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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3
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1answer
14 views

States in a $C^*$-algebra bounded?

A functional $\phi$ on a $C^*$-algebra $A$ with unit element, i.e. $\phi: A \rightarrow \mathbb{C}$ is called a state if $\phi(T^*T) \ge 0$ for all $T \in A$ and $\phi( \operatorname{id}) = 1.$ Now, I ...
2
votes
0answers
21 views

Operator norm of matrix of scalars regarded as matrix with entries in the unitization of a Banach algebra

Let $A$ be a non-unital Banach algebra, and let $A^+=\{(a,z):a\in A,z\in\mathbb{C}\}$ with product $(a,z)(b,w)=(ab+zb+wa,za)$ and norm $||(a,z)||=||a||+|z|$. Equip $M_n(A^+)$ with the operator norm by ...
2
votes
1answer
57 views

Pythagorean theorem does require the Cauchy-Schwarz inequality?

In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved ...
2
votes
0answers
16 views

A condition on quotient norms on quotient Banach algebras

Let $A$ be a non-unital Banach algebra, and let $A^+$ be the unitization of $A$ consisting of elements of the form $(a,z)$ where $a\in A$ and $z\in\mathbb{C}$ with multiplication ...
0
votes
1answer
23 views

‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎

Let ‎$‎‎A$ ‎be a unital ‎‎‎$‎‎C^*$-algebra. ‎‎ Assume that ‎$‎‎a\in A$ ‎is a ‎‎normal ‎and ‎invertible element ‎i.e ‎‎$‎‎aa^*=a^*a$ ‎and ‎‎$‎‎aa^{-1}=a^{-1}a=1$‎.‎ ‎let $‎‎C^*({a}) $ be the ...
0
votes
1answer
37 views

Why the set of pure state ‎is ‎weak* ‎compact?

Let ‎$‎‎A$ ‎be a‎ ‎C*-algebra‎. ‎ ‎$‎S(A)‎$ ‎is ‎the ‎set ‎of ‎state ‎on ‎‎$‎‎A$ and $‎‎PS(A)$ ‎is ‎the ‎set ‎of ‎pure ‎state ‎on ‎‎$‎‎A$. ‎ ‎ I ‎know ‎that ‎if ‎‎$‎‎A$ ‎is ‎unital ‎then ‎‎$‎‎S(A)$ ...
0
votes
0answers
16 views

Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
7
votes
1answer
219 views

Part of Lomonosov's Invariant Subspace Theorem

Let $X$ be a complex Banach space of infinite dimension, let $T\in\mathcal{B}(X)\backslash\{0\}$ be compact. Define $$\Gamma := \{S\in\mathcal{B}(X)\,|\,S\circ T=T\circ S\}$$and define, for each ...
1
vote
1answer
21 views

If $X$ is maximal ideal then it consists of non-invertible elements?

I'm reading through a paper where I came across the following theorem Let $A$ be a commutative complex Banach algebra with unit element $e$. Theorem: A subspace $X\subset A$ of codimension $1$ is ...
5
votes
1answer
97 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
0
votes
0answers
21 views

Local banach algebra without zero divisors

I need to construct example of banach algebra with unique nontrivial maximal ideal and without zero divisors. I think it is must be a subalgebra of $\mathbb{C}[[z]]$, but I could not build anything.
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0answers
20 views

postliminal $C^*$-algebra

A ‎$‎‎C^*$-algebra ‎‎$‎‎A$ ‎is ‎said ‎to ‎be ‎postliminal ‎if ‎for ‎every ‎non-zero ‎irreducible ‎representation ‎‎$‎(H,‎\varphi‎)‎$ ‎we ‎have ‎‎$‎‎K(H)‎\subseteq‎ ‎‎\varphi‎(A)‎$‎ ‎ In ‎Murrphy's ...
0
votes
0answers
24 views

The image of a continuous derivation on a Banach algebra is contained in the kernel of a character.

It is known that if $D$ is a continuous derivation on a commutative Banach algebra $\mathcal{A}$, then for any nonzero character $\theta$ on $\mathcal{A}$ we have $D(\mathcal{A})\subseteq ker ...
2
votes
1answer
21 views

property of the Gelfand transform: why does an isometry map closed sets to closed sets?

The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof: Here are my questions: Why $\Gamma:\mathfrak{U}\to ...
0
votes
0answers
14 views

Homotopy of bounded homomorphisms between Banach algebras

Let $A$ and $B$ be Banach algebras. Say that two bounded homomorphisms $\phi_0$ and $\phi_1$ from $A$ to $B$ are homotopic if there is a path $(\phi_t)_{t\in[0,1]}$ of bounded homomorphisms from $A$ ...
0
votes
0answers
15 views

Normal positive functional on Von Neumann algebras

Let $A$ be Von Neumann algebra. A positive linear functional ‎$‎‎‎\varphi‎$ on $A$ ‎is ‎said ‎to ‎be ‎normal ‎if ‎for ‎any ‎self‎adjoint and increasing nets such that ‎$‎‎u_{\alpha‎}\longrightarrow ...
0
votes
0answers
14 views

multiplication on the second dual of a Banach *-algebra

I am looking for an Arense regular Banach *-algebra $A$, whose multiplication on $A^{**}$ is not $w^*$-separately continuous.
2
votes
0answers
24 views

Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...
1
vote
2answers
55 views

Classification of Banach Algebras?

Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras? If yes, please provide a reference where I can read ...
1
vote
1answer
52 views

States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
0
votes
0answers
23 views

Well-definedness of the logarithm in a Banach-algebra

Let $x \in \mathcal{A}$ be an element of a unital Banach-algebra $\mathcal{A}$ and assume $\sigma(x) \subset \{ z \in \mathbf{C} : \vert 1 - z \vert < 1 \}$, where $\sigma(x)$ denotes the spectrum ...
0
votes
1answer
27 views

closed convex hull of projection

$1$:I know that if ‎$‎‎F$ is a ‎locally convex ‎compact ‎space ‎then ‎‎$‎‎‎\overline{co}(‎Ext (F))=F$‎ ($Ext$: means extreme point) $2$:I ‎know ‎that ‎if ‎‎$‎‎M$ ‎is a ‎Von ‎Neumann ‎algebra ‎then ...
0
votes
1answer
30 views

*-isomorphism and spectrum

‎‎‎$A$ is a ‎‎‎‎$‎‎C^∗$-algebra and $P(A)$ is a set of projection of it. Assume that $A$ ‎admits a‎ ‎strictly ‎positive ‎element ‎‎‎‎‎$a$ ‎such ‎that ‎‎‎‎‎$‎‎‎‎σ(a)‎-\{‎0\}$ ‎is ‎discrete‎. I want to ...
0
votes
1answer
24 views

projecton and positive element in $C^*$algebras

Let ‎$‎‎A$ ‎be ‎a‎ ‎$‎‎C^*$-algebra.‎$‎‎p\in A$ ‎is a ‎‎projections. ‎‎‎ Assume ‎that ‎‎$‎‎a$ ‎is a element in‎$‎‎ Ball(A_+)$ ‎such ‎that ‎‎$‎‎a‎\leq p‎$‎ Q:May I‎ ‎say ‎‎$‎‎ap=pa$?why?‎ ‎
1
vote
1answer
58 views

If an operator have only Real eigenvalues + symmetric then it's self-adjoint?

I know that if an operator is self-adjoint then has Real eigenvalues but I'm not sure about the converse i.e. if it has only Real eigenvalues and is symmetric then the operator is selfadjoint. Is that ...
2
votes
2answers
46 views

‎strictly ‎positive elements

Let ‎$‎‎A$ ‎be a ‎‎‎‎$‎‎C^*$-algebra‎. ‎$‎‎a\in A^+$ ‎is ‎strictly ‎positive in ‎$‎‎A$‎ ‎if ‎‎$‎‎‎\overline{aAa}=A‎$‎‎ *I know that if $A$ is unital, $a\in A^+$ is strictly positive iff $a\in ...
0
votes
0answers
25 views

strictly positive elments $a$ when $‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete

If ‎$‎‎A$ is a ‎‎$‎‎C^*$-algebra ‎and it ‎admits a‎ ‎strictly ‎positive ‎element ‎‎$‎‎a$ ‎such ‎that ‎‎$‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete‎ then‎ Q1:‎$‎‎A$ admits ‎an ‎approximate ‎unit ...
1
vote
0answers
22 views

Spectrum of unitary elements of a Banach algebra

Unitary elements of a Banach space have been defined in this paper as follows: Let $A$ be a Banach space and $a\in A, \|a\|=1$. Let $S_{a}=\{f\in A':\|f\|=1=f(a)\}$. Then $a$ is said to be ...
0
votes
0answers
23 views

Every positive functional achieves its norm on the identity - converse?

Let $X$ be a unital $C^*$ algebra. I can show that if $f$ is a positive linear functional, i.e. $f(xx^*)\geq0$, then $f$ is continuous and achieves its norm at the identity. Is the converse also true ...
1
vote
1answer
17 views

Involutions on semisimple Banach algebras are continuous

Let $A$ be a semisimple Banach algebra, $ x \mapsto x^{*} $ be an involution on $A$. Then it is continuous. I've looked through some reference books and webpages, all of them prove this by using ...
0
votes
1answer
33 views

How can I prove the following question

Let $(A,+,.,*,\|.\|)$ denotes complex Banach algebra such that $\|.\|$ norm on $A$ satisfies $$\|f*g\|| \leq \| f\|.\|g\|$$ and $e$ is the identity element. How can I prove that if $\| x\|<1$ ...
2
votes
1answer
41 views

What is the dual of the disc algebra viewed as a Banach space?

Let $A$ be the disc algebra, i.e., $A=\{f\in C(\bar{U}):f \text{ is holomorphic in }U\}$, where $U$ is the unit disc in the complex plane. The norm considered is the supremum norm. Are there any ...
4
votes
0answers
22 views

Examples of a Banach space with an algebra structure having only left continuity

There is a theorem (see for example, Rudin's Functional Analysis, theorem 10.2 ) that if $A$ is a Banach space with an algebra structure, such that both left and right multiplication are continuous, ...
0
votes
1answer
34 views

Gelfand transform of $a \in A$ vanishes at infinity?

