The c-star-algebras tag has no wiki summary.

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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 $K_1(-)$. ...

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**1**answer

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### Given a C-star dynamical system and a subgroup of the acting group, is the corresponding map on crossed product algebras necessarily an injection

Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and ...

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vote

**1**answer

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### k_0 group for $M(J\otimes K)=0$

Let J be any C*-algebra and K be the C*-algbra of compact operators on a separable, infinite dimensional Hibert space.
How to show $K_0(M(J\otimes K))=0$, where M denotes Multiplier algebra

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**1**answer

387 views

### Finite dimensionality of certain $C^{\star}$-algebras

In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:
Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be ...

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### What does the representation theory of the reduced C*-algebra correspond to?

Let $G$ be a locally compact group. The group C*-algebra $C^* (G)$ is designed to come with a natural bijection between its (nondegenerate) representations and the (strongly continuous, unitary) ...

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**1**answer

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### What is the commutative analogue of a C*-subalgebra?

Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff ...

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votes

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### Ideal of “Compact Operators” in a W*-algebra which gives the sigma-strong-* topology.

In the case of bounded operators on a Hilbert space $\mathcal{H}$, $L(\mathcal{H})$, there are multiple descriptions of the $\sigma$-strong-* topology, namely:
1) As given by seminorms ...

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### Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. ...

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### Conjugacy classes and reduced group $C^*$-algebra of an amenable group

The reduced $C^*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star_{red} G$. On the other ...

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**1**answer

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### Surjective *-homs between multiplier algebras

Let A and B be C*-algebras, and let $\phi:A\rightarrow B$ be a surjective *-homomorphism. Then $\phi$ is non-degenerate, and so we can extend it to *-homomorphism between the multiplier algebras: ...

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924 views

### The functoriality of group C* algebra structure

Hi!
Let G,H be discrete groups and f:G->H be any homomorphism of these groups. I have three questions about it:
1) how to prove the functoriality of the construction of universal C*-algebra of ...

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**0**answers

305 views

### Is there a good reference for studying the ideal structure of group C* algebras?

Is there a good reference for studying the ideal structure of group C* algebras?
Thanks.

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**2**answers

797 views

### Relative K-theory and split exact sequences of C* algebras

Let $A$ be a C* algebra, $J$ an ideal, $\pi: A \to A/J$ the quotient map. Recall that the relative K theory group $K_0(A, A/J)$ consists of equivalence classes of triples $(p,q,x)$ where $p$ and $q$ ...

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### Must we close weakly to apply the spectral theorem?

Let $H$ be an infinite dimensional separable complex Hilbert space. All C*-subalgebras of $B(H)$ are assumed to be non-degenerate.
The spectral projections of a self-adjoint element $T$ of $B(H)$ lie ...

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votes

**2**answers

1k views

### Simple inequality in C*-algebras

Sorry the title is a bit vague. Let A be a C*-algebra, and let x and y be positive elements in A. Is it true that $$ \|x-y\|^2 \leq \|x^2-y^2\|? $$
Well, yes. But the proof I have is a bit of a ...

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**1**answer

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### Algebraic properties of the algebra of continuous functions on a manifold.

Does the algebra of continuous
functions from a compact manifold to
$\mathbb{C}$ satisfy any specific
algebraic property?
I'm not sure what kind of algebraic property I expect, but I feel ...

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votes

**2**answers

320 views

### Kernel projections in the universal representation.

Let $A \subseteq \mathcal B(\mathcal H)$ be a unital C*-algebra in its universal representation. The GNS representation $\pi_\mu\colon A \rightarrow \mathcal B(\mathcal H_\mu)$ with base state $\mu$ ...

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873 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 ...

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**1**answer

443 views

### Approximate unit for the algebra C*(h) consisting of projectors

Let E be a Hilbert C*-module over some C*-algebra and let $h \in K(E)$. Due to B. Blackadar's, "K-Theory for Operator algebras" Thm. 17.11.4 for a separable C*-algebra $A$, represented by ...

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### Schwartz kernel theorem for A-linear operators

Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...

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**1**answer

561 views

### Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one

The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then ...

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### Obstructions to existence of finitely summable spectral triples

Connes proved in his beautiful paper "Compact metric spaces, Fredholm modules, and hyperfiniteness" published in 1989 that if $(A,H,D)$ is a finitely summable spectral triple with a unital ...

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### Does equality of the operator norm and the cb norm for every bimodule map over a C*-subalgebra imply that the subalgebra is matricially norming?

In this post, without further mention all C*-algebras are assumed to have an identity element and subalgebras inherit the identity.
Question: Let $\mathcal{C}$ be a C*-subalgebra of $\mathcal{B}$. ...

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### Is the group von Neumann algebra construction functorial?

Let $G$ be a group and $CG$ the complex group algebra over the field $C$ of complex number. The group von Neumann algebra $NG$ is the completion of $CG$ wrt weak operator norm in $B(l^2(G))$, the set ...

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### Definition of a von Neumann algebra

Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in ...

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votes

**2**answers

614 views

### Hilbert $C^*$-modules and approximate units

Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...

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### Saito-Wright definition of Rickart C*-algebras

A C*-algebra is Rickart if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $R(S)=${$a\in A\mid xa=0\, \forall ...

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votes

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### Gelfand duality in NCG

In non-commutative geometry, Gelfand duality is the construction of multiplicative linear functionals of a commutative C*-algebra, which can be viewed as the space of all its irreducible complex ...

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### What morphisms / Morita equivalences induce the 2-periodicity isomorphisms of KK-theory?

In Kasparov's paper, the canonical isomorphisms $KK_* \rightarrow KK_{*+2k}$ are defined rather implicitely (by tensoring and stabilization).
Are there morphisms of $C^*$-algebras which induce them ...

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### ubiquity, importance of path algebras

I work in planar algebras and subfactors, where the idea of path algebras on a graph (alternately known as graph algebras, graph planar algebras, etc.) is quite useful. The particular result I'm ...

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votes

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### Spectra of $C^*$ algebras

Gelfand-Naimark structure theorem for $C^* $ algebras gives a canonical isometric * isomorphism between any commutative unital $C^* $ algebra $A$ and the algebra of continuous complex-valued functions ...

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### Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...

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votes

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### Normal operators and it's spectrum in C*-algebras

If $A$ is a C*-algebra and $n$ is a normal element of $A$, then we have: (By Gelfand duality for example.)
$\operatorname{spec}( |N| ) = | \operatorname{spec}(N) | := \left\{ | \lambda | ; \lambda ...

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### Do torsion-free groups give projectionless group ($C^\ast$) algebras?

One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...

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**1**answer

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### Sources for exact triangles in triangulated categories.

The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides ...