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

1
vote
0answers
15 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
2
votes
1answer
68 views

Does *-operator be automatically continuous

In the C*-algebras, does the * -operator be automatically continuous? I think it is yes, because C*-algebras are semisimple, from the Johnson Theorem, it must be automatically continuous. Am I right? ...
2
votes
1answer
55 views

What's the difference between a Banach Algebra and a C*-algebra?

I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at C*-algebras based on my interests. ...
6
votes
2answers
196 views

Universal separable Banach algebras

The well-known Banach–Mazur theorem says that $C([0, 1])$ is a universal separable Banach space, in the sense that if $X$ is any separable Banach space then there is a map $f : X \to C([0, 1])$ which ...
0
votes
0answers
39 views

What is unit ball in the weak star topology of a Banach space? [on hold]

Let X is Banach space with dual $X^{*}$. What is unit ball of $X^{*}$in the weak star topology?
0
votes
0answers
6 views

What are the appropriate morphisms for forming inductive limits of Banach algebras?

For Banach algebras, if we take the morphisms to be bounded homomorphisms, the inductive limit construction may not result in a Banach algebra. (I remember seeing this fact but I don't know a specific ...
1
vote
2answers
25 views

How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
2
votes
0answers
22 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
3
votes
0answers
18 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
2
votes
0answers
13 views

How are $M_n(A)$ and $M_n(\mathbb{C})\otimes A$ identified as Banach algebras?

When $A$ is a Banach algebra, one can make $M_n(A)$ into a Banach algebra by equipping it with various norms. One can also make the algebraic tensor product $M_n(\mathbb{C})\otimes A$ into a Banach ...
3
votes
3answers
60 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...
7
votes
2answers
102 views

Is every Hilbert space a Banach algebra?

Let $H$ be a Hilbert space. Could we say that, always there is a multiplication on $H$, that makes it into a Banach algebra? If not, under which conditions does it exist?
3
votes
1answer
36 views

The same topologies

Let $L^1 (\mathbb{Z})$ be the space of all functions $f:\mathbb{Z}\rightarrow \mathbb{C}$ such that $\left\{\|f\|=\sum_{k\in \mathbb{Z}}|f(k)|<\infty\right\}$. Clearly, $L^1 (\mathbb{Z})$ is a ...
3
votes
1answer
22 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
0
votes
2answers
18 views

Can we embed unital Banach algebras into semi-simple ones?

A Banach algebra is (Jacobson) semi-simple if the intersection of all maximal left ideals is the zero ideal. Take a unital abelian Banach algebra $B$. Can we embed it unitally into an abelian ...
0
votes
0answers
7 views

Norm on $M_n((M_k(A))^+)$ where $A$ is a Banach algebra

If $A$ is a Banach algebra (either unital or non-unital), can we norm $M_n((M_k(A))^+)$ by regarding matrices in there as matrices in $M_{nk}(A^+)$ and taking one of the commonly used norms on the ...
-1
votes
2answers
48 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 ...
4
votes
1answer
51 views

Understanding the bidual of a $C^*$-algebra as a $C^*$-algebra

I have a lot of problems trying to understand the double dual of a $C^*$-algebra. Let $A$ be a $C^*$-algebra, I read that if you endow the bidual Banach space $A^{**}$ of $A$ with the weak-*topology, ...
4
votes
0answers
32 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
3
votes
1answer
55 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
8
votes
3answers
243 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
vote
1answer
23 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 ...
1
vote
1answer
26 views

cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
1
vote
1answer
17 views

Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
3
votes
1answer
59 views

Banach algebra with left or right minimal ideal without minimal bi-ideal

Let $A$ be a Banach algebra( or a ring). A left ideal $I$ of $A$ is called a left minimal ideal, if $I\neq\{0\}$ and there is no any other non-zero left ideal of $A$ completely lies in $I$. With a ...
0
votes
0answers
14 views

Operator norm on $M_n(A)$ by acting on $A^n$ or on $(A^+)^n$

If $A$ is a nonunital Banach algebra, we can let $M_n(A)$ act on $A^n$ or on $(A^+)^n$ where $A^+$ denotes the unitization of $A$, and get the respective operator norms on $M_n(A)$. Is there any ...
0
votes
1answer
20 views

$p$-operator space property

If $S,T,U,V\in B(L_p(X,\mu))$, $p\in[1,\infty)$, and we regard $\begin{pmatrix} S & T \\ U & V \end{pmatrix}$ as an operator on $B(L_p\oplus_p L_p)$, then supposedly we have ...
0
votes
1answer
17 views

Comparing norms - $M_{kn}(A)$ versus $M_k(M_n(A))$

I am having to deal with the problem of passing between $M_{kn}(A)$ and $M_k(M_n(A))$ where $A$ is a Banach algebra, removing parentheses in one direction, and adding parentheses in the opposite ...
2
votes
1answer
32 views

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 ...
1
vote
0answers
25 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. ...
1
vote
1answer
125 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
0answers
16 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 ...
2
votes
1answer
313 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
1
vote
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 ...
1
vote
0answers
25 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 ...
1
vote
1answer
24 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 ...
5
votes
0answers
41 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 ...
1
vote
1answer
24 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 ...
2
votes
0answers
71 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
42 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}} = ...
1
vote
1answer
24 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. ...
0
votes
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 ...
3
votes
2answers
53 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 ...
0
votes
0answers
15 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 ...
1
vote
1answer
22 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 ...
2
votes
0answers
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: ...
3
votes
1answer
34 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, ...
1
vote
1answer
49 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 ...
2
votes
0answers
18 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 ...
2
votes
1answer
65 views

Character space of $L^{1} (\mathbb Z)$

I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$. Are the Gelfand and norm topologies equal, on the character space of $L^{1} (\mathbb ...