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
31 views

Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
2
votes
2answers
100 views

Is there any multiplicative linear functional on B(H)?

If A is a Banach algebra, we say that $\Phi: A \longrightarrow \mathbb{C}$ is a multiplicative linear functional if $\Phi$ is nontrivial, linear and $\Phi(xy)=\Phi(x)\Phi(y)$. It is easy to see that ...
1
vote
0answers
106 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
0
votes
1answer
54 views

The map $T\longmapsto \|T\|$ is not continuous in the strong operator topology of $\mathscr B(H)$

In the context of Strong and Weak operator topologies on $\mathscr B(H)$ there is an statement that says: the map on $\mathscr B(H)$ that $T\longmapsto \|T\|$ is not continuous in the strong operator ...
1
vote
1answer
57 views

an exercise about the spectrum of an element in Banach algebra.

An exercise of Banach algebra, section of spectrum has wanted the proof of this statement: Let $A$ be a Banach algebra and $x\in A$. Show that for every open set $U$ in $\mathbb{C}$ that contains ...
2
votes
1answer
63 views

Spectral radius of a normal element in a Banach algebra

I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ...
0
votes
1answer
144 views

Spectrum of an element of a non unital C*-algebra

I know that spectrum of an element $x$ of a unital C*-algebra is nonempty. I like to find an example of a non unital C*-algebra that has an element with empty spectrum, if it exists. Motivation I ...
3
votes
3answers
173 views

Spectral radius of an element in a C*-algebra

The following is an proposition of Takesaki's Operator Theory: For any element $x$ of a Banach algebra ${\cal A}$, we have $$||x||_{sp}=\lim_{n\to \infty}||x^n||^{\frac{1}{n}}$$ Proof: My ...
1
vote
1answer
51 views

Norm of a character in a non-unital Banach algebra

Let $\cal A$ be an abelian non-unital Banach algebra and $h:{\cal A}\to {\Bbb C}$ be a homomorphism. If ${\cal A}$ has an approximate identity $\{e_i\}$ such that $||e_i||\leq 1$ for all i, then ...
0
votes
0answers
29 views

Locally Compact Groups - Reference Request

I start reading an article about locally compact groups $G$ and the group algebra $L^1(G)$,and I need a good book to introduce myself to these concepts. Can you help please? Thanx
1
vote
1answer
78 views

Subalgebra generated by selfadjoint operator $A_0\in\mathscr{L}(H,H)$

Let $\mathscr{L}(H,H)$ be the Banach algebra of bounded operators defined on a complex Hilbert space $H$ and let $B(A_0)$ be the subalgebra generated by the selfadjoint operator $A_0$, i.e. ...
2
votes
1answer
67 views

Algebra of bounded functions on a completely regular space

Let $T$ be a completely regular topological space, i.e. a topological space satisfying axiom $T_1$ and such that for any closed set $F\subset T$ and any $t_0\in T\setminus F$ there is a continuous ...
0
votes
1answer
48 views

When Heine - Borel theorem holds

If $\cal A$ is an abelian Banach algebra, then its maximal ideal space $\Omega$ is a compact Hausdorff space. In the proof of this theorem, the author says, since $\Omega \subset ball \cal A^*$, it ...
1
vote
1answer
82 views

Example of a proper, dense ideal in a unital Banach algebra

I know that if ${\cal A}$ is a non- unital Banach algebra, then ${\cal A}$ may contain some proper, dense ideals. I need an example of a proper, dense ideal in a unital Banach algebra or show that it ...
0
votes
1answer
26 views

Ideal $I$ contained in a non-trivial ideal and quotient algebra

I read, in Tikhomirov's appendix to Kolmogorov-Fomin's Элементы теории функций и функционального анализа, that in order that the ideal $I$ be contained in a non-trivial ideal $I'\subset X$, it is ...
0
votes
1answer
40 views

$\lambda\in \sigma(x)\Rightarrow \lambda^{n}\in\sigma(x^{n})$ in Banach algebra

Let $\sigma(x)$ be the spectrum of an element $x$ of a unitary Banach algebra $X$ with the unity $e\in X$. I read that, since $\lambda e-x$ divides $\lambda^n e-x^n$, we have $\lambda\in ...
2
votes
1answer
47 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
4
votes
1answer
134 views

Greatest open ball of invertible elements in a Banach algebra

Let $a$ be an invertible element of a Banach algebra $A$. Then we know that also each $a+b$ with $b\in A$ and $||b||<||a^{-1}||^{-1}$ is invertible. Now my question is whether ...
2
votes
1answer
36 views

Condition in the definition of Banach star algebra

Here the definition of Banach star algebra is given as Banach algebra with an involution. In the book by Murphy for example, it is given as Banach algebra with an involution plus the condition that ...
1
vote
1answer
81 views

The supremum norm is submultiplicative

Is the following proof correct: Let $X$ be compact a compact Hausdorff space and $C(X)$ the continuous functions $f: X \to \mathbb{C}$ on X. We can equip $C(X)$ with the (edit: sorry, semi-)norm ...
3
votes
0answers
80 views

Prove that a norm makes a space Banach

I have to prove that if $A$ is a C*-Algebra then the algebra $A_1$ obtained adjoining the identity is a C*Algebra too (with the usual algebraic operation defined). I have any problem in all the ...
0
votes
0answers
15 views

Prove a condition for a Banach algebra [duplicate]

