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) ...

learn more… | top users | synonyms

0
votes
1answer
59 views

Density of linear span of idempotents in $L^{\infty}$

How do I show that the linear span of idempotents is dense in $L^{\infty}(\Omega,\mu)$ where $(\Omega,\mu)$ is a measure space? I don't really have any idea how to do this. Does it involve ...
0
votes
2answers
87 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 ...
2
votes
1answer
104 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
0
votes
1answer
46 views

(Commutative Banach algebra) Prove that $G(\mathcal A)$ be an open set in $\mathcal A$.

UPDATE I have a problem: Let $\mathcal A$ be a commutative Banach algebra. Denote $G(\mathcal A)$ is the set of all invertible elements in $\mathcal A$. Prove the following assertions: ...
0
votes
0answers
33 views

Identity in Banach Algebras

This is an extract from Douglas: "Banach Algebra Techniques in operator theory". "For Banach algebras and, in particular, for $C(X)$ the importan idea is that of multiplicative linear functional. ...
3
votes
1answer
212 views

Banach Algebra: $\sigma(xy)\cup\{0\} = \sigma(yx)\cup\{0\}$

It is Rudin excercise 10.4 where we aim to prove $\sigma(xy)\cup\{0\} = \sigma(yx)\cup \{0\}$ for elements $x,y\in A$ a Banach-algebra.( $\sigma$ being the spectrum) In (a) we prove that $e-yx$ ...
3
votes
1answer
70 views

$L(\ell_{p})$ contains only one proper closed ideal

I am trying to solve the following problem: Show that if $1<p<\infty$ and $T:\ell_{p}\rightarrow\ell_{p}$ is not compact then there is a complemented infinite dimensional subspace $E$ of ...
1
vote
1answer
98 views

If a field contains the complex field, then it is $\mathbb{C}$

This question is originated from a book by Gaal, (Linear Analysis and Representation Theory). Theorem 7 from section 6, chapter 1 reads as follows, and I quote: "Theorem 7: Let $A$ be a complex, ...
1
vote
1answer
41 views

Non-modular maximal ideal in abelian Banach algebra

Let $A$ be the disk algebra (i.e. the algebra of all functions that are continuous on the closed unit disk and analytic on the open unit disk) and let $A_{0}=\{f\in A:f(0)=0\}$. Then $A_{0}$ is a ...
3
votes
0answers
50 views

Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of Banach algebras. It's a standard fact that if $f$ is has closed range, i.e. $A_1$ is embedded as a closed subalgebra of $A_2$, then for every $x ...
0
votes
1answer
35 views

Help me please proving the theorem

I am reading the book of Walter Rudin Functional Analysis and the page 235 was given theorem 10:12 which is as follows: Theorem: If $A$ is a Banach algebra, then $G(A)$ is a open subset of $A$, and ...
3
votes
1answer
126 views

A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form ...
2
votes
0answers
64 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
0
votes
0answers
31 views

Diagonalizing operator over $L^2(\mathbb{T})$

I've been asked to diagonalise an operator on $L^2(\mathbb{T})$, given by $Tf(z) = f(z^{-1}$). I know that I'm expected to find a $U$ such that $TU = UM_f$, where $M_f$ is the multiplication operator, ...
1
vote
0answers
75 views

Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
-1
votes
0answers
188 views

Positive elements in a C*-algebra [closed]

Prove that if $a$ is an element in a $C^*$-algebra $A$, then $a$ is positive if and only if $f(a) \geq 0$ for every state $f$ on $A$.
1
vote
0answers
15 views

Closure of the set of fredholm perturbation

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
8
votes
1answer
222 views

Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the ...
5
votes
1answer
103 views

Gelfand transform is an isometry

I'm having a bit of trouble showing that the Gelfand transform $A \rightarrow C(\operatorname{sp}(A))$ is isometric iff $\|x^2\| = \|x\|^2$ for a general unital commutative Banach algebra. For a $C^*$ ...
1
vote
0answers
37 views

Definite positive measure and GNS representation

Let $G$ be a locally compact group. Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$. In [D, p ...
-1
votes
1answer
59 views

Is power of convergence nets, convergence?

Let $A$ be a Banach algebra and $(f_{\alpha})$ is a net in $A$ and convergence in norm to $f$. Is $(f^{n}_{\alpha})$ convergence in norm to $f^{n}$ for every $n$ in $\mathbb{N}$?
5
votes
1answer
96 views

Example of a singular element which is not a topological divisor of zero

We know that every topological divisor of zero in a commutative Banach algebra is singular. I need an example of a singular element which is not a topological divisor of zero.
2
votes
0answers
45 views

Character space of $L^{1} (\mathbb Z)$

I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$. Are the Gelfand and norm topologies equal, on the character space of $L^{1} (\mathbb ...
4
votes
0answers
117 views

Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
4
votes
1answer
143 views

Uniqueness of the involution on a $C^*$-algebra

indication please Let $A$ be a C*-algebra. Suppose that there exists on $A$ another involution $x\rightarrow x^{\#}$ such that $||xx^{\#}||=||x||^2$, for all $x\in A$. Prove that $x^{\ast}=x^{\#}$, ...
0
votes
1answer
56 views

Roots in Banach algebras.

I'm studying positive functionals on normed algebras and I got stuck in the following problem: Let $A$ be a unital Banach algebra, and $x\in A$ be such that $\Vert x\Vert <1$. Then the series ...
1
vote
1answer
36 views

A question about Banach algebras: showing that $\operatorname{Sp}a \subset D_o \cup D_1$

Maybe this problem be easy for a person that have study in Banach Algebra; please give me a hint. Let $e=0$ or $1$, and $a$ be an arbitrary element in a Banach algebra $A$. Let $D_o$ and $D_1$ be the ...
1
vote
0answers
71 views

Commutative Noetherian Banach algebra.

Prove that: 1) Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. 2) Every commutative real unital Noetherian Banach algebra ...
4
votes
2answers
116 views

Commutative unital Banach algebra with nilpotent elements

What would be a concrete example of a commutative unital Banach algebra that contains nilpotent elements?
3
votes
1answer
101 views

Show that: If $A$ is an arbitrary abelian Banach algebra, its spectrum is totally disconnected

I don't know how should I start to show: If $A$ is an arbitrary abelian Banach algebra in which the idempotents have dense linear span, its specrum (the space of characters on $A$) is totally ...
2
votes
1answer
109 views

Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge (or monotonous converge) to $1$ in $A^+$, does $\{x_n\}$ must be the ...
10
votes
1answer
335 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 ...
7
votes
3answers
521 views

Banach Algebra counterexample

Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof) Thank you very much :)
3
votes
1answer
115 views

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by ...
2
votes
1answer
129 views

Constructing a functions with Gelfand Naimark

If $X$ and $Y$ are compact Hausdorff spaces, show that for any algebra homomorphism $$ F:C(Y) \to C(X) $$ there exists a continuous function $f:X\to Y$ such that $$ F(\phi)=\phi \circ f, \forall \phi ...
2
votes
1answer
183 views

Application of Stone-Weierstrass with a non-unital algebra

Let $X$ be a locally compact Hausdorff space. We say that a function $f\colon X \to \mathbb{R}$ vanishes at $\infty$ if for each $\epsilon >0$ there exists a compact $K_\epsilon \subset X$ such ...
2
votes
1answer
46 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 ...
8
votes
1answer
299 views

Understanding topological divisor of zero

I am reading this paper also here where in the Theorem 2.1 term topological divisor of zero has been used. I have gone through wiki articles where it has been mentioned that An element $z$ of a ...
1
vote
0answers
80 views

Approximation of certain continuous functions by analytic functions

Let $f\in C(S^{1},M_{n}(\mathbb{C}))$ be a unitary. Does there exist an analytic unitary function $g$ from $S^{1}$ to $M_{n}(\mathbb{C})$ that approximates $f$?
0
votes
0answers
52 views

Resolution of Identity

I am reading the twelfth chapter of Rudin's Functional Analysis and am unable to understand two comments he makes in passing in the section on Resolution of Identity. Let E be a resolution of identity ...
1
vote
0answers
49 views

Resolution of Identity

I am reading the twelfth chapter of Rudin's Functional Analysis and am unable to understand two comments he makes in passing in the section on Resolution of Identity. Let E be a resolution of identity ...
0
votes
3answers
99 views

The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$

On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by ...
1
vote
0answers
85 views

Semisimple commutative Banach algebra

On a semisimple commutative Banach algebra all Banach algebra norms are equivalent. Is this true without assuming semisimplicity?
1
vote
0answers
53 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
2
votes
1answer
186 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
2
votes
1answer
91 views

Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?

Let $A$ be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in $A$ such that its spectrum is infinite.
2
votes
1answer
58 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
1
vote
1answer
42 views

Real commutative Banach algebra with identity

I am looking for example of real commutative Banach algebra with identity which does not admit a nonzero real multiplicative linear functional
-1
votes
1answer
80 views

Spectrum of a unitary

I have a unitary element $v$ in $C(S^{1}, \mathbb{C})$ with full spectrum (the whole circle). Is it possible to construct another unitary $u$ in $C(S^{1}, \mathbb{C})$ out of $v$ such that the ...
1
vote
1answer
48 views

Absolute value of an element in a C*-algebra

Is absolute value of a partial isometry a partial isometry itself?