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2
votes
0answers
139 views

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 ...
2
votes
1answer
186 views

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 ...
9
votes
0answers
328 views

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 ...
1
vote
0answers
152 views

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 ...
5
votes
0answers
561 views

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 ...
4
votes
1answer
614 views

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 ...
1
vote
0answers
155 views

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?
2
votes
2answers
314 views

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 ...
4
votes
2answers
296 views

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 ...
22
votes
1answer
1k views

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 ...
4
votes
1answer
188 views

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 ...
0
votes
1answer
220 views

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 ...
10
votes
2answers
519 views

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 ...
7
votes
4answers
664 views

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 ...
4
votes
1answer
307 views

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 ...
6
votes
1answer
372 views

$c_0$-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 ...
1
vote
1answer
338 views

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 ...
1
vote
2answers
329 views

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 ...
6
votes
0answers
633 views

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 ...
4
votes
1answer
467 views

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 ...
3
votes
1answer
233 views

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 $M_n$, ...
7
votes
3answers
624 views

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 ...
2
votes
1answer
524 views

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 ...
5
votes
2answers
540 views

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 $G$), ...
2
votes
0answers
133 views

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 ...
0
votes
0answers
269 views

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 ...
1
vote
1answer
435 views

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 $e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$, ...
7
votes
1answer
378 views

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 ...
8
votes
1answer
338 views

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 ...
5
votes
1answer
253 views

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 ...
3
votes
1answer
388 views

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 ...
4
votes
3answers
632 views

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 ...
2
votes
2answers
645 views

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 ...
1
vote
1answer
408 views

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 ...
2
votes
2answers
189 views

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 ...
54
votes
2answers
4k views

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 ...
35
votes
4answers
3k views

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 ...
11
votes
3answers
871 views

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 ...
0
votes
1answer
907 views

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 ...
4
votes
3answers
365 views

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 ...
16
votes
1answer
1k views

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 ...