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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Lifting idempotents from a quotient of a Banach algebra

In a quotient of a Banach algebra $A$, if an invertible element is connected to the identity by a continuous path of invertibles, then it can be lifted to an invertible element in $A$. Is there an ...
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14 views

Show that the ideal generated by an inner function is closed.

Suppose $H^\infty=\{f\in H(U):\exists M<\infty,\,\forall z\in U,\,|f(z)|\le M\}$ is the set of bounded holomorphic functions in the unit disk. It is a Banach algebra with pointwise multiplication. ...
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12 views

Involution on the set of all multipliers of $A$ ($A$ is a $C^*$-algebra)

Let $A$ be a $C^*$-algebra. $M(A)$ denotes the set of all multipliers of $A$, i.e. $m\in M(A)$ means that there is a map $m^*:A\to A$ such that $m(a)^*b=a^*m^*(b)$ for all $a,b\in A$. I want to know ...
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11 views

Bounded v.s. completely bounded homomorphisms between $L^p$ operator algebras

Take an $L^p$ operator algebra to mean a closed subalgebra $A\subset B(L^p(X,\mu))$ for some ("nice") measure space $(X,\mu)$, $p\in[1,\infty)$. Equip the matrix algebra $M_n(A)$ with the norm ...
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1answer
29 views

Show that Quotient of a C*-algebra is a C*-algebra in its own right

Let $A$ be a commutative C*-algebra over $\mathbb{C}$ and let $J$ be an ideal of $A$ such that $J\in Closed(A)$ and that contains $x^*$ if it contains $x$, for all $x\in A$. I know that $A/J$ is also ...
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1answer
20 views

Operator norm on $M_n(A)$ where $A$ is a Banach algebra

On $M_n(\mathbb{C})$, if we take the operator norm by acting on $\mathbb{C}^n$ where $||(z_1,\ldots,z_n)||=\max_{1\leq i\leq n}||z_i||$, then for $[a_{ij}]\in M_n(\mathbb{C})$ we have ...
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20 views

Ideals of the operator algebra

Let $A$ be a Banach algebra. Is there any relation ship between two-sided closed ideals of $A$ and two-sided closed ideals of the operator algebra $\mathscr B(A)$? Is there any characterization for ...
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29 views

Finite dimensionality of some subspace of convolution Banach algebra $L^1(G)$

Let $G$ be a locally compact group (not only compact group) with the left Haar measure $\lambda$. Consider the convolution Banach algebra $L^1(G,\lambda)$. For which $f\in L^1(G,\lambda)$ the ...
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1answer
22 views

application of positive linear functionl

The space $L^{1}(\mathbb R)$ is a commutative Banach algebra under convolution. A linear functional $F$ on $L^{1}(\mathbb R)$ is positive if $F(f^*f)\geq 0$ for $f\in L^{1}(\mathbb R).$ (where ...
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41 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 ...
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1answer
41 views

How to apply Cauchy's formula in proof 10.13 of Rudin's FA

Let $A$ be a Banach algebra and $x\in A$. In part of prood 10.13 of Rudin's Functional Analysis (page 254), where he is trying to prove that $$\rho(x) = \lim_{n\to\infty}||x^n||^{\frac{1}{n}} = ...
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1answer
19 views

Orthogonal elements in a unitization of a $C^*$-algebra

Let $A$ ne a $C^*$-algebra and $a,b\in A$ self-adjoint. a and b are orthogonal, iff $ab=0$. Let A be nonunital and denote $A_1$ it's unitization, i.e., $A_1\cong A\oplus\mathbb{C}$ as vector spaces. ...
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1answer
37 views

How to prove that $\Delta( A)$ with the Gelfand topology is compact and Hausdorff?

