Questions tagged [noncommutative-geometry]

Question 1

Grothendieck dissolved the classical notion of a point into a functor of points $$ h_X : (\mathbf{Sch})^{\mathrm{op}} \longrightarrow \mathbf{Set}, \qquad h_X(S) = \mathrm{Hom}(S, X), $$ re-...

Question 2

I was reading this paper by G. Tabuada. Here in section 2.2 author ask to notice every quasi equivalence of dg categories is a Morita equivalence. From the definition it seems that quasi equivalence ...

Question 3

I'm reading Elements of Noncommutative Geometry by Gracia-Bondía, Várilly and Figueroa, and in the first section of Chapter $1$ they are talking about continuous functions on locally compact spaces. ...

Question 4

Let $f: R \to S$ be a homomorphism of commutative rings. If helpful, one can assume that $R$ is a subring of $S$. Now, let $A$ be an $S$-algebra and $M$ an $A$-bimodule. The Hochschild cohomology ...

Question 5

Consider a compact Hausdorff space space $X$, a $\sigma-$algebra of Borel sets $\mathcal{B}(X)$ and probabilistic Radon measure $p$ on $X$. Let $L^\infty(X,\mathcal{B}(X),p;\mathbb{C})$ be a ...

Question 6

I am studying this survey by Keller and I am stuck on page 11 (chapter 3.8 about Morita equivalence) where he has the notation of the tensor product between two dg modules. Let $A$ and $B$ be small dg ...

Question 7

I have a problem with understanding how does the duality between categories of localizable measurable spaces and von Neumman algebras work. To sum up: A localizable measurable space is a measurable ...

Question 8

We know $A$ a $*$-subalgebra of $B(H)$ is strongly dense in its double commutant $A''$. Are there particular conditions on $A$ or $H$ which allow us to strengthen this to one of the following: $A$ is ...

Question 9

We know for $A$ a $*$-subalgebra of $B(H),$ $A$ is strongly dense in its double commutant $A''.$ Consider the following situation: $H$ is separable, $(e_n)$ an orthonormal basis on it. Let $B(H)_\...

Question 10

Let $A$ be a nonunital $C^*$-algebra. Is there a useful characterization of which (closed 2-sided) ideals $I$ of $A$ are such that $A/I$ is unital? In the commutative case, this happens if and only if ...

Question 11

Let $A$ be a $C^*$ - algebra. Definition, Let $p \in A$. $p$ is a projection if and only if, $ p^* = p $ and $ p^2 = p $. =========================== Let $G$ be a topological group which is locally ...

Question 12

It is a problem that comes from Alain Connes' article Non-commutative Differential Geometry. In the proof of Corollary 2 in Part I, Section 6, we need a property that $\mathrm{Trace}(T-D^{-1}TD)=\...

Question 13

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two separable Hilbert spaces. Let $\mathcal{L} ( \mathcal{H} )$ be the $C^*$ - algebra of bounded linear operators on $\mathcal{H}$, a separable Hilbert ...

Question 14

Let $\mathfrak A$ be a $C^\ast$-algebra and $\phi\colon\mathfrak A\to\mathbb C$ a continuous positive function whose restriction to any commutative sub-$C^\ast$-algebra is linear. Is $\phi$ linear on ...

Question 15

Let $A$ and $B$ be two $C^*$-algebras, and let $A \odot B $ be the algebraic tensor product of $A$ and $B$ as a vector space. The vector space $A \odot B$ is a $^*$-algebra defined by the following ...

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Questions tagged [noncommutative-geometry]

To "enrich" as opposed to "dissolve" the point?

quasi equivalence implies Morita equivalence

When $X$ is locally compact but not compact, why is the set of continuous functions $f:X\to\mathbb{C}$ "too big"?

Base-change for Hochschild Cohomology?

A noncommutative analog of algebra of continuous and bounded functions.

Tensor product of dg modules over dg categories

Recovering localizable measure from null sets of localizable measurable space.

Are $*$-Subalgebras Ever Dense in Their Double Commutant in Other Topologies?

Real *-Subalgebras of the Real Analog of $B(H)$ and their Strong Closures

When is a quotient of a $C^*$-algebra unital?

Does the $C^$-algebra $ C_{r}^ (G) $ contain projections? [closed]

Does $\mathrm{Trace}(D^{-1}TD)=\mathrm{Trace}(T)$ hold for unbounded operators?

Isomorphism between $\mathcal{L} ( \mathcal{H}_1 \otimes \mathcal{H}_2 )$ and $ \mathcal{L} ( \mathcal{H}_1 ) \otimes \mathcal{L} ( \mathcal{H}_2 )$.

Does linearity on all commutative subalgebras imply linearity on the whole algebra?

Bijection between minimal $C^*$-tensor product representations, and pairs of its correspondants restriction representations.

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