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6
votes
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
3answers
317 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
841 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
435 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 ...
10
votes
0answers
980 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), ...
2
votes
1answer
545 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 ...
3
votes
0answers
230 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 ...
9
votes
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 $\mathcal{B}$. ...
13
votes
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 ...
7
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
606 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 ...
6
votes
0answers
344 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 ...
6
votes
3answers
1k views

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 ...
6
votes
0answers
211 views

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 ...
14
votes
7answers
1k views

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 ...
4
votes
6answers
937 views

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 ...
11
votes
1answer
633 views

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 ...
3
votes
2answers
778 views

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 ...
6
votes
2answers
540 views

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 ...
6
votes
1answer
745 views

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 ...