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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When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
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132 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 ...
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1answer
19 views

Example of Banach algebra with empty character space

Recently I read through a proof that the character space of a commutative unital Banach algebra is non-empty. Since all the assumptions are necessary it should be possible to give an example of a ...
4
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1answer
77 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 ...
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1answer
28 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 ...
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2answers
52 views

Space of Functions: 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 ...
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1answer
32 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 ...
10
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1answer
385 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 ...
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0answers
46 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 ...
3
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1answer
61 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 ...
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14 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 ...
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89 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 ...
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1answer
23 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 ...
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0answers
19 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
38 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
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2answers
34 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 ...
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1answer
48 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 ...
9
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1answer
262 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 ...
5
votes
1answer
161 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 $x^{\ast}=x^{\#}$, ...
2
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2answers
82 views

Is the group algebra separable?

Let $G$ be a locally compact Hausdorff group. Is the group algebra $L^1(G)$ separable? Would you please help me or introduce references that can help me. Thanks
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58 views

Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of unital 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 ...
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1answer
86 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 $f$?
2
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1answer
31 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
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57 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 ...
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votes
1answer
38 views

Convergence of product of convergent sequences [closed]

How can we prove the following theorem about term-by-term products of convergent sequences? Theorem: If $(a_n)$ and $(b_n)$ are sequences in normed algebra $\mathcal {A}$ and $a,b\in\mathcal{A}$ such ...
6
votes
3answers
85 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
1
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1answer
22 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 ...
5
votes
1answer
114 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.
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21 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 ...
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1answer
60 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
1answer
28 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 ...
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0answers
29 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 ...
5
votes
1answer
40 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 ...
6
votes
1answer
175 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
2
votes
1answer
43 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
68 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
2answers
50 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 ...
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1answer
61 views

Amenability of finite dimensional norm algebras

Let $(\cal A,\|\cdot\|)$ be a finite dimensional norm algebra (Banach Algebra). Can we say any thing about the amenability of $\cal A$. What if we impose some extra conditions on $\cal A$, say ...
4
votes
2answers
47 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
3
votes
1answer
82 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 $F:[0,1]^n \mapsto [0,1]^n$, ...
3
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1answer
47 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, $f_{y}(x)= f(x-y)$, ...
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vote
0answers
27 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 ...
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votes
1answer
36 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
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1answer
41 views

In unital Banach algebra $r(a^n) = (r(a))^n$

I tried to prove the following: If $A$ is a unital Banach algebra and $r(a)$ denotes the spectral radius then $r(a^n) = (r(a))^n$. Could somebody please tell me if I got this proof right? Thanks. ...
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0answers
29 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 ...
3
votes
2answers
207 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
1
vote
1answer
24 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 ...
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votes
1answer
62 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: ...
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votes
1answer
112 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 ...
1
vote
1answer
50 views

Spectrum: Polynomials

It is written in Bratteli-Robinson that some simple transformations yield the relations: $$\sigma(a+A)=a+\sigma(A)$$ $$\sigma(A^n)\subseteq\sigma(A)^n$$ The latter one is deduced by the ...