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1
vote
1answer
415 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
190 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 ...
62
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 ...
40
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
955 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
968 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
376 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 ...
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 ...