The banach-algebras tag has no usage guidance.

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### Maximal spectrum of a complex, unital and commutative Banach-algebra

Let $A$ be a complex, unital and commutative Banach-algebra.
Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff?
...

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### Density of adjoint operators

I am interested in operators on a non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{\ast\ast})$ by $\Phi(T)=...

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### Is an ultrapower of a faithful Banach algebra always faithful?

Let $A$ be an infinite dimensional faithful Banach algebra and let $\mathcal U$ be a free ultrafilter. Is the ultrapower $(A)_{\mathcal U}$ faithful?

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### Subalgebras of $B(E)$

Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...

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### Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...

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### Banach Algebras and the peripheral spectrum

Here is a little theorem that I'm trying to prove. I haven't seen it in literature before, but I think the applications will be quite useful, particularly in the context of Banach algebras.
Denote ...

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### Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all $x\...

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### When is a Banach Algebra $C^\star$

I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...

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### Injective modules over Fourier algebra

Is there any article on injective modules over Fourier Algebras?
Do we have anything about injectivity of $A(G)$ as a $A(G)$-bimodule?

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### Ring isomorphism $\Phi \colon C(X) \to C(Y)$ and zero dimensionality of $X$

We denote the ring of all continuous real-valued functions on $X$ by $C(X)$. The ring of all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of $...

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### Terminology: Banach spaces equipped with continuous associative product?

This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach algebra/...

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### Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?

Here's a research problem, which I think interesting.
Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n \...

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### Exotic uniform algebras

The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region ...

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### Second adjoint operators on non quasi-reflexive Banach spaces

I am interested in 'algebraic-density'-type properties of second adjoint operators in the algebra of bounded operator on a second dual of a Banach space. Incidentally, I have a problem with ...

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### C*-algebras with bizzarre structure of projections

This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C*-algebra such that its projections do not form a lattice (...

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### Measures on general topological groups

I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...

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### Reasonable crossnorm on Banach algebra tensor product constructed from isometric non-degenerate representations

The following is an abstraction of a more specific problem I've been grappling with, so I might give some extra unnecessary information. (The `bounded approximate
left identities' assumption is ...

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### $c_0$-direct sum of $\mathcal{K}(\mathcal{H})$

Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...

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### Algebraically simple Banach algebras

There are plenty of semi-simple Banach algebras - this broad class includes C*-algebras and algebras of bounded operators on a given Banach space. On the other hand, it seems unlikely to me that there ...

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### Reversed disc algebra?

Take $U=\mathbb{D}_2\setminus \overline{\mathbb{D}_1}$ ($\mathbb{D}_r$ is the open disc centered at 0 with radius $r$) and consider the space $A(U)$ of all functions on $\overline{U}$ which are ...

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### injectivity of Fourier transform: is there algebraic proof?

Let $L$ be the Banach algebra of $L^1$-functions from $\mathbb{R}$ to $\mathbb{C}$ with $L^1$-norm and convolution as algebra multiplication. Assume that we knew that the homomorphisms from $L$ to $\...

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### Unbounded representations of Banach algebras

Can a representation of a Banach algebra be unbounded?
To clarify, I'm not asking about a representation as unbounded operators, but
rather a homomorphism $\pi: A \to B(H)$ for some Hilbert space $H$,...

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### Ultraproduct of n-dimensional Banach spaces and algebras

Hi, I am interested in the following question:
Fix $n$.
Let $M_n$ be matrix algebra over the field of complex numbers with normalized trace $tr_n$. Let $M_n^{\omega}$ be an ultrapover of $M_n$, ...

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### Preduals of B(E)

For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always ...

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### spectra of sums in (Banach) algebras

A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".
Answer there led me to the following question.
If for ...

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### Which Banach algebras are group algebras?

Given a locally compact Hausdorff group $G$, one can construct several Banach star-algebras using $G$ (and its associated Haar measure): $L^1 (G)$, $M(G)$ (regular complex measures on $G$), $L^{\infty}...

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### Holomorphic stability of inverse limit of pre-$C^*$-algebras

Let A be a C*-algebra and let At be a set of dense *-subalgebras of A, stable under holomorphic functional calculus on A, which are also Banach algebras complete with respect to the norms ||$\cdot$||t....

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### Lifting of product of a Banach algebra

Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...

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### Is exp(rA) = (exp(A))^r for real r and A in a Banach space?

Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
...

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### Gelfand theory Problem

I have 2 problems in Gelfand theory. I shall be thankful for any
answers.
1)What is the gelfand spectrum of l^1(N)?
A few of
the elements are evaluations of functions(defined below) on closed
unit ...

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### Injectivity for bimodules and Hochschild cohomology

Let $A$ be a Banach algebra and let $X$ be an $A$-bimodule. Is there a notion of (relative) injectivity for $X$ which would imply that $\mathcal{H}^n(A,X)$ vanishes for all $n\ge 1$? Here $\mathcal{H}^...

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### Does there exist any massive proper $C^*$-subalgebra?

Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...

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### Factorization in the Wiener algebra on the unit disc.

Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum_{k\ge0}|a_k|<\infty$$ where
$$f(z)=\sum a_kz^k$...

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### Set of invertible operators in B(H) is connected. Is it true? Is there a reference?

Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...

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### Are there good inequalities on the norm?

It's well known that in a Hilbert space, good inequalities exist concerning the norm due to the existence of inner product.Now let X be a general Banach algebra, are there good inequalities concerning ...

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### Completely equivalent operator norms on $*$-Banach algebras.

Let $A$ be an algebra over $\mathbf{C}$ with an involution operator, and let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two $equivalent$ operator norms, making $A$ into a $*$-Banach algebra (we denote them as ...

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### Is this a correct interpretation of support in coarse geometry?

Let $X = \mathbb{R}^n$, and consider a nondegenerate representation $\rho: C_0(X) \to B(H)$ where $B(H)$ is the algebra of bounded operators on a separable Hilbert space. The support of a vector $v \...

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### Norms of Commutators

If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...

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### Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...

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### The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$

Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...

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### spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]

Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(...

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### uniformity for Banach algebras - a three space property?

Let $A$ be a commutative, unital Banach algebra and $I \subset A$ an ideal such that $I$ with the relative norm is a uniform Banach algebra and $A / I$ with the quotient norm is uniform as well.
Does ...

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### In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...