How do I prove: $\Delta( A)$ with the Gelfand topology is compact and Hausdorff. I've tried proving it closed, but I'm having difficulties with how to begin writing a proof. And I have no idea ...
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1answer
37 views

Describe the GNS construction [closed]

Question: Describe the GNS construction for the C$^*$-algebra $ C[0, 1]$ and for the positive linear functional $\phi $ given by $\phi(f) = f (0)$. What should i do? Should I describe Hilbert space ...
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1answer
33 views

Two-sided closed ideals of $C(X,M_2(\mathbb C))$

Let $X$ be compact and Hausdorff space. I know all closed ideals of $C(X)$. I want to substitute $\mathbb C$ by $M_2(\mathbb C)$. What can we say about two-sided closed ideals of $C(X,M_2(\mathbb ...
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2answers
48 views

Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
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0answers
14 views

C*−algebras and operator theory, murphy [duplicate]

Do you know how to solve this exercise? (Murphy, $C*$−algebras and operator theory, $2^{nd}$ chapter, 1st exercise) Let $A$ be a Banach algebra such that for all a\in A the implication $Aa=0$ or ...
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1answer
13 views

intersection of a multiplier algebra with a commutant of a $C^*$-algebra

I have a question about multiplier algebras and commutants of $C^*$-algebras in general. First of all, the question is related to this structure theorem about completely positive order zero maps (you ...
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16 views

example of the arens multiplication; I want to unterstand the construction

I want to understand the double dual as a $C^*$-algebra of a given $C^*-$algebra $A$ but my first problem is to understand the arens multiplication on the double dual $A^{**}$ (considered as Banach ...
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1answer
31 views

Completely positive, orthogonaly preserving maps

First of all, here is the setting for my problem: Definition: Let $A$ be a $C^*$-Algebra, $a,b\in A$. We say, a and b are orthogonal, if $ab=ba=a^*b=ab^*=0$. Proposition: Let $A$ be a $C^*$-Algebra, ...
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24 views

equivalence of properties. Is the restriction in (ii) redundant?

I have a question about the claim, which I found in a paper: Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. The following are equivalent: ...
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1answer
19 views

Jordan-homomorphism; equivalent properties

Let $\phi:A\to B$ a linear map between $C^*$-algebras $A$ and $B$. I want to know, why the following properties are equivalent: $(i) \phi(ab+ba)=\phi(a)\phi(b)+\phi(b)\phi(a)$ and $(ii) ...
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31 views

Maximal ideal space of $L^{\infty}(m)$ separable?

Let $m$ denote the Lebesgue-measure on the unit interval and let $L^{\infty}(m)$ be the set of measurable essentially bounded functions on that interval (i.e. $f \in L^{\infty}(m)$ if and only if ...
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1answer
24 views

Elements near the identity of a linear subspace

I am currently trying to understand a proof and ran into the following problem. The proof states (everything takes place in a commutative, unital Banach-Algebra): A linear subspace $X$ with ...
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1answer
25 views

On existence of square root of positive elements of a unital $C^*$-algebra

Given a unital $\mathcal{C}^*$-algebra $A$ and a positive element $a \in A$, I am trying to prove the existence of a square root $a^{\frac{1}{2}}$ i.e. a positive element $b \in A$ such that $b^2 = ...
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0answers
23 views

The Gelfand and norm topologies are equal on the character space of $L^1(\mathbb Z)$

We know that the character space of the Banach algebra $L^{1}(\mathbb Z)$ is homeomorphic to the unit circle $\mathbb T$, but I can't show that the Gelfand and norm topologies are equal on that.
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1answer
42 views

Algebra with element having empty spectrum?

The definition of the spectrum makes sense for any algebra. I guess we can go to the unitization to make sense of it even non-unital algebras. Recalling the well-known fact that for normed algebras, ...
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53 views

Prove that disk algebra is isomorphic to the closure of $\mathbb{C}(z)$ in $C(\mathbb{T})$.

Let $D = \{ z \in \mathbb{C} : |z| < 1 \}$ be the open unit disk and $\mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}$ its boundary. We will naturally write $\bar{D}$ for its closure $\{ z \in ...
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1answer
49 views

Countable set in a Banach space which spans densely?