Let $A$ be a non-unital commutative Banach algebra defined over the field of complex numbers. Given $a \in A$, one defines the function $\widehat{a}:\Phi_A\to{\mathbb C}$ by ...
1
vote
1answer
34 views

A criterion for commutative Banach algebra

Suppose $A$ is a Banach algebra and there exists $C>0$ such that $\|xy\| \leqslant C\|yx \|$ for all $x,y \in A$. I am trying to show that in this case $A$ is commutative. It is easy to show that ...
1
vote
1answer
30 views

Left multiplication operator in Banach Algebra

Suppose $A$ is a Banach algebra and $x \in A$. Consider the left multiplication operator $M_x$: $$ M_x(y) = xy. $$ Assume that $M_x$ is invertible. Does that imply that $x$ is invertible in $A$?
1
vote
1answer
49 views

Positive logarithm in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible. I just see that the usual $\log$ function is continuous on ...
0
votes
1answer
21 views

Invertible elements in Banach algebra.

Suppose $A$ is a Banach algebra. Is it true that a) $x$ and $xy$ are invertible, then so is $yx$; b) $xy$ and $yx $ are invertible, then so are $x$ and $y$?
1
vote
1answer
16 views

Spectral radius in Banach algebra is commutative

I want to show that for a Banach algebra $A$ and elements $x,y \in A$, we have $$ r_A(xy) = r_A(yx), $$ where $r_A$ is a spectral radius. This is how I am trying to do that: $$ r_a(xy) = \lim_{n ...
2
votes
2answers
52 views

Normed algebra with bounded multiplication

Sometimes one finds the following definition of a normed algebra: This is an algebra with a norm on the underlying vector space such that there is a constant $K \geq 0$ such that $|x \cdot y| \leq K ...
1
vote
1answer
26 views

Maximal ideal in commutative Banach algebras. Why commutative?

I am having some trouble in understanding where is used the fact that the algebra is taken to be commutative in the following Theorem. Let $\mathcal A$ be a commutative Banach algebra (over ...
1
vote
1answer
18 views

Do non-unital Banach subalgebras of $B(L^p)$ have contractive approximate identities?

As stated in the title, I would like to know whether for $p\in[1,\infty)$, a Banach subalgebra of $B(L^p(X,\mu))$ (or a Banach algebra isometrically isomorphic to such a subalgebra) has a contractive ...
2
votes
1answer
26 views

Norm on reduced crossed product - $C^*$ version v.s. $L^p$ version

Let $(G,A,\alpha)$ be a $C^*$-dynamical system where $G$ is a countable discrete group. When defining the reduced crossed product, one can proceed as follows: Let $\pi$ be a faithful representation ...
1
vote
1answer
26 views

How to handle direct sums and unitizations of $L^p$ operator algebras?

Let $p\in[1,\infty)$. An $L^p$ operator algebra refers to a Banach algebra that is isometrically isomorphic to a closed subalgebra of $B(L^p(X,\mu))$ for some ($\sigma$-finite) measure space ...
1
vote
0answers
46 views

Rudin - Functional Analysis, exercise 1 - chapter 10,

Use the identity $(xy)^n = x(yx)^{n-1}y$ to prove that $xy$ and $yx$ always have the same spectral radius. Both $x,y$ belongs to a Banach Algebra. My attempt: $(\lambda e - xy) = \sum_{j=0}^n ...
2
votes
1answer
103 views

Convolution algebra $L^1(G)$ for non sigma-finite $G$

Let's assume that $G$ is locally compact and Hausdorff topological group, hence it carries a Haar measure, $\mu$. We can than consider space of integrable functions $L^1(G)$ (class of functions to be ...
2
votes
1answer
46 views

What's the point with the Gelfand–Naimark theorem?

Does anybody can explain me in plain english what's the real point with the Gelfand–Naimark Theorem. I know it's crucial, but I think I'm missing how much it's crucial.
1
vote
0answers
26 views

If $\mathbb{A}$ and $\mathbb{B}$ are unital banach algebras, and $f: \mathbb{A} \to \mathbb{B}$

If $\mathbb{A}$ and $\mathbb{B}$ are unital banach $\mathbb{C}$ algebras, and $f: \mathbb{A} \to \mathbb{B}$ a unital algebra homomorphism$\mathbb{C}$ algebra morphism, and $\text{Sp}(\mathbb{A})$ ...
0
votes
1answer
26 views

Proving $l^{1}(\mathbb{Z})$ with $\ast$ is a commutative Banach algebra

Define the $\ast$ on $l^{1}(\mathbb{Z})$ by $$(x\ast y)(n)=\sum_{k=-\infty}^{\infty}x(n-k)y(k),\qquad\text{with } n\in\mathbb{Z}\text{ and } x,y\in l^{1}(\mathbb{Z})$$ I want to show that this ...
0
votes
1answer
17 views

Computing spectra in Banach algebras

In general, computing the spectrum of a specific element in a Banach algebra can be very difficult. What are some of the less obvious tricks that you've encountered?