Tagged Questions

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: [tag:banach-algebras], [tag:von-neumann-algebras], [tag:operator-algebras], [tag:spectral-theory].

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389 views

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 ...
2
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0answers
238 views

Six term exact sequence In E-theory

I just want to know whether the two six term exact sequences in E-theory is true for nonseparable C*-algebras. We know already if the first varible is complex number, then we get six term exact ...
4
votes
1answer
399 views

Crossed product of a non unital C*-algebra

Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*} &\alpha:C_0(X)\rightarrow C_0(X)\\\ &\alpha(f)=f\circ T \end{align*} is an automorphism. ...
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1answer
454 views

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 ...
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2answers
645 views

Solving the equation $xax=b$ in a C*-algebra.

Let $a, b\in A_+$ be positive elements of some C*-algebra $A$. Assume furthermore that $a$ is invertible. Is it true that $$ \exists! x\in A_+\quad:\quad xax=b\quad ? $$ Already in the case ...
1
vote
1answer
255 views

Products on the K-theory of graded C*-algebras

One can define products on the K-theory of graded C*-algebras as in http://web.me.com/ndh2/math/Papers_files/Higson,%20Guentner%20-%202004%20-%20Group%20C*-algebras%20and%20K-theory.pdf on page 152, ...
6
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2answers
968 views

Space of compact operators

I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach ...
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1answer
260 views

The stabilized homotopy category of graded C* algebra

Hi everyone On page 147 of the note "Group C*-Algebras and K-theory" by N.Higson and E.Guentner there are something about the stabilized homotopy category of graded C* algebra, which is a category ...
2
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1answer
344 views

When C(K) is closed in sigma strong topology?

Fix a compact Hausdorff space $K$ and think about $C(K)$ as a C*-algebra acting on a Hilbert space $H$. Suppose that $C(K)$ is closed in $\mathcal{B}(H)$ in: $\sigma$-strong $\sigma$-strong* ...
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0answers
170 views

Grading on Multiplier Algebras

A graded C*-algebra A is inner-graded if there exists a self-adjoint unitary $\varepsilon$ in the multiplier algebra M(A) of A which implements the grading automorphism $\alpha$ on A: ...
3
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1answer
298 views

Graded $C^*$-algebras can be faithfully represented on a graded Hilbert space

Hi everyone I try to use GNS-construction to show every graded C*-algebras can be faithfully represented on a graded Hilbert space. If $A$ is a graded C*-algebra with grading automorphism $\alpha$ of ...
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2answers
653 views

Why is this a conditional expectation into the fixed point algebra?

Let $A$ be a C*-algebra and let $\alpha$ be an action of the circle group $S_1$ on $A$ (Gauge action). We define the following map: $$E:A\rightarrow A;\quad E(a):=\int\alpha_t(a)\textrm{d} t.$$ My ...
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1answer
578 views

When are certain group C*-algebras exact?

This is somewhere between a "reference request" and "ask an expert", but I hope it is not too trivial or off-topic. Anyway. There has been a lot of attention given to showing that for certain ...
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vote
1answer
622 views

Strict positivity in dense subalgebras of $C^{*}$-algebras

Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let ...
5
votes
1answer
281 views

Is the unitary group of $l^2(A)$ with the strict topology contractible?

Let $A$ be a $C^*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary ...
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2answers
1k views

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 ...
4
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0answers
358 views

What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces. Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$. I think this induces a fiber ...
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4answers
2k views

Reference: Learning noncommutative geometry and C^* algebras

I am starting to study noncommutative geometry and C^* algebras so my question is Does anyone knows a good reference on this subject? I would like a basic book with intuitions for definitions and ...
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0answers
171 views

Fredholmness and invertibility in a C* algebra generated convolution-type operators

Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
8
votes
1answer
501 views

topology on the automorphism group of a C* algebra

Let $A$ be a $C^*$-algebra. The group $Aut(A)$ of $\ast$-automorphisms of $A$ is usually equipped either with the pointwise norm topology, i.e. the topology generated by the semi-norms $\lVert \varphi ...
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2answers
532 views

Do separable $C^*$-algebras form a set?

The question is in subject. Update: See Andreas Thom's answer.
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1answer
271 views

Is every action from an amenable group amenable on a unital $C^*$-algebra?

Is every action from an amenable group amenable on a unital $C^*$-algebra?
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2answers
737 views

Range of completely positive projection

Let $A$ be a C*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^*$-algebra? In the case where ...
2
votes
2answers
354 views

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 $K_1(-)$. ...
7
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1answer
429 views

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 ...
1
vote
1answer
209 views

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
6
votes
1answer
390 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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votes
2answers
709 views

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) ...
11
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1answer
806 views

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 ...
4
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3answers
386 views

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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3answers
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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 ...
3
votes
1answer
509 views

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: ...
6
votes
2answers
968 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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0answers
307 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.
8
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2answers
813 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 $q$ ...
12
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0answers
667 views

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 ...
6
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2answers
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 ...
9
votes
1answer
1k views

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 ...
3
votes
2answers
326 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 $\mu$ ...
11
votes
3answers
915 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 ...
4
votes
1answer
462 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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0answers
1k views

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), ...
1
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1answer
575 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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0answers
231 views

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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1answer
1k views

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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3answers
1k views

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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votes
3answers
1k views

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 ...
5
votes
2answers
630 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 ...
7
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0answers
374 views

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 ...