the author claims that$$u = {\rm strong} - \lim_{\epsilon\to 0} ...
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C*-Algebras: Contractive Morphism

Problem Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with $\mathbb{1}_\mathcal{A}\in\mathcal{A}$. Consider an algebraic morphism $\pi:\mathcal{D}\subseteq\mathcal{A}\to\mathcal{B}$ with ...
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0answers
23 views

Show that $\pi(M)'' = \pi(M'')$

Let $M$ is a $*-$ subalgebra of $B(H)$. Let $\bar H$ denote the direct sum $\sum H_i$ where $\{H_i\}$ is a family of replicas of $H$. Define $$\pi :x\in B(H) \to \bar x \in B(\bar ...
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1answer
25 views

Literature: Derivations in C*-Algebras

Do you have some nice reference for dynamical systems in C*-algebras (including discussion of their derivations!) like notes, papers, books, etc.?
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21 views

$(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$

Please give me a hint to prove $(\lim x_n)y = \lim (x_n y)$ when $x_n, y \in B(H)$ for every n. Thanks in advance.
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1answer
41 views

If $A$ is a $*-$ Banach algebra then $\bar A^{wot} = \bar A^{weak^*}$?

If $A$ is a $*-$ subalgebra of $B(H)$, then clearly $\bar A^{weak^*}\subset \bar A^{wot}$ (wot means weak operator topology). Also on every bounded subset of $A$, two topologies equal. Now my question ...
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1answer
21 views

$A y= b$ in $C(X)$

Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of all complex valued continuous functions on $X$. If $A\in C(X)^{m\times n}$, $b\in C(X)^{m\times 1}$, and for all $x\in ...
2
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1answer
65 views

Are the invertible elements of a Banach algebra closed in the set of left-invertible elements?

Let $A$ be a unital Banach algebra. Denote by $\mathrm{Inv}(A)$ the invertible elements in $A$, and $\mathrm{Inv}_\ell(A)$ the left-invertible elements. That is, $a \in \mathrm{Inv}_\ell(A)$ if and ...