Let $\mathcal{C}(\mathbf{T})$ be the algebra of continuous complex functions on the unit circle $\mathbf{T}$. Consider the following two statements: The $*$-subalgebra generated by $1$ and $z$ spans ...
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2answers
36 views

Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
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1answer
41 views

$||a||\leq \sup_{||b||\leq 1} ||ab||$ in a C*-algebra

I would like to prove that, if $a$ is an element in a C*-algebra then $$\|a\|\leq \sup_{\|b\|\leq 1} \|ab\|$$ It is obvious if the algebra is unital. What if it is not?
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34 views

Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
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33 views

positive elements in $L(H)$

At the moment I'm studying positive elements in $C^*$-algebras. Let $H$ be a complex Hilbert space, $L(H)$ the linear bounded operators $H\to H$ and $T^*=T$. Then: $$\sigma(T)\subseteq [0,\|T\| ]\; ...
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34 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
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1answer
19 views

positive linear maps which are involutive

Let $S$ be an operator system, $B$ a $c^*$-algebra and $\phi:S\to B$ a positive linear map. Then $\phi$ is involutive, i.e. $\phi(x^*)=\phi(x)^*$ for all $x\in S$. I want to prove this claim but I'm ...
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1answer
28 views

positive linear maps of $c^*$-algebras are bounded

Let $A, B$ be $c^*$-algebras and $\phi:A\to B$ a positive, linear map. Then $\phi$ is bounded. Proof: It is sufficient to proof boundedness of $\phi$ on the unitarization (I missunderstood that, see ...
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16 views

Are two operator norms on $M_n(A)$ equivalent?

If $A$ is a Banach algebra, then $M_n(A)$ can be given the operator norm as operators on $A\oplus_p\cdots\oplus_p A$ ($1<p<\infty$) to make it a Banach algebra. If in addition $A$ is an operator ...
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0answers
20 views

Applications of $C^\ast$ algebras in differential topology

I was wondering if there were any useful ways $C^\ast$ algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable.
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2answers
56 views

Can $ {L^{1}}(G) $ be a $ C^{*} $-algebra?

Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ ...
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1answer
25 views

A property of a ideal of Banach algebras

Let $B$ be a Banach algebra and $A$ be a bi-ideal of it. Suppose that for any $b\in B$, $Ab=\{0\}$ implies $b=0$. Now could we say that for some $c\in B$ if $cA=\{0\}$ then $c=0?$
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1answer
39 views

Banach algebra norms on $M_n(A)$

Let $A$ be a Banach algebra (not sure whether I need $A$ to be unital). I saw the claim that all Banach algebra norms on $M_n(A)$ with continuous projections on entries are equivalent. How does one ...
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13 views

When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
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32 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
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29 views

Determining primitive ideal space of C* algebra

What is the general way of determining the space of primitive ideals of the C* algebra if there is any? Thanks.
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28 views

Continuous homorphism from $(\mathbf{R},+)$ to group of invertible elements in Banach algebra is differentiable

Let $A$ be a Banach algebra with $1$ and $\varphi\colon \mathbf{R}\to A$ be continuous such that $\varphi(0)=1$ and $\varphi(x+y)=\varphi(x)\varphi(y)$ for each $x,y\in \mathbf{R}$. The claim is ...
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33 views

if $f$ is in Banach space, then $\nabla f $ is in the dual space?

I am not very deep in advanced real analysis. Could you help me decipher the following two phrases hold? 1) if $f$ is in Banach space $\mathcal{B}$, then $\nabla f $ is in the dual space ...
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1answer
29 views

Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
2
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2answers
91 views

Algebra is Generated by nilpotent Lie Algebra

$X$ Banach space. In $B(X)$, we can define a Lie product $[ , ]:[T_1,T_2]=T_1T_2-T_2T_1$ for any $T_1,T_2 \in B(X).$ Let $\mathcal{L}$ Lie Algebra. $\mathcal{L}^1=\mathcal{L}$ , $ ...
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1answer
38 views

realizing/ understanding $C^*(\phi_g(C([0,1])))$ and "support projection of an element of a $C^*$-algebra

someone asked me the following question but I don't know the answer, but and I'm interested in it too. Let $C([0,1])=\{f:[0,1]\to\mathbb{C}; \text{f is continuous}\}$, $g\in C([0,1])$ be a positive ...
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0answers
28 views

Banach algebras for which the Gelfand transform is 1-1 but not onto

Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is $\{0\}$) but not onto? Context Page 96 of Kaniuth's ...