Can anyone help me by providing a detailed verification of the following theorem? **Let $\mathcal{A}$ be a Banach algebra.If all $a,b\in\mathcal{A}$ goes $$\Vert ab \Vert= \Vert a \Vert\Vert b ...
0
votes
2answers
66 views

Function Spaces: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
3
votes
1answer
76 views

Prove a condition for a Banach algebra to be isometrically isomorphic to $\mathbb C$

Can anyone help me by providing a detailed verification of the following theorem? Let $\mathcal{A}$ be a Banach algebra. If there exists $M<+\infty$ so that $$\Vert a \Vert\Vert b \Vert\leq M ...
0
votes
1answer
26 views

Explaining the theorem

By searching this url http://www2.math.ou.edu/~cremling/teaching/lecturenotes/fa-new/ln7.pdf in google given Theorem 7.4. and given its proff, but I do not understand very well this proof because it ...
2
votes
0answers
65 views

Riesz Projection in Functional Analysis.

By definition the Riesz projection of a Banach algebra element $a$ associated with a complex number $\alpha$ is given by $p(\alpha , a)= \frac{1}{2\pi i} \int_{\Gamma} ( \mu -a)^{-1} \,\, ...
2
votes
2answers
56 views

If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element of Banach algebra $\mathcal{A}$ with unit $e$.

I was reading an article yesterday which was silent on the algebra of Banach. In that article was provided this example If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element ...
2
votes
2answers
60 views

Why is the Gelfand transform injective?

There is a theorem that proves that if $A$ is a commutative C*-Algebra, the Gelfand map is an isometric *-isomorphism of $A$ onto $\hat{A}$ i.e. the spectrum of $A$. (Theorem 1.1 in Averson's "An ...
1
vote
1answer
70 views

Characterizing C* algebra generated by elements.

Let $A$ be a C*-algebra and $A_0\subset A$. Then it is known that the $\mathbb{C}$-algebra generated by $A_0$ (i.e. the intersection of all sub-$\mathbb{C}$-algebras containing it ) is just the ...
2
votes
1answer
50 views

Extension theory and automorphism extension

My question is motivated by the following two posts On finite 2-groups that whose center is not cyclic and Automorphisms of group extensions Question: Assume that $A,B,C$ are there algebraic ...
3
votes
2answers
39 views

Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
0
votes
0answers
23 views

Continuity of $\zeta(a)$: is my proof correct?

Let $A$ be a unital Banach algebra and define $\displaystyle \zeta (a) = \inf_{c \in A: \|c\| =1}\|ac\|$. I tried to prove $|\zeta (a) - \zeta (b)| \le \|a-b\|$ for all $a,b \in A$, could someone ...
2
votes
0answers
73 views

Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
-1
votes
1answer
69 views

How to prove that this inequality holds

Let $A$ be a unital Banach algebra. I wanted to prove the following inequality but didn't manage: $$ \begin{align} \left | \|a\| - \inf_{d \in A: \|d\| = 1}\|bd\| \right | \le \inf_{\|d\|=1} \left ...
1
vote
0answers
41 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
1
vote
1answer
33 views

Show that P is an L-projection iff $P^{*}$ is an M-projection

I have started reading "M-ideals in Banach spaces and Banach algebras", but I stuck on the first page. It says that "there is an obvious duality between L- and M- projections: P is an L-projection ...
5
votes
1answer
55 views

On the spectrum of a product in a Banach algebra, in specific case

Let $A$ be a Banach algebra, and suppose that $a,b\in A$ have spectra that satisfy: $\sigma(a) \subset U$, and $\sigma(b)\subset U$, where $U$ is the open right half-plane of complex numbers with ...
4
votes
2answers
75 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
5
votes
1answer
114 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 ...
4
votes
1answer
63 views

C*-algebra representations

Let A be a C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space. Putting $\phi=\sum_n 2^{-n} \phi_n$, can you show that $\phi$ is a state and the representation $\pi_\phi$ ...
3
votes
1answer
50 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
3
votes
1answer
280 views

How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
1
vote
0answers
32 views

For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
9
votes
1answer
366 views

Maximal ideals and maximal subspaces of normed algebras

This is a kind of "prove or give a counter-example" question, and I'm having some difficults with it: By a maximal ideal $I$ of an algebra $A$, we mean an ideal $I\neq A$ which is not properly ...
0
votes
0answers
34 views

Follow up on star algebra (proof verification)

I previously asked this question about a proof of the following claim: If $A$ is a commutative non-unital non-zero $C^\ast$ algebra then $\Omega (A)$ is not empty. In the meantime I believe to ...
0
votes
1answer
38 views

Where is commutativity of $b$ needed?

I have a question about the following proof: If $e^{ia}-e^{i\lambda}=(a-\lambda)be^{i\lambda}$ and $(a-\lambda)$ is not invertible then $(a-\lambda)x$ is not invertible for all $x$. Why "since $b$ ...
1
vote
1answer
37 views

Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
0
votes
2answers
116 views

Positive Elements: Characterization

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Define positive elements as: $$A\geq0\iff\sigma(A)\geq0\quad(A=A^*)$$ Positive elements can be characterized by: $$A\geq0\iff A=B^*B$$ ...
2
votes
1answer
124 views

Norm on unitisation of a $C^\ast$ algebra

In the theory of $C^\ast$ algebras there exists the following theorem: If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
5
votes
1answer
168 views

Positive Operators: Definition?

Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: ...