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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Norm of a unital homomorphism

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two unital $C^*$-algebras and $\varphi : \mathcal{A}_1 \to \mathcal{A}_2$ a unital $*$-homomorphism, i.e. a linear map such that $\varphi(xy)=\varphi(x)\...
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What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [on hold]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
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Ideal generated by a set of singular elements in a Banach algebra.

Let $A$ be a commutative unital Banach Algebra. Suppose for every $a\in A$, $\|a\|=1$, I get a singular element $b_a$. I know that each such $b_a$ is contained in a proper maximal ideal of $A$. Is it ...
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27 views

Dirac functional embedding

I got the following statements to show. Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with ...
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Relationship between spectral rays commuting

Show sing binomial Newton that if $S$ and $T$ are continuous operators on Banach space $B$ such that $ST = TS$ then $$r_\sigma(T + S) \leq r_\sigma(T) + r_\sigma(S)$$ where $r_\sigma$ is spectral ...
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Proof that group of invertible elements in a Banach algebra have 1 or infinite connected components?

I'm trying to reconcile this proof that I've read that a group of invertible elements in a commutative (complex) Banach algebra have 1 or infinite connected components with this example I'm looking at....
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Exercise 16 - chapter 11 - From Rudin's Functional Analysis

I'm trying to solve this problem, which comes from the book mentioned in the title. Suppose $A$ is a Banach algebra, $m$ is an integer, $m\geq2$, $K<\infty$, and $$\|x\|^m \leq K\|x^m\| $$ ...
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Approximate unit for an ideal of a C* algebra.

Suppose I have a C* algebra $A$ and an ideal $J$ with approximate unit $\{e_{\lambda}\}$. Let $x\rightarrow{}q(x)$ denote the projection onto $A/J$. I want to show that $\lim_{\lambda}||x-xe_{\lambda}...
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Let $e_0$ be minimal projection, why $E_{e_0}$ is a Hilbert space?

‎A nonzero element $e \in \mathcal{K}(\mathcal{H})$ is called a projection‎, ‎if it is‎ ‎self adjoint and idempotent‎. ‎In addition‎, ‎if $e\mathcal{K}(\mathcal{H})e = \mathbb{C}e$ then‎, ‎it is ...
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Definition of C$^\ast$-algebra: which conditions can be deduced from the others?

A C$^\ast$-algebra is a Banach algebra $A$ with an involution, i.e. a map $\ast$ such that: $(x^\ast)^\ast=x$ for all $x\in A$; $(x+y)^\ast=x^\ast+y^\ast$ for all $x,y\in A$; $(ax)^\ast=\overline ax^...
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Are compact operators trace class operators?

We say that $A\in B(\mathcal{H})$ is a trace class operator, if $\sum_{i\in I}\langle|A|e_i,e_i\rangle<\infty$,$\hspace{0.1cm}$ such that {$e_i; i\in I$} is a orthonormal bass for Hilbert space $\...
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32 views

Decomposition of spectrum

We know that if X is a banach space and T be an element in Banach Algebra B(X) then the union of residual spectrum continuous spectrum and point spectrum is spectrum of Banach algebra i.e σ(T) is the ...
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28 views

Let $A$ be a unital Banach algebra and $J$ maximal ideal of $A$. Why is $\overline{J}$ an ideal of $A$?

Let $A$ be a unital Banach algebra with maximal ideal $J$. Why is it true that $\overline{J}$ is also an ideal of $A$, i.e. the closure of a maximal ideal of $A$ is an ideal of $A$?
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Why does $a(1_A-x)=(1_A-x)a=1_A$

I am working through the following theorem and proof, but I struggle to understand the last part. Theorem: Let $A$ be a Banach algebra with unit element $e$, $x \in A$ and $||x||< 1$. Then ...
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34 views

Projective tensor product continuous?

For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V):=B(V,V)$ are linear bounded endomorphisms with operator norm? If so, ...
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109 views

Are there bounded nonconvergent sequences satisfying this recurrence relation?

