3
votes
1answer
66 views

Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
0
votes
1answer
61 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
3
votes
0answers
87 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
-1
votes
1answer
41 views

Polynomial Calculus on Spectrum: well defined?

Consider a bounded operator over a Banach space: $T\in\mathcal{B}(E)$ Apply polynomial calculus on the the chosen operator: $p(T),p\in\mathbb{C}[X]$ Why do we need to prove that when two polynomials ...
0
votes
1answer
53 views

Root of polynomial implies vanishing remainder. Application to spectral theory!

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
2
votes
1answer
67 views

Convergence Radius => Nonanalytic

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
1
vote
0answers
25 views

Show that $B(X)$ is semisimple for a Banach space $X$ [duplicate]

Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$. I ...
0
votes
1answer
66 views

boundary of a spectrum proof

Let $A$ be a closed unital subalgebra of banach algebra $B$. Prove that ${\delta}{\sigma}_{B}(x)$ is contained in ${\delta}{\sigma}_{A}(x)$ for every $x$ in $B$.
0
votes
1answer
57 views

homomorphism or not

Let $T$ be a bounded operator on $H$ and fix a vector $x\in H$. Define $f$ on the space of polynomials in $T$ by $f(p(T))=p(x)$. Is $f$ a homomorphism? Initally I thought it obvious but the subtelty ...
0
votes
1answer
33 views

Gelfand Transform in a specific case

What is the gelfand transform of an operator in the algebra generated by a bounded normal operator and it's adjoint? Thanks
0
votes
0answers
54 views

Properties of an additive mappings which preserves projections

Let $A$ and $B$ be two $C^{*}$-algebras and $\Phi:A\longrightarrow B$ be an additive map which satisfies $\Phi(0)=0$, $\Phi(I)=I$ and $\Phi$ preserves projections, (i.e, $\Phi(P)=Q$ where $Q$ is also ...
0
votes
1answer
97 views

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the ...
0
votes
2answers
90 views

Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
10
votes
1answer
361 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
2
votes
1answer
47 views

limit of evaluated automorphisms in a Banach algebra

Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible ...
0
votes
1answer
60 views

The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$

Suppose $x$ is invertible in the unital Banach algebra $A$. How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$
4
votes
1answer
148 views

A problem on bounded invertible linear operator in Banach space

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
1
vote
0answers
181 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
2
votes
1answer
60 views

Spectrum in Hilbert space

Let $H$ be a Hilbert space and $T\in B(H)$ be normal. Want to show that if $\lambda$ is an isoloated point in $\sigma(T)$ then $\lambda$ is an eigenvalue of $T$.I have no idea how to do this one. ...
0
votes
1answer
39 views

$‎\sigma(a)=‎\sigma‎(b)‎$‎‎, ‎‎if ‎‎‎$‎a,b$‎ ‎‎are unitarily equivalent

‎Let ‎$‎A$ be a *-algebra and ‎$‎a,b$ are ‎unitaril‎y equivalent ‎in ‎‎$‎A$ ( i.e. there exists a unitary ‎$‎u$ of ‎$‎A$ s.t ‎$b=uau^{*‎}‎$ ‎‎).‎ ‎I ‎want ‎to ‎prove ‎that ...
0
votes
0answers
43 views

$\widehat{a}: \Omega(A)‎\rightarrow‎ \mathbb{C}~,~\tau‎ \mapsto \tau(A)‎‎ $

Suppose that $A$ is abelian Banach algebra for which the space $\Omega(A)$ is non-empty. If $a \in A$, we define the function $\widehat{a}‎‎$ by $$\widehat{a}: \Omega(A)‎\rightarrow‎ ...
1
vote
1answer
34 views

Gelf‎and ‎representation ‎Theorem

In ‎proof ‎of "‎‎Gelf‎and ‎representation ‎Theorem‎" ‎(see 1.3.6 Theorem of Murphy's book )‎, I ‎am ‎understanding ‎that ‎why ‎the ‎map $$ A ‎‎\rightarrow‎ ‎C_{0}(‎\Omega(A)‎)~ , ‎~‎‎a‎‎\rightarrow‎ ...
1
vote
1answer
158 views

‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$ ,then $\bigcap_{v\in K(X)}\sigma(u+v) =\cdots$

‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$,why the following equality is true? $$\bigcap_{v\in K(X)}\sigma(u+v) =\sigma(u) \setminus \{\lambda \in\mathbb{C}\mid u - ...
-1
votes
1answer
101 views

The ‎inclusion relation $\sigma(ab) ‎\subseteq ‎\sigma(a)‎\sigma(b)$ is not true for all Banach algebras

Let ‎‎$‎A$ ‎be a‎ ‎unital ‎abelian‎ ‎Banach ‎algebra. ‎Give ‎me ‎an ‎example ‎that two ‎following ‎inclusion ‎relations ‎is ‎not ‎true ‎for ‎all ‎Banach ‎algebras‎ $$\sigma(a+b) ‎\subseteq ...
1
vote
0answers
46 views

The hermitian element $h=\sum_{n=1}^\infty \frac{p_{n}}{3^{n}}$ generates $C_{0}(\Omega)$‎

‎Please help me to solve the following problem‎ : Let $\Omega$ be a locally compact Hausdorff space‎, ‎and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections ...
1
vote
1answer
40 views

A Linear map $‎u : X ‎\longrightarrow ‎Y‎‎$ ‎ ‎is ‎not ‎bounded ‎below ‎‎iff ‎there ‎is …

