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63
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 ...
47
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 ...
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 ...
17
votes
1answer
676 views

Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
17
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 ...
14
votes
2answers
433 views

Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes. Let $ ...
12
votes
2answers
377 views

$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group. Denote the ...
12
votes
1answer
372 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ ...
11
votes
2answers
593 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 ...
11
votes
3answers
984 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 ...
10
votes
3answers
1k views

Projections in Banach spaces

Dear All, I am absolutely lost in the following problem: Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm ...
9
votes
1answer
186 views

Theory of C* algebras over other fields than the complex numbers

How much of the theory of C*-algebras holds when the complex numbers are replaced by different (algebraically closed) field (possibly with a distinguished ordered subfield that satisfies the same ...
9
votes
2answers
385 views

Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?

Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by ...
9
votes
0answers
349 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 ...
8
votes
1answer
357 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 ...
7
votes
3answers
644 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 ...
7
votes
4answers
741 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 ...
7
votes
1answer
262 views

characterization of commutative Banach algebras

Let $A$ be a Banach algebra with the following property: For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges. ...
7
votes
1answer
559 views

When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous

It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space. I would like to know if the weakened module version of this question is answered. More precisely: ...
7
votes
1answer
240 views

Masas in second duals of Banach algebras

Lel $B$ be a Banach algebra and give $B^{**}$ one of the Arens products in order to make it a Banach algebra. Then the canonical embedding $\kappa\colon B\to B^{\ast\ast}$ is a homomorphic embedding ...
7
votes
1answer
487 views

Banach Algebra Counterexample

Can someone give me an example of a Banach Algebra which does not have an isometric representation in a Hilbert Space ? (if possible, can you add a proof or a reference ? ) Thank you very much ! ...
7
votes
1answer
386 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 ...
6
votes
1answer
396 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 ...
6
votes
1answer
263 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 ...
6
votes
0answers
192 views

What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...
6
votes
0answers
709 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 ...
5
votes
2answers
241 views

When is the group algebra $L^1(G)$ semisimple?

Let $G$ be locally compact group. Define group algebra as $$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$ with convolution product. When is the group algebra $L^1(G)$ ...
5
votes
2answers
469 views

When is it $C(X)$?

Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that ...
5
votes
2answers
597 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 $G$), ...
5
votes
1answer
221 views

$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$

Let $F_{2}$ be the free group with two generators. Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way. ...
5
votes
1answer
197 views

Can we extend a multiplicative linear functional of a closed left ideal on whole of the algebra?

Let B be a closed left ideal of a Banach algebra A. Also, B has a right approximate identity (in B). If g is a nonzero multiplicative linear functional on B, can we always extend g to a ...
5
votes
1answer
254 views

NonCommutative Baire theorem

The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is ...
5
votes
0answers
566 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
3answers
661 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 ...
4
votes
1answer
216 views

C*-Algebras: Dynamics vs. Derivations

Problem Given a C*-algebra $\mathcal{A}$. Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$. (More precisely, strongly continuous ...
4
votes
3answers
377 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 ...
4
votes
1answer
351 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 ...
4
votes
2answers
303 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 ...
4
votes
1answer
488 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 ...
4
votes
1answer
413 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
1answer
657 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 ...
4
votes
1answer
194 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 ...
4
votes
0answers
189 views

Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...
4
votes
0answers
163 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
3
votes
1answer
389 views

Example of an infinite dimensional reflexive Banach algebra

If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach ...
3
votes
2answers
131 views

Root of positive function in Fourier algebra

Let $G$ be a locally compact group, let $A(G)$ be the Fourier algebra of $G$. We think of $A(G)$ as a subalgebra of $C_0(G)$. Question 1: Let $f\in A(G)$ be a function that is pointwise positive. ...
3
votes
2answers
305 views

On hyperplanes of $L\infty$

Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is: 1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" ...
3
votes
1answer
113 views

Characterizations of Wiener algebra

The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that $$ \mathcal W\subset ...
3
votes
1answer
138 views

When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?

I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without ...
3
votes
1answer
95 views

Regular commutative Banach algebras which are not closed under complex conjugate

Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some ...