In this question, we ask about convergent sequences $(p_n)$ satisfying $p_{i} = 2p_{i+6} - {1 \over 2}p_{i+1} - {1 \over 4}p_{i+3} - {1 \over 8}p_{i+7} - {1 \over 16}p_{i+15} - {1 \over 32}p_{i+31} - ...
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58 views

Injectivity of the evaluation map from holomorphic functions to a Banach algebra

In Functional Analysis, we have covered Functional Calculus, that is, a way to associate, once having fixed a Banach algebra $A$ and an element $a\in A$, an element $\tilde f(a)\in A$ to every $f$ ...
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What are the Hermitian idempotent matrices with respect to $l_{1}$ norm

Let $A=(M^{n}(\mathbb{C}), \|\cdot\|_{1})$. The numerical range of $a\in A$ is defined as $V(a)=\{f(a):f\in A',\|f\|=1=f(1)\}$. $a\in A$ is said to be Hermitian if $V(a)\subseteq \mathbb{R}$. $a\in A$ ...
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28 views

Spectral radius of an element in a Banach algebra

I want to solve this problem: Let $A$ be an unital Banach algebra and $a\in A$ then \begin{equation} r(a)<1 \Longrightarrow \lim_{n\to \infty} a^n=0 \Longrightarrow (1-a)^{-1}=\sum_{n=0}^\infty ...
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The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
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37 views

Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ is not connected. Prove that $A$ contains a nontrivial idempotent $z$.

While trying to solve the exercise below, I came up with a wrong conclusion, but I can't see why it's wrong. Also I'm accepting suggestions to get the right solution. This is the problem 17 from ...
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15 views

Smallest closed unital subalgebra containing an element has connected resolvent.

Suppose I have $A$ a commutative, unital Banach algebra. Let $C$ be the smallest closed unital subalgebra containing $a\in{}A$ (i.e. the closure of polynomials in $a$). I want to show that the ...
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Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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55 views

Rudin functional Analysis chapter $10$, exercise $13$

This is Rudin's functional Analysis chapter $10$, exercise $13$. I am confused about the notation $\sigma_A(f)$, what does that mean?(What role does the subscript $A$ play here). And can someone ...
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23 views

Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra

In my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I ...
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A complex unital algebra which is a Banach space is also a Banach algebra

In my functional analysis class I got stuck on this question on Banach algebras: Let $ \mathbb{A} $ be a complex unital algebra (it has a unit i.e. a multiplicative identity) and $ \mathbb{A} $ is ...
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8 views

Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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Gelfand transform on functions

The Gelfand transformation identifies function spaces $C_0(X)$ for locally compact Haussdorff $X$ with commutative $C^*$ Algebras. Additionally there is a statement that if $f: X \to Y$ is a proper ...
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Toeplitz operators on $\ell_p$ modulo compact operators

There is the well-known extension of $C^*$-algebras $0\rightarrow K\rightarrow\mathcal{T}\rightarrow C(S^1)\rightarrow 0$ where $\mathcal{T}$ is the Toeplitz algebra (generated by the unilateral shift ...
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Is the hermitian condition a must?

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$ In this post, I saw a comment stating that "More generally, if $a$ is normal then $∥a^n∥=∥a∥^n$ for each positive ...
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Are there finite dimensional $\mathbb{R}$-algebras which are not Banach algebras?

It is known that not every algebra (over a ground field $\mathbb{R}$ or $\mathbb{C}$) is a Banach algebra. It might be a silly question, but are there examples of finite dimensional ($\mathbb{R}$- or $...
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1answer
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Every Jordan function $\phi$ on $A$ is multiplicative.

I am reading the following proposition by : FF. Bonsall and J. Duncan, Complete Normed Algebras, pg. 79 Definition: A Jordan function on $A$ is a nonzero linear functional $\phi$ on $A$ such that ...
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Question on maximal left ideals.