Do you help me to: c‎hecking ‎that a‎‎ ‎linear ‎map ‎‎$‎u : X ‎\longrightarrow ‎Y‎‎$ ‎between ‎Banach ‎spaces ‎is ‎not ‎bounded ‎below ‎if ‎and ‎only ‎if ‎there ‎is a‎ ‎sequence ‎of ‎unit ‎vector ...
2
votes
2answers
116 views

‎‎If $A$ contains ‎an ‎idempotent $e‎$ (‎‎$‎e‎\neq ‎‎0,1‎‎$‎) , then $‎\Omega(A)‎$ ‎is ‎disconnected

If $A$‎ ‎be a‎ ‎unital ‎abelian ‎Banach ‎algebra ‎and ‎contains ‎an ‎idempotent $e$‎ ‎(that ‎is ‎‎$‎e=‎e‎^{‎2‎}‎‎$‎) ‎other ‎than $0$‎ ‎and $1$‎ ,‎ ‎then help me to show that ‎‎$‎\Omega(A)‎$ ‎is ...
-3
votes
1answer
104 views

If a,b ‎are ‎unitary ‎equivalent,‎Dose ‎ ‎‎$‎\sigma(a)=‎\sigma(b)‎$‎ is true?

‎‎Let A‎ ‎is ‎an ‎unital‎‎ ‎algebra ‎and ‎‎$ ‎Ad‎~u:‎‎‎A\rightarrow ‎A~,~a‎\mapsto~‎uau‎^{*}‎‎$ ‎and u‎ ‎is ‎unitary ‎element ‎of A‎(‎$‎uu‎^{‎*‎}=‎u‎‎^{*}‎u=1‎$‎), ‎if ‎‎$‎b=‎uau‎^{‎*‎}‎‎$ ‎(a,b ‎are ...
6
votes
0answers
127 views

invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
5
votes
1answer
317 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
4
votes
1answer
190 views

$\sigma(x)$ has no hole in the algebra of polynomials

Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras. This amounts to prove that ...
2
votes
0answers
39 views

Initial topology of the spectrum mapping $\sigma$

Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$. The collection of compact subsets ...
2
votes
2answers
90 views

Multiplication operators

Consider a commutative Banach algebra $A$ and the Banach algebra of bounded operators $B(A)$ on $A$. Associate to each $a\in A$ the multiplication operator $T_ax =ax$ ($x\in A$). Is always the mapping ...
4
votes
1answer
218 views

Schwarz inequality for unital completely positive maps

I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper: Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between ...
5
votes
2answers
203 views

Why are compact operators 'small'?

I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it. I know that compact operators map bounded sets to totally bounded ones, that ...
2
votes
0answers
51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
2
votes
2answers
91 views

What does $(B+I)/I\sim B/(B\cap I)$ tell us?

Let $A$ be a $C^*$-algebra in which $B$ is a $C^*$-subalgebra and $I$ is a closed ideal. In several books on $C^*$-algebras I have encountered the following: $(B+I)/I$ is $*$-isomorphic to ...
1
vote
1answer
106 views

Every ideal has an approximate identity?

Averson's 1970 paper on extensions of $C^*$-algebras seems to assume that every ideal has an approximate identity. However, I am a little bit suspicious here, since he does not assume the closeness ...
0
votes
2answers
211 views

Linear functionals can be decomposed as linear combinations of positive ones?

I am reading Arveson's Notes on Extensions of $C^*$-algebras. In proving theorem 1, he needs to establish some results concerning bounded linear functionals. However, he said it suffices to prove for ...
3
votes
2answers
134 views

If $a\ge 0$ and $b\ge 0$, then $\sigma(ab)\subset\mathbb{R}^+$.

This is an exercise in Murphy's book: Let $A$ be a unital $C^*$-algebra and $a,b$ are positive elements in $A$. Then $\sigma(ab)\subset\mathbb{R}^+$. The problem would be trivial if the algebra ...
1
vote
1answer
98 views

References on Algebraic Operators

Let $\mathcal{H}$ be a Hilbert space and $d$ is an inner derivation on $\mathcal{L}(\mathcal{H})$. An operator $T\in\mathcal{L}(\mathcal{H})$ is algebraic if $p(T)=0$ for some polynomial $p$. In ...
5
votes
1answer
70 views

Unital nonabelian banach algebra where the only closed ideals are $\{0\}$ and $A$

This is a problem in exercise one of Murphy's book Find an example of a nonabelian unital Banach algebra $A$, where the only closed ideals are $\{0\}$ and $A$. But does such an algebra exist at ...
1
vote
0answers
66 views

When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
2
votes
1answer
292 views

The spectra of weighted shifts

Since weighted shifts are like the model-operators in operator theory and people have been studying them for so long, I think there should be quite a large literature on the spectra of such operators. ...
4
votes
1answer
144 views

A subset of $\bar{S}\backslash S$ contains an open ball in $\bar{S}$? (operator theory)

E and S are subsets of a metric space. $E$ is a subset of $\bar{S}\backslash S$. Then $\overline{E}\subset(\overline{S}\backslash S^{o})$, but I wonder whether there is some condition that guarantees ...
3
votes
1answer
282 views

Reflexive Banach algebras?

I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces. For a Banach space $X$, we investigate its dual $X'$ ...
11
votes
1answer
687 views

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
6
votes
1answer
444 views

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
9
votes
1answer
338 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from ...
4
votes
2answers
310 views

If $(I-T)^{-1}$ exists, can it always be written in a series representation?

If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$. Thinking in terms of a ...