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Results tagged with c-star-algebras
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user 763
A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: [banach-algebras], [von-neumann-algebras], [operator-algebras], [spectral-theory].
6
votes
Comparison between the operator norm and the $L^1$ norm on group algebras
This is only a partial answer, but it shows that if G has an element of infinite order then the weakest form of Q2 has a negative answer.
We use the so-called Rudin-Shapiro polynomials (really due to …
- 25.8k
7
votes
Accepted
Closed prime ideal in $C[0, 1]$
No. Note that an ideal in a commutative ring with identity is prime if and only if the quotient ring is an integral domain.
Now consider $C[0,1]$. It is known that the closed ideals in this Banach alg …
- 25.8k
4
votes
Jordan isomorphisms of type I von Neumann algebras
I have not read through all the details carefully, but I think your questions can probably be answered using the results in Kadison's 1951 paper "On isometries of operator algebras". That paper actual …
- 25.8k
10
votes
Accepted
$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras
Yes to both.$\newcommand{\Cst}{{\rm C}^*}$
The standard example for the first is: take a discrete group $G$ and let $A$ be its full $\Cst$-algebra, $B$ its reduced $\Cst$-algebra. There is a canonica …
- 25.8k
5
votes
1
answer
177
views
Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor p...
Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces.
Given two $\Cst$-al …
- 25.8k
5
votes
Accepted
When do completely positive maps have a closed image?
This is not a complete answer to the original question, since the original question has a rather open-ended phrasing; but I think it addresses Diego's main points, and shows that even quite well-behav …
- 25.8k
1
vote
Is the algebra of bounded operators stable?
In response to Matt "calling my bluff" I realised that my claim takes some work to justify. The following is very much "thinking aloud through a head cold" so is not a polished exposition, and I have …
- 25.8k
9
votes
Is the algebra of compact operators flat?
Assuming that your $\otimes$ denotes min-tensor product of ${\rm C}^\ast$-algebras, then the answer to the question in the body of your post -- which is NOT the same as the question in the title of yo …
- 25.8k
4
votes
center of a $C^*$-algebra
I think you can obtain an example by modifying the classical Toeplitz algebra.
$\newcommand{\H}{\mathbf{H}}$
$\newcommand{\bT}{\mathbf{T}}$
$\newcommand{\cT}{\mathcal T}$
$\newcommand{\KH}{{\mathcal …
- 25.8k
8
votes
Accepted
A precise definition of contractible Banach algebras
$A$ is contractible if $H^1(A,X)=0$ for all Banach $A$-bimodules $X$ (here $H^1$ denotes continuous Hochschild cohomology for Banach algebras, as defined in the works of Johnson or Helemskii). It is a …
- 25.8k
20
votes
Accepted
Are algebraically isomorphic $C^*$-algebras $*$-isomorphic?
Answering the question in the body of the original post, which seems to be more restricted than the implicit question in the title of the post....
The answer is YES. See
L. Terrell Gardner, On is …
- 25.8k
3
votes
Accepted
Existence of an integrable representation
[Thanks to Loren Spice for fixing the references and pointing out a silly error/mis-statement in the original version of this answer.]
The answer to Q1 is no for $G={\mathbb Z}$, since all its irre …
- 25.8k
2
votes
Suppose that $a \mu = \mu a$ for all $a$ in $C^*$-algebra $A$. Then $\mu \in Z(A^{**})$
I think this follows from the fact that multiplication in $A^{**}$ is separately weak-star continuous, i.e.
for all $a\in A^{**}$ the function $A^{**} \to A^{**}$, $b\mapsto ab$, is weak-star to weak …
- 25.8k
11
votes
Accepted
Is a C*-algebra with an isomorphic predual a von Neumann algebra?
Via my colleague Garth Dales, some observations which answer your question in the negative, even in the abelian case:$\newcommand{\N}{{\mathbb N}}$
We know that $K$ is hyper-Stonean iff $C(K)$ is …
- 25.8k
3
votes
Accepted
busby invariant of extensions of $C^*$-algebras
The idempotents in the corona algebra are very restricted, because they lift to functions$\newcommand{\Real}{{\bf R}}\newcommand{\veps}{\varepsilon}$ $f\in C_b(\Real)$ such that $f^2-f\in C_0(\Real)$. …
- 25.8k