Suppose I have a unital Banach algebra $A$ and a maximal left ideal $L$ and an element $a$ such that $La\subset{}L$. I want to show that there exists (uniqueness is easy) some complex $\lambda$ such ...
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calculating the abstract index of $C(T)$

Consider the following definition in operator theory: I'm reading an example of the abstract index of $\mathcal{A}$ in Zhu's An Introduction to Operator Albebras: Here $G_0(\mathcal{A})$ is the ...
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27 views

When is the set $\{ x^* x : x \in A\} $ a cone in a *-algebra?

So in a $C^*$-algebra, every positive element can be written as $x^*x$, and the set of positive elements form a cone. What if we remove all information about the norm? Say $A$ is an algebra, with $*:...
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Submultiplicative Hilbert space norm on $B(H)$

Let $H$ be a complex Hilbert space and let $B(H)$ denote the space of bounded linear operators $H \to H$ equipped with operator norm: $$ \lVert T \rVert = \sup\big\{ \lVert Tx \rVert \: : \: \lVert x \...
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2answers
63 views

Show that $\exp(-\lambda x) \cdot\exp(\lambda x)=1$ using the power series

Let $A$ be a commutative Banach algebra. Consider the exponential function $$\exp(\lambda x) = \sum_{n=1}^\infty\frac{(\lambda x)^n}{n!},$$ where $x \in A$ and $\lambda \in \mathbb C$. We can easily ...
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1answer
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Why if $X \subset A$ has codimension $1$, the only subspace properly containing $X$ is $A$?

I saw the following question a while ago on Math.SE This question. The answer provided seems to give a satisfactory result, but one thing in the answer I can not quite see. The author of the answer ...
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46 views

Why is the canonical approximate identity positive?

Let $\{u_n\}$ be an approximative identity for C*-algebra A. Why the element $1-u_n$ is also positive? I'm asking about it because in some theorem we use something like this: $\mid \mid a-u_n a \mid \...
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Matrices over tensor product of Banach algebras

Suppose that $A$ is a Banach algebra, or even a $C^*$-algebra, and $B$ is a closed subalgebra of $B(L^p)$ for some $L^p$ space. In particular, $M_n(B)$ has a canonical matrix norm. Is there some ...
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44 views

Algebra of compact operators on $\ell_p$

Are the algebras of compact operators $K(\ell_p)$ and $K(\ell_q)$ isomorphic as Banach algebras for $1\leq p<q<\infty$?
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28 views

Explicit example of Gel'fand transform

I would like to determine explicitly the Gel'fand transform for the (commutative unital) Banach algebra of $2 \times 2$ matrices of the form $$\pmatrix{ a && b \\ 0 && a}, \quad a,b \...
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34 views

Why the set of all bounded operators on Hilbert space is a prime?

In abstract algebra, a nonzero ring $R$ is a prime ring if for any two elements $a$ and $b$ of $R$, $arb=0$ for all $r$ in $R$ implies that either $a=0$ or $b=0$. Or for any two ideals $A$ and $B$ of $...
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1answer
40 views

Norm of a character in a non-unital Banach algebra without approximate identity

As is shown here, the norm of a character in a non-unital Banach algebra with an approximate identity is $1$. I wonder if this result still holds for general non-unital Banach algebras. Let $A$ be ...
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Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ \int_0^1\|f(...
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1answer
27 views

Applying Complex Stone-Weierstrass Theorem for Compact Subsets

The complex Stone-Weierstrauss theorem can be applied to compact spaces. However, say I have an open set $D\subset \mathbb{C}$, and I want to approximate a holomorphic function $f$ on $D$ via ...
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2answers
48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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1answer
31 views

One-sided identities in Banach algebras

What is an example of a Banach algebra with a left identity but with no right identities? Is there an example of an operator algebra with this property?
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Norms on matrices over Banach algebras

Let $A$ be a unital Banach algebra. Is there a sequence of norms $||\cdot||_n$ such that $(M_n(A),||\cdot||_n)$ is a Banach algebra for each $n$, identity matrices of all sizes have norm 1, the ...