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Questions tagged [c-star-algebras]

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].

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.

For bounded operators on a given Hilbert space, C*-algebras characterize topologically closed subalgebras of ${\mathcal B}({\mathcal H})$ (in operator norm), also closed under taking the adjoint operator. C*-algebras are at the heart of and are extensively used in .

Other related tags: , , , .

853 questions
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Non degenerate representations for C*-algebras

Hi! While studying C*-algebras I found 2 different definitions for non degenerate representations (-homomorphisms $\pi:\mathcal{A} \rightarrow B(\mathcal{h})$ where $\mathcal{A}$ is a C-algebra and $B(\mathcal{h})$ is the space of bounded linear…
Alessandro Gentile
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Why C*-algebras is not as popular as other areas of pure mathematics?

I am applying for graduate school in pure mathematics and I recently got very interested in C*-algebra. I am definitely wrong but I get the feeling that C*-algebras is not as popular as other areas of pure mathematics like number theory, analysis,…
Nina Wang
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A variant of the Stone-Weierstrass theorem?

I would like to ask specialists in C*-algebras if the following variant of the Stone-Weierstrass theorem is true. Suppose $A$ is a C*-algebra and $C$ is its center. Since $C$ is a commutative C*-algebra, there exists a compact space $T$ such that…
Sergei Akbarov
  • 7,274
5
votes
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unitization-process of unital- and non-unital $C^*$-algebras

I have a small question about unitization of (unital) $C^*$-algebras. I first asked on math.stackexchange because it is basic theory, but I still have no suitable answer, the link…
Sabrina Gemsa
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Identifying the multiplier $C^*$-algebra $M(C_0(X) \otimes B)$

Is there an accessible proof for the following fact? If $A=C_0(X)$ with $X$ locally compact Hausdorff and $B$ is a $C^\ast$-algebra then $M(A\otimes B)$ is the set of bounded strictly continuous functions $X \to M(B)$. Denote the set of bounded…
user167952
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1 answer

short question about biduals of $C^\ast$-algebras

Let $A$ be a $C^\ast$-algebra. Consider the canonical embedding $A\to A^{**},\; a\mapsto i(a)$, such that $i(a)(a^*)=a^*(a)$ for all $a\in A$. Here is $A^{**}$ considered as a Banach space. It's well known that $i(A)$ is weak$*$-dense in $A^{**}$. I…
Sabrina Gemsa
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1 answer

proper action and amenable action

We say that an action of a (discrete) group G on a locally compact space X is called proper if the map from $G\times X$ to $X\times X$ defined by $(g,x)\mapsto (gx,x)$ is proper. Why is a proper action amenable? (see On the Baum-Connes assembly map…
m07kl
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Continuity of functors under inductive sequence of $C^*$-algebras.

We know the fact that $K_0(-)$ and $K_1(-)$ are continuous under inductive sequence of $C^*$-algebras (in fact inductive system), i.e. $K_0(\varinjlim A_n)=\varinjlim K_0(A_n)$ similar for $K_1(-)$. In fact it is also true that…
m07kl
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1 answer

Multiple $C^*$ structures?

In principle, associative *-algebras can be equipped with multiple norms. Can this be done in such a way that (after closure in the respective norm) they are turned into $C^*$-algebras in multiple, not essentially equivalent ways?
Arnold Neumaier
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map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$?

Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^*)=f(a)^*$ for all $a\in A$, $f(1_A)=1_B$ and $f(a)f(b)=0$ for all $a,b\in A_{sa}$ with $ab=0$. Follows $f(a^2)=f(a)^2$ for all $a\in A$? I have tried to proof…
Sabrina Gemsa
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2
votes
1 answer

Bratteli diagram decided by AF-algebras

In general, an AF-algebra can has some different Bratteli diagrams. We can add some identical arrows to make the Bratteli diagrams different, but it is too trivial, are any good examples? For which AF-algebras, its Bratteli diagram can be unique…
Strongart
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1 answer

Pairwise orthogonal projections in C*-algebras

Is every non-zero projection in a C*-algebra $A$ a supremum or infimum (at least majorized by / majorizes) a family of pairwise orthogonal non-zero projections in $A$? PS. Are there any cheap ways to generate abelian sub C*-algebras of a given…
Joachim Gryn
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Representation of a C*-algebra

Let $A$ be a C*-algebra. It seems that for given an *-representation $\pi$ of $A$, there is unique central projection $z_{\pi}$ in $A^{**}$ such that $\pi$ is just (unitary equivalent to) $\rho_z$ where $$\rho_z:A\to A^{**} : a\to az$$ It means…
ABB
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Uniform structure on the Banach bundle generated by a Banach module

The construction used in the Dauns-Hofmann theorem defines a Banach bundle $\pi:X\to M$ that corresponds to any $C^*$-subalgebra $A$ lying in the center of a $C^*$-algebra $B$ (this is described for example in the monography by M.Dupre and…
Sergei Akbarov
  • 7,274
1
vote
1 answer

When are tracial states continuous in the strict topology?

Let $A$ be a separable (non-unital) C*-algebra. Let $\tau: A\to\mathbb{C}$ a tracial state. Consider the strict topology on $A$, i.e. $a_n\to a$ iff $a_nx\to ax$ and $xa_n\to xa$ for all $x\in A$. It is clear that this is the same as the usual…
Gabor Szabo
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