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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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
23 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 ...
3
votes
1answer
27 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
17 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 ...
2
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0answers
44 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
17 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 ...
1
vote
1answer
38 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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1answer
16 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 ...
0
votes
0answers
13 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
20 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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0answers
25 views

C*-Algebras: Group vs. Derivation

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: ...
0
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1answer
21 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 ...
1
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1answer
52 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, ...
0
votes
2answers
40 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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1answer
23 views

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}??
0
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1answer
31 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 ? ...
0
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1answer
75 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 ...
0
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1answer
36 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 ...
0
votes
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 $\varphi$ ...
0
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1answer
31 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 ...
0
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1answer
25 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 ...
0
votes
1answer
54 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 ...
2
votes
1answer
30 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 $$u = strong - \lim_{\epsilon\to 0} ...
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votes
2answers
72 views

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
21 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 ...
0
votes
1answer
23 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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0answers
18 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.
1
vote
1answer
37 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 ...
2
votes
1answer
20 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
votes
1answer
54 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 ...
2
votes
1answer
81 views

Locally Compact Stone-Weierstrass Theorem

STATEMENT: Let $X$ be a locally compact Hausdorff space, and let $A = C_∞(X)$ be the algebra of continuous real-valued functions on $X$ that vanish at infinity, as above, equipped with the supremum ...
2
votes
0answers
38 views

Generators: Analytic Vectors

Given a Banach space $E$. Consider semigroup $T:\mathbb{R}_+\to\mathcal{B}(E)$. Define its derivative: $$A_Tx:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)$$ (The domain being those elements whose limit ...
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1answer
45 views

Why is $\widehat{a}$ bijective?

I am trying to understand the proof of the following theorem: Let $A$ be a unital Banach algebra generated by $1$ and $a$. Then $A$ is abelian and the map $\widehat{a}:\Omega(A) \to \sigma (a), \tau ...
3
votes
1answer
40 views

Comparing weak and weak operator topology

We can compare topologies on $B(H)$. For instance, Sot topology is stronger than wot topology or $\sigma-$ weak topology is equivalent to weak* topology. I would like to compare wot topology and weak ...
5
votes
0answers
192 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
0
votes
1answer
27 views

Where is a mistake in my proof concerning the spectrum of elements of a unital Banach algebra?

I was going to prove: Let $A$ be a unital Banach algebra. Then $$\sigma(a) = \{\tau(a) \mid \tau \in \Omega (A)\}$$ and I started the following argument: Let $\lambda \in \sigma (a)$ and let ...
1
vote
0answers
32 views

The spectrum of an element in a non-unital Banach algebra

Let $A$ be a commutative non-unital Banach algebra. Let $\widetilde{A}$ denote its unitisation. I am trying to understand the proof of $$ \sigma (a) = \{\tau (a) \mid \tau \in \Omega (A)\}\cup ...
2
votes
1answer
37 views

Restriction of a spectral measure

Let $x$ be a self-adjoint operator on $H$. By spectral theorem, there is a spectral measure $\mu$ correspondence to $*-$ homomorphism $\pi:C(\sigma(x)) \to B(H)$ such that $x=\int_{-||x||}^{||x||} ...
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1answer
124 views

An abelian Banach algebra without characters

Can one give an example of an abelian Banach algebra with empty character space? Such algebra must be necessarily non-unital. I couldn't find any examples of such algebras. Thanks!
0
votes
0answers
31 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
0
votes
1answer
78 views

A lemma by Foguel and Weiss [1973]

so I am reading Krengel's text on Ergodic theorems. And the next lemma bugs me as for the proof of it. It's by Foguel and Weiss. Statement: If $P_1, P_2$ are commuting elements of a Banach algebra ...
3
votes
1answer
23 views

Natural *-isomorphism

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$. My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that ...
2
votes
1answer
53 views

The set of analytic functions on unit circle is not a C*-algebra

Let $\mathbb{D}$ be the open unit disc on the complex plane and consider the set $$A=\{f\in C({\rm cl}\, {\Bbb D})\colon f \text{ is an analytic function on } {\Bbb D}\}.$$ It is certainly closed ...
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1answer
23 views

Is $C(\Omega)$ a C*-algebra if $\Omega$ is not locally compact, nor compact?

We always say if $\Omega$ is compact or locally compact, then C(\Omega) is a C*-algebra. Now is $C(\Omega)$ a C*-algebra if $\Omega$ is not compact nor locally compact? If not, I want to know which ...
2
votes
1answer
36 views

spectral projection of an element in a C*-algebra

I'm studying Takesaki's Operator theory and I preferred "spectral projection "in the page 43 of this book while he didn't speak about it before. I searched it, but I could not find it. Please explain ...
2
votes
1answer
40 views

Positive linear functional on an involutive Banach algebra

Why is every positive linear functional on an involutive Banach algebra with a bounded approximate continuous?
2
votes
1answer
37 views

Approximate unit of an involutive Banach algebra

I know that every C*-algebra has an approximate unit. I have two questions: why we cannot show that every involutive Banach algebra has an approximate unit? I need an example of an involutive Banach ...
1
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1answer
32 views

Positive elements in a Banach algebra

Let $A$ be a unital Banach algebra. If $a$ is an element of $A$ with $||1-a||_{sp}<1$, then there exists $b\in A$ such that $b^2=a$. Furthermore, if $A$ is an involutive Banach algebra and if $a$ ...
2
votes
1answer
36 views

Cyclic representation on $L^2(\mu)$

Show that if $(X,\Omega,\mu)$ is a $\sigma-$ finite measure space and $H=L^2(\mu)$, then $\pi:L^\infty(\mu)\to B(H)$ defined by $\pi(\phi)=M_\phi$ is a cyclic representation and find all the cyclic ...
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vote
0answers
35 views

Approximate unit of a separable C*-algebra

The following is a corollary of Takesaki's Operator Theory: My question: I do not know why the author says"there exists an n such that $||x(1-v_n)^\frac{1}{2}||<\epsilon$" . Please help me to ...